Recent research on computing the diameter of geometric intersection graphs has made significant strides, primarily focusing on the 2D case [Duraj et al., 2024; Hsien-Chih Chang et al., 2024; Chan et al., 2025] where truly subquadratic-time algorithms were given for simple objects such as unit-disks and (axis-aligned) squares. However, in three or higher dimensions, there is no known truly subquadratic-time algorithm for any intersection graph of non-trivial objects, even basic ones such as unit balls or (axis-aligned) unit cubes. This was partially explained by the pioneering work of Bringmann et al. [Karl Bringmann et al., 2022] which gave several truly subquadratic lower bounds, notably for unit balls or unit cubes in 3D when the graph diameter Δ is at least Ω(log n), hinting at a pessimistic outlook for the complexity of the diameter problem in higher dimensions. In this paper, we substantially extend the landscape of diameter computation for objects in three and higher dimensions, giving a few positive results. Our highlighted findings include: 1) A truly subquadratic-time algorithm for deciding if the diameter of unit cubes in 3D is at most 3 (Diameter-3 hereafter), the first algorithm of its kind for objects in 3D or higher dimensions. Our algorithm is based on a novel connection to pseudolines, which is of independent interest. 2) A truly subquadratic time lower bound for Diameter-3 of unit balls in 3D under the Orthogonal Vector (OV) hypothesis, giving the first separation between unit balls and unit cubes in the small diameter regime. Previously, computing the diameter for both objects was known to be quadratic hard when the diameter is Ω(log n) [Karl Bringmann et al., 2022]. 3) A near-linear-time algorithm for Diameter-2 of unit cubes in 3D, generalizing the previous result for unit squares in 2D [Karl Bringmann et al., 2022]. 4) A truly subquadratic-time algorithm and lower bound for Diameter-2 and Diameter-3 of rectangular boxes (of arbitrary dimension and sizes), respectively.

For a metric space (X, d), a family ℋ of locality sensitive hash functions is called (r, cr, p₁, p₂) sensitive if a randomly chosen function h ∈ ℋ has probability at least p₁ (at most p₂) to map any a, b ∈ X in the same hash bucket if d(a, b) ≤ r (or d(a, b) ≥ cr). Locality Sensitive Hashing (LSH) is one of the most popular techniques for approximate nearest-neighbor search in high-dimensional spaces, and has been studied extensively for Hamming, Euclidean, and spherical geometries. An (r, cr, p₁, p₂)-sensitive hash function enables approximate nearest neighbor search (i.e., returning a point within distance cr from a query q if there exists a point within distance r from q) with space O(n^{1+ρ}) and query time O(n^ρ) where ρ = (log 1/p₁)/(log 1/p₂). But LSH for hyperbolic spaces ℍ^d remains largely unexplored. In this work, we present the first LSH construction native to hyperbolic space. For the hyperbolic plane (d = 2), we show a construction achieving ρ ≤ 1/c, based on the hyperplane rounding scheme. For general hyperbolic spaces (d ≥ 3), we use dimension reduction from ℍ^d to ℍ² and the 2D hyperbolic LSH to get ρ ≤ 1.59/c. On the lower bound side, we show that the lower bound on ρ of Euclidean LSH extends to the hyperbolic setting via local isometry, therefore giving ρ ≥ 1/c².

Computing the diameter of the intersection graphs of objects is a basic problem in computational geometry. Previous works showed that the complexity of computing the diameter mainly depends on the object types: for unit disks and squares in 2D, the problem is solvable in truly subquadratic time [Chan et al., 2025], while for other objects, including unit segments and equilateral triangles in 2D or unit balls and axis-parallel unit cubes in 3D, there is no truly subquadratic time algorithm under the Orthogonal Vector (OV) hypothesis [Bringmann et al., 2022]. We undertake a comprehensive study of computing the diameter of geometric intersection graphs for various types of objects. We discover many new irregularities, showing that the landscape is extremely nuanced: the source of hardness is a combination of the object type, the true diameter value, and how the objects intersect with each other. Our highlighted results for the 2D case include: 1) The diameter of non-degenerate, axis-aligned line segments can be computed in truly subquadratic time. Previous hardness result [Bringmann et al., 2022] for line segments applies only to degenerate instances. On the other hand, for the degenerate case, we show that a truly subquadratic time algorithm exists when the true diameter is constant. 2) An almost-linear-time algorithm for unit-square graphs of constant diameter. Previous algorithms [Duraj et al., 2024; Chan et al., 2025] rely on succinct representation assuming bounded VC-dimension; for such a strategy Ω(n^{7/4}) time is an inherent barrier. 3) An Õ(n^{4/3})-time algorithm to decide if the diameter of a unit-disk graph is at most 2. This improves upon the recent algorithm with running time Õ(n^{2-1/9}) [Chan et al., 2025]. 4) Deciding if the diameter of intersection graphs of fat triangles or line segments is at most 2 is truly subquadratic-hard under fine-grained complexity assumptions. Previous lower bounds [Bringmann et al., 2022] only hold when deciding if diameter is at most 3. Our findings are presented in a pair of papers. This paper focuses solely on the 2D case, while the companion paper is devoted to higher-dimensional cases.

We study the Patrol Security Game (PSG), a robotic patrolling problem formulated as an extensive-form Stackelberg game, in which the attacker strategically selects the timing, location, and duration of an attack. The defender’s goal is to compute an infinite-horizon patrolling policy that minimizes the attacker’s expected payoff. By restricting the defender’s strategy to a time-homogeneous first-order Markov chain, we show that PSG can be reformulated as a combinatorial minimax problem. We prove that the optimal strategy under zero-penalty scenarios corresponds to minimizing either the expected hitting time or return time, depending on the attacker’s visibility model. These optimal policies are closed-form and can be computed efficiently. On the other hand, in high-penalty cases, we observe that the patrolling schedule with high randomness can minimize the attacker’s expected gain. However, in general, the minimax objective becomes non-convex. To address this, we introduce a bi-criteria optimization framework that jointly considers the expected maximum reward (EMR) and entropy rate of the patrolling policy. We propose three graph-based algorithms and a deep reinforcement learning model to efficiently balance these two objectives. Each algorithm demonstrates distinct strengths under different configurations, such as varying penalty scales and cost function settings. The extensive experiments on both synthetic and real-world crime datasets validate the effectiveness of our approaches, demonstrating superior performance and scalability compared to state-of-the-art baselines.

This work proposes a class of differentially private mechanisms for linear queries, in particular range queries, that leverages correlated input perturbation to simultaneously achieve unbiasedness, consistency, statistical transparency, and control over utility requirements in terms of accuracy targets expressed either in certain query margins or as implied by the hierarchical database structure. The proposed Cascade Sampling algorithm instantiates the mechanism exactly and efficiently. Our theoretical and empirical analysis demonstrates that we achieve near-optimal utility, effectively compete with other methods, and retain all the favorable statistical properties discussed earlier.

Change point detection aims to identify abrupt shifts occurring at multiple points within a data sequence. This task becomes particularly challenging in the online setting, where different types of change can occur, including shifts in both the marginal and joint distributions of the data. In this paper, we address these challenges by tracking the Riemannian geometry of correlation matrices, allowing Riemannian metrics to compute the geodesic distance as an accurate measure of correlation dynamics.

We give the first truly subquadratic time algorithm, with ${O^{\ast}}\left( {{n^{2 - 1/18}}} \right)$ running time, for computing the diameter of an n-vertex unit-disk graph, resolving a central open problem in the literature. Our result is obtained as an instance of a general framework, applicable to different graph families and distance problems. Surprisingly, our framework completely bypasses sublinear separators (or r-divisions) which were used in all previous algorithms. Instead, we use low-diameter decompositions in their most elementary form. We also exploit bounded VC-dimension of set systems associated with the input graph, as well as new ideas on geometric data structures. Among the numerous applications of the general framework, we obtain:1)An $\tilde O\left( {m{n^{1 - 1/(2d)}}} \right)$ time algorithm for computing the diameter of m-edge sparse unweighted graphs with constant VC-dimension d. The previously known algorithms by Ducoffe, Habib, and Viennot [SODA 2019] and Duraj, Konieczny, and Potępa [ESA 2024] are truly subquadratic only when the diameter is a small polynomial. Our result thus generalizes truly subquadratic time algorithms known for planar and minor-free graphs (in fact, it slightly improves the previous time bound for minor-free graphs).2)An $\tilde O\left( {{n^{2 - 1/12}}} \right)$ time algorithm for computing the diameter of intersection graphs of axis-aligned squares with arbitrary size. The best-known algorithm by Duraj, Konieczny, and Potępa [ESA 2024] only works for unit squares and is only truly subquadratic in the low-diameter regime.3)The first algorithms with truly subquadratic complexity for other distance-related problems, including all-vertex eccentricities, Wiener index, and exact distance oracles. In particular, we obtain the first exact distance oracle with truly subquadratic space and $\tilde O(1)$ query time for any sparse graph with bounded VC-dimension, again generalizing previous results for planar and minor-free graphs.

Hierarchical clustering is a fundamental unsupervised machine learning task with the aim of organizing data into a hierarchy of clusters. Many applications of hierarchical clustering involve sensitive user information, therefore motivating recent studies on differentially private hierarchical clustering under the rigorous framework of Dasgupta's objective. However, it has been shown that any privacy-preserving algorithm under edge-level differential privacy necessarily suffers a large error. To capture practical applications of this problem, we focus on the weight privacy model, where each edge of the input graph is at least unit weight. We present a novel algorithm in the weight privacy model that shows significantly better approximation than known impossibility results in the edge-level DP setting. In particular, our algorithm achieves $O(\log^{1.5}n/\varepsilon)$ multiplicative error for $\varepsilon$-DP and runs in polynomial time, where $n$ is the size of the input graph, and the cost is never worse than the optimal additive error in existing work. We complement our algorithm by showing if the unit-weight constraint does not apply, the lower bound for weight-level DP hierarchical clustering is essentially the same as the edge-level DP, i.e. $\Omega(n^2/\varepsilon)$ additive error. As a result, we also obtain a new lower bound of $\tilde{\Omega}(1/\varepsilon)$ additive error for balanced sparsest cuts in the weight-level DP model, which may be of independent interest. Finally, we evaluate our algorithm on synthetic and real-world datasets. Our experimental results show that our algorithm performs well in terms of extra cost and has good scalability to large graphs.

Randomized dimensionality reduction is a widely-used algorithmic technique for speeding up large-scale Euclidean optimization problems. In this paper, we study dimension reduction for a variety of maximization problems, including max-matching, max-spanning tree, as well as various measures for dataset diversity. For these problems, we show that the effect of dimension reduction is intimately tied to the *doubling dimension* $\lambda_X$ of the underlying dataset $X$---a quantity measuring intrinsic dimensionality of point sets. Specifically, the dimension required is $O(\lambda_X)$, which we also show is necessary for some of these problems. This is in contrast to classical dimension reduction results, whose dependence grow with the dataset size $|X|$. We also provide empirical results validating the quality of solutions found in the projected space, as well as speedups due to dimensionality reduction.

Graph Neural Networks (GNNs) have shown remarkable success across various scientific fields, yet their adoption in critical decision-making is often hindered by a lack of interpretability. Recently, intrinsic interpretable GNNs have been studied to provide insights into model predictions by identifying rationale substructures in graphs. However, existing methods face challenges when the underlying rationale subgraphs are complex and varied. In this work, we propose TopInG: Topologically Interpretable Graph Learning, a novel topological framework that leverages persistent homology to identify persistent rationale subgraphs. TopInG employs a rationale filtration learning approach to model an autoregressive generating process of rationale subgraphs, and introduces a self-adjusted topological constraint, termed topological discrepancy, to enforce a persistent topological distinction between rationale subgraphs and irrelevant counterparts. We provide theoretical guarantees that our loss function is uniquely optimized by the ground truth under specific conditions. Extensive experiments demonstrate TopInG’s effectiveness in tackling key challenges, such as handling variform rationale subgraphs, balancing predictive performance with interpretability, and mitigating spurious correlations. Results show that our approach improves upon state-of-the-art methods on both predictive accuracy and interpretation quality.

Given a weighted graph G = (V,E,w), a (β, ε)-hopset H is an edge set such that for any s,t ∈ V, where s can reach t in G, there is a path from s to t in G ∪ H which uses at most β hops whose length is in the range [dist_G(s,t), (1+ε)dist_G(s,t)]. We break away from the traditional question that asks for a hopset H that achieves small |H| and small diameter β and instead study the sensitivity of H, a new quality measure. The sensitivity of a vertex (or edge) given a hopset H is, informally, the number of times a single hop in G ∪ H bypasses it; a bit more formally, assuming shortest paths in G are unique, it is the number of hopset edges (s,t) ∈ H such that the vertex (or edge) is contained in the unique st-path in G having length exactly dist_G(s,t). The sensitivity associated with H is then the maximum sensitivity over all vertices (or edges). The highlights of our results are: - A construction for (Õ(√n), 0)-hopsets on undirected graphs with O(log n) sensitivity, complemented with a lower bound showing that Õ(√n) is tight up to polylogarithmic factors for any construction with polylogarithmic sensitivity. - A construction for (n^o(1), ε)-hopsets on undirected graphs with n^o(1) sensitivity for any ε > 0 that is at least inverse polylogarithmic, complemented with a lower bound on the tradeoff between β, ε, and the sensitivity. - We define a notion of sensitivity for β-shortcut sets (which are the reachability analogues of hopsets) and give a construction for Õ(√n)-shortcut sets on directed graphs with O(log n) sensitivity, complemented with a lower bound showing that β = Ω̃(n^{1/3}) for any construction with polylogarithmic sensitivity. We believe hopset sensitivity is a natural measure in and of itself, and could potentially find use in a diverse range of contexts. More concretely, the notion of hopset sensitivity is also directly motivated by the Differentially Private All Sets Range Queries problem [Deng et al. WADS 23]. Our result for O(log n) sensitivity (Õ(√n), 0)-hopsets on undirected graphs immediately improves the current best-known upper bound on utility from Õ(n^{1/3}) to Õ(n^{1/4}) in the pure-DP setting, which is tight up to polylogarithmic factors.

The Johnson-Lindenstrauss (JL) lemma is a cornerstone of dimensionality reduction in Euclidean space, but its applicability to non-Euclidean data has remained limited. This paper extends the JL lemma beyond Euclidean geometry to handle general dissimilarity matrices that are prevalent in real-world applications. We present two complementary approaches: First, we show how the JL transform can be applied to vectors in pseudo-Euclidean space with signature $(p,q)$, providing theoretical guarantees that depend on the ratio of the $(p, q)$ norm and Euclidean norm of two vectors, measuring the deviation from Euclidean geometry. Second, we prove that any symmetric hollow dissimilarity matrix can be represented as a matrix of generalized power distances, with an additional parameter representing the uncertainty level within the data. In this representation, applying the JL transform yields multiplicative approximation with a controlled additive error term proportional to the deviation from Euclidean geometry. Our theoretical results provide fine-grained performance analysis based on the degree to which the input data deviates from Euclidean geometry, making practical and meaningful reduction in dimensionality accessible to a wider class of data. We validate our approaches on both synthetic and real-world datasets, demonstrating the effectiveness of extending the JL lemma to non-Euclidean settings.

Motivated by the problem of estimating bottleneck capacities on the Internet, we formulate and study the problem of vantage point selection. We are given a graph G = (V, E) whose edges E have unknown capacity values that are to be discovered. Probes from a vantage point, i.e, a vertex v ∈ V, along shortest paths from v to all other vertices, reveal bottleneck edge capacities along each path. Our goal is to select k vantage points from V that reveal the maximum number of bottleneck edge capacities. We consider both a non-adaptive setting where all k vantage points are selected before any bottleneck capacity is revealed, and an adaptive setting where each vantage point selection instantly reveals bottleneck capacities along all shortest paths starting from that point. In the non-adaptive setting, by considering a relaxed model where edge capacities are drawn from a random permutation (which still leaves the problem of maximizing the expected number of revealed edges NP-hard), we are able to give a 1-1/e approximate algorithm. In the adaptive setting we work with the least permissive model where edge capacities are arbitrarily fixed but unknown. We compare with the best solution for the particular input instance (i.e. by enumerating all choices of k tuples), and provide both lower bounds on instance optimal approximation algorithms and upper bounds for trees and planar graphs.

Finding the diameter of a graph in general cannot be done in truly subquadratic assuming the Strong Exponential Time Hypothesis (SETH), even when the underlying graph is unweighted and sparse. When restricting to concrete classes of graphs and assuming SETH, planar graphs and minor-free graphs admit truly subquadratic algorithms, while geometric intersection graphs of unit balls, congruent equilateral triangles, and unit segments do not. Unit-disk graphs is one of the major open cases where the complexity of diameter computation remains unknown. More generally, it is conjectured that a truly subquadratic time algorithm exists for pseudo-disk graphs where each pair of objects has at most two intersections on the boundary. In this paper, we show a truly-subquadratic algorithm of running time O^~(n^{2-1/18}), for finding the diameter in a unit-disk graph, whose output differs from the optimal solution by at most 2. This is the first algorithm that provides an additive guarantee in distortion, independent of the size or the diameter of the graph. Our algorithm requires two important technical elements. First, we show that for the intersection graph of pseudo-disks, the graph VC-dimension - either of k-hop balls or the distance encoding vectors - is 4. This contrasts to the VC dimension of the pseudo-disks themselves as geometric ranges (which is known to be 3). Second, we introduce a clique-based r-clustering for geometric intersection graphs, which is an analog of the r-division construction for planar graphs. We also showcase the new techniques by establishing new results for distance oracles for unit-disk graphs with subquadratic storage and O(1) query time. The results naturally extend to unit L₁ or L_∞-disks and fat pseudo-disks of similar size. Last, if the pseudo-disks additionally have bounded ply, we have a truly subquadratic algorithm to find the exact diameter.

The hereditary discrepancy of a set system is a certain quantitative measure of the pseudorandom properties of the system. Roughly, hereditary discrepancy measures how well one can $2$-color the elements of the system so that each set contains approximately the same number of elements of each color. Hereditary discrepancy has well-studied applications e.g. in communication complexity and derandomization. More recently, the hereditary discrepancy of set systems of shortest paths has found applications in differential privacy [Chen et al.~SODA 23]. The contribution of this paper is to improve the upper and lower bounds on the hereditary discrepancy of set systems of unique shortest paths in graphs. In particular, we show that any system of unique shortest paths in an undirected weighted graph has hereditary discrepancy $\widetilde{O}(n^{1/4})$, and we construct lower bound examples demonstrating that this bound is tight up to hidden $\text{polylog } n$ factors. Our lower bounds apply even in the planar and bipartite settings, and they improve on a previous lower bound of $Ω(n^{1/6})$ obtained by applying the trace bound of Chazelle and Lvov [SoCG'00] to a classical point-line system of Erdős. As applications, we improve the lower bound on the additive error for differentially-private all pairs shortest distances from $Ω(n^{1/6})$ [Chen et al.~SODA 23] to $Ω(n^{1/4})$, and we improve the lower bound on additive error for the differentially-private all sets range queries problem to $Ω(n^{1/4})$, which is tight up to hidden $\text{polylog } n$ factors [Deng et al.~WADS 23].

We introduce \textbf{N}on-\textbf{Euc}lidean-\textbf{MDS} (Neuc-MDS), which extends Multidimensional Scaling (MDS) to generate outputs that can be non-Euclidean and non-metric. The main idea is to generalize the inner product to other symmetric bilinear forms to utilize the negative eigenvalues of dissimiliarity Gram matrices. Neuc-MDS efficiently optimizes the choice of (both positive and negative) eigenvalues of the dissimilarity Gram matrix to reduce STRESS, the sum of squared pairwise error. We provide an in-depth error analysis and proofs of the optimality in minimizing lower bounds of STRESS. We demonstrate Neuc-MDS's ability to address limitations of classical MDS raised by prior research, and test it on various synthetic and real-world datasets in comparison with both linear and non-linear dimension reduction methods.

We propose new differential privacy solutions for when external invariants and integer constraints are simultaneously enforced on the data product. These requirements arise in real world applications of private data curation, including the public release of the 2020 U.S. Decennial Census. They pose a great challenge to the production of provably private data products with adequate statistical usability. We propose integer subspace differential privacy to rigorously articulate the privacy guarantee when data products maintain both the invariants and integer characteristics, and demonstrate the composition and post-processing properties of our proposal. To address the challenge of sampling from a potentially highly restricted discrete space, we devise a pair of unbiased additive mechanisms, the generalized Laplace and the generalized Gaussian mechanisms, by solving the Diophantine equations as defined by the constraints. The proposed mechanisms have good accuracy, with errors exhibiting sub-exponential and sub-Gaussian tail probabilities respectively. To implement our proposal, we design an MCMC algorithm and supply empirical convergence assessment using estimated upper bounds on the total variation distance via L-lag coupling. We demonstrate the efficacy of our proposal with applications to a synthetic problem with intersecting invariants, a sensitive contingency table with known margins, and the 2010 Census county-level demonstration data with mandated fixed state population totals.

Structural balance theory studies stability in networks. Given a $n$-vertex complete graph $G=(V,E)$ whose edges are labeled positive or negative, the graph is considered \emph{balanced} if every triangle either consists of three positive edges (three mutual ``friends''), or one positive edge and two negative edges (two ``friends'' with a common ``enemy''). From a computational perspective, structural balance turns out to be a special case of correlation clustering with the number of clusters at most two. The two main algorithmic problems of interest are: $(i)$ detecting whether a given graph is balanced, or $(ii)$ finding a partition that approximates the \emph{frustration index}, i.e., the minimum number of edge flips that turn the graph balanced. We study these problems in the streaming model where edges are given one by one and focus on \emph{memory efficiency}. We provide randomized single-pass algorithms for: $(i)$ determining whether an input graph is balanced with $O(\log{n})$ memory, and $(ii)$ finding a partition that induces a $(1 + \varepsilon)$-approximation to the frustration index with $O(n \cdot \text{polylog}(n))$ memory. We further provide several new lower bounds, complementing different aspects of our algorithms such as the need for randomization or approximation. To obtain our main results, we develop a method using pseudorandom generators (PRGs) to sample edges between independently-chosen \emph{vertices} in graph streaming. Furthermore, our algorithm that approximates the frustration index improves the running time of the state-of-the-art correlation clustering with two clusters (Giotis-Guruswami algorithm [SODA 2006]) from $n^{O(1/\varepsilon^2)}$ to $O(n^2\log^3{n}/\varepsilon^2 + n\log n \cdot (1/\varepsilon)^{O(1/\varepsilon^4)})$ time for $(1+\varepsilon)$-approximation. These results may be of independent interest.

Heart rate, a commonly accessible health data from most wearables, carries rich information of a person’s well-being, yet remains of limited deep health applications, due to the lack of groundtruth of health events and their impact on heart rate patterns. Specifically, standard health analytics usually are designed based on well-modeled health conditions thus known data patterns and rich training data. To bridge the gap, we propose HeartInsightify, an exploratory framework that facilitates the process of deriving health-relevant measurable indicators from longitudinal heart rate data, without any of the above knowledge. HeartInsightify focuses on comparative and qualitative study, using model-free statistical methods such as conformal prediction, to study similarities, perform clustering and detect outliers, and build multi-resolutional data summaries, allowing human experts to efficiently examine and verify their health relevance. We conduct extensive experiments to evaluate HeartInsightify using individuals’ free-living heart rate data collected through Fitbit over 6 years. We illustrate the process of analyzing heart rate data for its health relevance and demonstrate the effectiveness of HeartInsightify. We envision that HeartInsightify lays the groundwork for personalized health analytics with continuous monitoring data from wearables.

Many data applications have certain invariant constraints due to practical needs. Data curators who employ differential privacy need to respect such constraints on the sanitized data product as a primary utility requirement. Invariants challenge the formulation, implementation, and interpretation of privacy guarantees. We propose subspace differential privacy, to honestly characterize the dependence of the sanitized output on confidential aspects of the data. We discuss two design frameworks that convert well-known differentially private mechanisms, such as the Gaussian and the Laplace mechanisms, to subspace differentially private ones that respect the invariants specified by the curator. For linear queries, we discuss the design of near-optimal mechanisms that minimize the mean squared error. Subspace differentially private mechanisms rid the need for post-processing due to invariants, preserve transparency and statistical intelligibility of the output, and can be suitable for distributed implementation. We showcase the proposed mechanisms on the 2020 Census Disclosure Avoidance demonstration data, and a spatio-temporal dataset of mobile access point connections on a large university campus.

We consider the following surveillance problem: Given a set $P$ of $n$ sites in a metric space and a set of $k$ robots with the same maximum speed, compute a patrol schedule of minimum latency for the robots. Here a patrol schedule specifies for each robot an infinite sequence of sites to visit (in the given order) and the latency $L$ of a schedule is the maximum latency of any site, where the latency of a site $s$ is the supremum of the lengths of the time intervals between consecutive visits to $s$. When $k=1$ the problem is equivalent to the travelling salesman problem (TSP) and thus it is NP-hard. We have two main results. We consider cyclic solutions in which the set of sites must be partitioned into $\ell$ groups, for some~$\ell \leq k$, and each group is assigned a subset of the robots that move along the travelling salesman tour of the group at equal distance from each other. Our first main result is that approximating the optimal latency of the class of cyclic solutions can be reduced to approximating the optimal travelling salesman tour on some input, with only a $1+\varepsilon$ factor loss in the approximation factor and an $O\left(\left( k/\varepsilon \right)^k\right)$ factor loss in the runtime, for any $\varepsilon >0$. Our second main result shows that an optimal cyclic solution is a $2(1-1/k)$-approximation of the overall optimal solution. Note that for $k=2$ this implies that an optimal cyclic solution is optimal overall. The results have a number of consequences. For the Euclidean version of the problem, for instance, combining our results with known results on Euclidean TSP, yields a PTAS for approximating an optimal cyclic solution, and it yields a $(2(1-1/k)+\varepsilon)$-approximation of the optimal unrestricted solution. If the conjecture mentioned above is true, then our algorithm is actually a PTAS for the general problem in the Euclidean setting.

Publishing data from IoT devices raises concerns of leaking sensitive information. In this paper we consider the scenario of publishing data on events with timestamps. We formulate three privacy issues, namely, whether one can tell if an event happened or not; whether one can nail down the timestamp of an event within a given time interval; and whether one can infer the relative order of any two nearby events. We show that perturbation of event timestamps or adding fake events following carefully chosen distributions can address these privacy concerns. We present a rigorous study of privately publishing discrete event timestamps with privacy guarantees under the Pufferfish privacy framework. We also conduct extensive experiments to evaluate utility of the modified time series with real world location check-in and app usage data. Our mechanisms preserve the statistical utility of event data which are suitable for aggregate queries.

Technology innovation has provided the opportunity to study the characteristics of natural human mobility. In this paper, we look at how to identify interesting clusters (by different individuals or other naturally defined groups) in a family of trajectory traces. We focus on coarse-grained, sparsely sampled trajectories inferred from sporadic occurrences in an unsupervised setting. This is a challenging setting due to difficulties in selecting features and similarity measures, and due to lack of prior knowledge of data distribution. We propose a non-parametric clustering algorithm, which makes little assumptions on prior knowledge of both data distribution and cluster properties. Our algorithm, Conformal DBSCAN, combines density-based DBSCAN clustering with the statistical conformal prediction framework. We first identify groups of highly similar trajectories as the initial seeds of clusters, similar to DBSCAN. Then we include additional trajectories that belong to this cluster, with a guaranteed statistical confidence level, derived by an improved conformal prediction framework. This allows the clustering algorithm to automatically adapt to different data distributions. Our algorithms are shown to significantly outperform alternative clustering algorithms on several artificial and real-world datasets.

In this paper, we propose co-evolution models for both dynamics of opinions (people's view on a particular topic) and dynamics of social appraisals (the approval or disapproval towards each other). Opinion dynamics and dynamics of signed networks, respectively, have been extensively studied. We propose a co-evolution model, where each vertex i in the network has a current opinion vector vi and each edge (i, j) has a weight wij that models the relationship between i, j. The system evolves as opinions and edge weights are updated over time by the following rules: Opinion dynamics: The opinion of agent i is updated as a linear combination of its current opinion and the weighted sum of neighbors' opinions with coefficients in matrix W = [wij]. Appraisal dynamics: The appraisal wij is updated as a linear combination of its current value and the agreement of the opinions of agents i and j. The agreement of opinion vi and vj is taken as the dot product vi · vj. We are interested in characterizing the long-time behavior of the dynamic model–i.e., whether edge weights evolve to have stable signs (positive or negative) and structural balance (the multiplication of weights on any triangle is non-negative). Our main theoretical result solves the above dynamic system with time-evolving opinions V(t) = [v1(t), …, vn(t)] and social tie weights W(t) = [wij(t)]n×n. For a generic initial opinion vector V(0) and weight matrix W(0), one of the two phenomena must occur at the limit. The first one is that both sign stability and structural balance (for any triangle with individual i, j, k, wijwjkwki ≥ 0) occur. In the special case that V(0) is an eigenvector of W(0), we are able to obtain the explicit solution to the co-evolution equation and give exact estimates on the blowup time and rate convergence. The second one is that all the opinions converge to 0, i.e., limt→∞ |V(t)| = 0. We also performed extensive simulations to examine how different initial conditions affect the network evolution. Of particular interest is that our dynamic model can be used to faithfully detect community structures. On real-world graphs, with a small number of seeds initially assigned ground truth opinions, the dynamic model successfully discovers the final community structure. The model sheds lights on why community structure emerges and becomes a widely observed, sustainable property in complex networks.

The presence of correlation is known to make privacy protection more difficult. We investigate the privacy of socially contagious attributes on a network of individuals, where each individual possessing that attribute may influence a number of others into adopting it. We show that for contagions following the Independent Cascade model there exists a giant connected component of infected nodes, containing a constant fraction of all the nodes who all receive the contagion from the same set of sources. We further show that it is extremely hard to hide the existence of this giant connected component if we want to obtain an estimate of the activated users at an acceptable level. Moreover, an adversary possessing this knowledge can predict the real status (“active” or “inactive”) with decent probability for many of the individuals regardless of the privacy (perturbation) mechanism used. As a case study, we show that the Wasserstein mechanism, a state-of-the-art privacy mechanism designed specifically for correlated data, introduces a noise with magnitude of order Ω(n) in the count estimation in our setting. We provide theoretical guarantees for two classes of random networks: Erdős-Rényi graphs and Chung-Lu power-law graphs under the Independent Cascade model. Experiments demonstrate that a giant connected component of infected nodes can indeed appear in real-world networks and a simple inference attack can reveal the status of a good fraction of nodes.

Motivated by indoor localization by tripwire lasers, we study the problem of cutting a polygon into small-size pieces, using the chords of the polygon. Several versions are considered, depending on the definition of the "size" of a piece. In particular, we consider the area, the diameter, and the radius of the largest inscribed circle as a measure of the size of a piece. We also consider different objectives, either minimizing the maximum size of a piece for a given number of chords, or minimizing the number of chords that achieve a given size threshold for the pieces. We give hardness results for polygons with holes and approximation algorithms for multiple variants of the problem.

This paper considers the problem of privately reporting counts of events recorded by devices in different regions of the plane. Unlike previous range query methods, our approach is not limited to rectangular ranges. We devise novel hierarchical data structures to answer queries over arbitrary planar graphs. This construction relies on balanced planar separators to represent shortest paths using O(logn) number of canonical paths, where n is the number of nodes in the graph. Pre-computed sums along these canonical paths allow efficient computations of 1D counting range queries along any shortest path. We make use of differential forms together with the 1D mechanism to answer 2D queries in which a range is a union of faces in the planar graph. The methods are designed such that the range queries could be answered with differential privacy guarantee on any single event, with only a poly-logarithmic error. They also allow private range queries to be performed in a distributed setup. Theoretical and experimental results confirm that the methods are efficient and accurate on real data and incur less error than competing existing methods.

Advances in wireless communication technology have allowed for the collection of large-scale human motion trajectories by recording the appearance of mobile devices within the neighborhood of wireless base stations. Such city-scale datasets pose new challenges on efficient data collection, analysis and similarity based queries. In this paper, we propose new partial similarity measures, categorized as time-sensitive, order-sensitive and order-insensitive ones, and show with real data that these partial similarity measures are more robust than classical measures and more suitable for generating meaningful query results in near-neighbor type of data mining applications. Further, the power of the partial similarity persists even with significant down-sampling. We presented rigorous analysis of the performance of partial similarity measures with subsampling. Our evaluation using real data shows high recall and precision (≥ 90%) with samples only in the order of 1% of the original data size.

Due to increasing concerns of data privacy, databases are being encrypted before they are stored on an untrusted server. To enable search operations on the encrypted data, searchable encryption techniques have been proposed. Representative schemes use order-preserving encryption (OPE) for supporting efficient Boolean queries on encrypted databases. Yet, recent works showed the possibility of inferring plaintext data from OPE-encrypted databases, merely using the order-preserving constraints, or combined with an auxiliary plaintext dataset with similar frequency distribution. So far, the effectiveness of such attacks is limited to single-dimensional dense data (most values from the domain are encrypted), but it remains challenging to achieve it on high-dimensional datasets (e.g., spatial data), which are often sparse in nature. In this paper, for the first time, we study data inference attacks on multi-dimensional encrypted databases (with 2-D as a special case). We formulate it as a 2-D order-preserving matching problem and explore both unweighted and weighted cases, where the former maximizes the number of points matched using only order information and the latter further considers points with similar frequencies. We prove that the problem is NP-hard, and then propose a greedy algorithm, along with a polynomial-time algorithm with approximation guarantees. Experimental results on synthetic and real-world datasets show that the data recovery rate is significantly enhanced compared with the previous 1-D matching algorithm.

System deployments of IoT systems have drawn research attention, because it is very challenging to meet both physical and cyber constraints in real systems. In this article, we consider the problem of deploying wireless camera networks inside a complex indoor setting for surveillance applications. We formulate the problem of the minimum connected guarding network whose objective is to place a minimum number of cameras satisfying both visual coverage of the domain and wireless network connectivity. We prove that finding the minimum connected guarding network is NP-hard in both the geometric and discrete settings. We also give a 2-approximation algorithm to the geometric minimum guarding network problem. Motivated by the connection of this problem with the watchman tour problem and the art gallery problem, we developed two algorithms to calculate the locations of camera deployment. By deploying a prototype testbed, we verify the feasibility of the system design. Using simulations on 20 real floor plans, we demonstrate that our solutions reduce the number of cameras by up to 28%, and reduce the number of relay nodes by up to 47%.

Beacon-based time-of-flight indoor localization systems have shown great promise for applications ranging from indoor navigation to asset tracking. In large-scale deployments, a major practical challenge is determining the placement of a minimal number of beacons that ensures full coverage -- each point in the domain has line-of-sight paths to enough beacons to uniquely localize itself. Three beacons with line-of-sight paths are always enough, but two beacons within line of sight may also work, given a favorable geometry. In this paper, we propose two beacon placement algorithms that leverage the floor plan geometry with provable theoretical guarantees. First, we present a greedy algorithm using properties of sub-modular functions to place O(OPT · ln m) beacons, where m is the number of discrete location points in the region that need to be localized, and OPT is the size of the optimal solution. Second, we present a random sampling algorithm that places O (OPT · log(OPT)) beacons while localizing all targets. We evaluate our algorithms on both real-world and randomly generated floor plans. Our algorithms place on an average 6 ~ 23% and 12% fewer beacons in real-world topologies and randomly generated floor plans respectively, as compared to prior work. We also present a study where we ask users to attempt to place nodes manually and discover that even humans that are well versed on the coverage problem find it hard to balance the trade-off between the number of beacons and area localized.

In this paper we propose a new membership attack method called co-membership attacks against deep generative models including Variational Autoencoders (VAEs) and Generative Adversarial Networks (GANs). Specifically, membership attack aims to check whether a given instance x was used in the training data or not. A co-membership attack checks whether the given bundle of n instances were in the training, with the prior knowledge that the bundle was either entirely used in the training or none at all. Successful membership attacks can compromise the privacy of training data when the generative model is published. Our main idea is to cast membership inference of target data x as the optimization of another neural network (called the attacker network) to search for the latent encoding to reproduce x. The final reconstruction error is used directly to conclude whether x was in the training data or not. We conduct extensive experiments on a variety of datasets and generative models showing that: our attacker network outperforms prior membership attacks; co-membership attacks can be substantially more powerful than single attacks; and VAEs are more susceptible to membership attacks compared to GANs.

A common approach to protect user's privacy in data collection is to perform random perturbations on user's sensitive data before collection in a way that aggregated statistics can still be inferred without endangering individual secrets. In this paper, we take a closer look at the validity of Differential Privacy guarantees, when sensitive attributes are subject to social contagion. We first show that in the absence of any knowledge about the contagion network, an adversary that tries to predict the real values from perturbed ones, cannot train a classifier that achieves an area under the ROC curve (AUC) above 1-(1-δ)/(1+eε), if the dataset is perturbed using an (ε,δ)-differentially private mechanism. Then, we show that with the knowledge of the contagion network and model, one can do substantially better. We demonstrate that our method passes the performance limit imposed by differential privacy. Our experiments also reveal that nodes with high influence on others are at more risk of revealing their secrets than others. Our method's superior performance is demonstrated through extensive experiments on synthetic and real-world networks.

With the popularity of portable wireless devices it is important to model and predict how information or contagions spread by natural human mobility - for understanding the spreading of deadly infectious diseases and for improving delay tolerant communication schemes. Formally, we model this problem by considering M moving agents, where each agent initially carries a distinct bit of information. When two agents are at the same location or in close proximity to one another, they share all their information with each other. We would like to know the time it takes until all bits of information reach all agents, called the flood time, and how it depends on the way agents move, the size and shape of the network and the number of agents moving in the network.

With the ubiquitous adoption of smartphones and mobile devices, it is now common practice for one's location to be sensed, collected and likely shared through social platforms. While such data can be helpful for many applications, users start to be aware of the privacy issue in handling location and trajectory data. While some users may voluntarily share their location information (e.g., for receiving location-based services, or for crowdsourcing systems), their location information may lead to information leaks about the whereabouts of other users, through the co-location of events when two users are at the same location at the same time and other side information, such as upper bounds of movement speed. It is therefore crucial to understand how much information one can derive about other's positions through the co-location of events and occasional GPS location leaks of some of the users. In this paper we formulate the problem of inferring locations of mobile agents, present theoretically-proven bounds on the amount of information that could be leaked in this manner, study their geometric nature, and present algorithms matching these bounds. We will show that even if a very weak set of assumptions is made on trajectories' patterns, and users are not obliged to follow any 'reasonable' patterns, one could infer very accurate estimation of users' locations even if they opt not to share them. Furthermore, this information could be obtained using almost linear-time algorithms, suggesting the practicality of the method even for huge volumes of data.

On-street parking is an essential component of parking infrastructure for smart cities, which allows users to park near their destinations for short term. However, due to limited capacity, saturated on-street parking becomes a serious and widespread problem for urban transportation systems. Greedily searching for an on-street parking spot in a saturated area is often a frustrating task for drivers, and cruising for vacant parking spots results in additional delays and impaired local circulation. With the recent development of networked smart parking meter, real-time city-wide on- street parking information becomes available for more efficient parking management. In this paper, we design an online parking guidance system that recommends parking spots in real-time based on the parking availability prediction. With a receding horizon optimization framework, our solution minimizes the user's driving and walking cost by adapting the spatiotemporally dynamic supply and demand in the local area, significantly reducing parking competitions in a timely manner. We implement and evaluate our solution with a dataset of 13,503,655 parking records collected from 5228 in-ground sensors distributed in the Australian city Melbourne. The evaluation results show that our approach achieves up to 63.8% delay reduction compared with existing solutions.

Being popular in language evolution, cognitive science, and culture dynamics, the Naming Game has been widely used to analyze how agents reach global consensus via communications in multi-agent systems. Most prior work considered networks that are symmetric and homogeneous (e.g., vertex transitive). In this paper we consider asymmetric or heterogeneous settings that complement the current literature: 1) we show that increasing asymmetry in network topology can improve convergence rates. The star graph empirically converges faster than all previously studied graphs; 2) we consider graph topologies that are particularly challenging for naming game such as disjoint cliques or multi-level trees and ask how much extra homogeneity (random edges) is required to allow convergence or fast convergence. We provided theoretical analysis which was confirmed by simulations; 3) we analyze how consensus can be manipulated when stubborn nodes are introduced at different points of the process. Early introduction of stubborn nodes can easily influence the outcome in certain family of networks while late introduction of stubborn nodes has much less power.

The maturing of mobile devices and systems provides an unprecedented opportunity to collect a large amount of real world human motion data at all scales. While the rich knowledge contained in these data sets is valuable in many fields, various types of personally sensitive information can be easily learned from such trajectory data. The ones that are of most concerns are frequent locations, frequent co-locations and trajectory re-identification through spatio-temporal data points. In this work we analyze privacy protection and data utility when trajectory IDs are randomly mixed during co-location events for data collection or publication. We demonstrate through both analyses and simulations that the global geometric shape of each individual trajectory is sufficiently altered such that re-identification via frequent locations, co-location pairs or spatial temporal data points is not possible with high probability. Meanwhile, a decent number of local geometric features of the trajectory data set are still preserved, including the density distribution and local traffic flow.

Robot Coverage Path planning (i.e., the process of providing full coverage of a given domain by one or multiple robots) is a classical problem in the field of robotics and motion planning. The goal of such planning is to provide nearly full coverage while also minimize duplicately visited area. In this paper, we focus on the scenario of path planning on general surface, including planar domains with complex topology, complex terrain, and general surface in 3D space. Our approach described in this paper adopts a natural, intrinsic and global parametrization of the surface for robot path planning, namely the holomorphic quadratic differentials. We give each point on the surface a uv-coordinates naturally represented by a complex number, except for a small number of zero points (singularities). We show that natural, efficient robot paths can be obtained by using such coordinate systems. The method is based on intrinsic geometry and thus can be adapted to general surface exploration in 3D.

Modern planetary-scale online services have massive data to transfer over the wide area network (WAN). Due to the tremendous cost of building WANs and the stringent timing requirement of distributed applications, it is critical for network operators to make efficient use of network resources to optimize data transfers. By leveraging software-defined networking (SDN) and reconfigurable optical devices, recent solutions design centralized systems to jointly control the network layer and the optical layer. While these solutions show it is promising to significantly reduce data transfer times by centralized cross-layer control, they do not have any theoretical guarantees on the proposed algorithms. This paper presents approximation algorithms and theoretical analysis for the online transfer scheduling problem over optical WANs. The goal of the scheduling problem is to minimize the makespan (the time to finish all transfers) or the total sum of completion times. We design and analyze various greedy, online scheduling algorithms that can achieve 3-competitive ratio for makespan, 2-competitive ratio for minimum sum completion time for jobs of unit size, and 3α-competitive ratio for jobs of arbitrary transfer size and each node having degree constraint d, where α = 1 when d = 1 and α = 1.86 when d ≥ 2. We also evaluated the performance of these algorithms and compared the performance with prior heuristics.

In this paper we study the following problem: given a set of m sensors that collectively cover a set of n target points with heterogeneous coverage requirements (target j needs to be covered every fjslots), how to schedule the sensor duty cycles such that all coverage requirements are satisfied and the maximum number of sensors turned on at any time slot is minimized. The problem models varied real-world applications in which sensing tasks exhibit high discrepancy in coverage requirements - critical locations often need to be covered much more frequently. We provide multiple algorithms with best approximation ratio of O (log n + log m) for the maximum number of sensors to turn on, and bi-criteria algorithm with (α, β)-approximation factors with high probability, where the number of sensors turned on is an α = O(δ(log (n) + log(m))/β)-approximation of the optimal (satisfying all requirements) and the coverage requirement is a β-approximation; δ is the approximation ratio achievable in an appropriate instance of set multi-cover. When the sensor coverage exhibits extra geometric properties, the approximation ratios can be further improved. We also evaluated our algorithms via simulations and experiments on a camera testbed. The performance improvement (energy saving) is substantial compared to turning on all sensors all the time, or a random scheduling baseline.

In this work, we study privacy preserving trajectory sensing and query when n mobile entities (e.g., mobile devices or vehicles) move in an environment of m checkpoints (e.g, WiFi or cellular towers). The checkpoints detect the appearances of mobile entities in the proximity, meanwhile, employ the MinHash signatures to record the set of mobile entities passing by. We build on the checkpoints a distributed data structure named the MinHash hierarchy, with which one can efficiently answer queries regarding popular paths and other traffic patterns. The MinHash hierarchy has a total of near linear storage, linear construction cost, and logarithmic update cost. The cost of a popular path query is logarithmic in the number of checkpoints. Further, the MinHash signature provides privacy protection using a model inspired by the differential privacy model. We evaluated our algorithm using a large mobility data set and compared with previous works to demonstrate its utilities and performances.

We study the r-gather clustering problem in a mobile and distributed setting. In this problem, nodes must be clustered into groups of at least r nodes each, and the goal is to minimize the diameter of the clusters. This notion of clustering is motivated by protecting user anonymity in location-based services or trajectory publication. Prior works on r-gather problems are centralized and cannot be easily adapted to the mobile setting. We describe a distributed algorithm that produces compact clusters, within an approximation factor 4 of the minimum cluster diameter possible. The algorithm can run on the mobile nodes and access points at the network edge locally, and can handle node mobility, rapidly switching cluster memberships as needed. The distributed approach naturally comes with the advantage of greater resilience and stability. Additionally, we show that it achieves local optimality; i.e., from the point of view of any particular node, the solution is nearly as favorable as possible, irrespective of the global configuration. We also show how to cluster trajectories with dynamic re-groupings. Further, we improve the theoretical hardness results for the problem in the Euclidean setting.

As sensor networks are increasingly deployed for critical applications, reliability and latency guarantee become more important than ever to meet industrial requirements. In this paper, we investigated the impact of link burstiness on stream scheduling using a data trace of 3,600,000 packets collected from an indoor testbed. We demonstrate that a good tradeoff between reliability and latency can be achieved by allocating certain time slots on each link for stream transmissions based on its burst length and frequency distributions. With this observation, we design transmission scheduling and routing algorithms for data streams to meet a specified reliability requirement while minimizing end-to- end latency. For the multi-stream scheduling problem, we prove its NP-hardness and design an algorithm that achieves the reliability guarantee and an O(log n) approximation of minimizing the maximum end-to-end latency for any stream. Trace- driven simulations show that our solution meets specified end-to-end reliability requirements with latency up to 9.18 times less than existing solutions.

We consider the problem of capacitated kinetic clustering in which n mobile terminals and k base stations with respective operating capacities are given. The task is to assign the mobile terminals to the base stations such that the total squared distance from each terminal to its assigned base station is minimized and the capacity constraints are satisfied. This paper focuses on the development of distributed and computationally efficient algorithms that adapt to the motion of both terminals and base stations. Suggested by the optimal transportation theory, we exploit the structural property of the optimal solution, which can be represented by a power diagram on the base stations such that the total usage of nodes within each power cell equals the capacity of the corresponding base station. We show by using the kinetic data structure framework the first analytical upper bound on the number of changes in the optimal solution, i.e., its stability. On the algorithm side, using the power diagram formulation we show that the solution can be represented in size proportional to the number of base stations and can be solved by an iterative, local algorithm. In particular, this algorithm can naturally exploit the continuity of motion and has orders of magnitude faster than existing solutions using min-cost matching and linear programming, and thus is able to handle large scale data under mobility.

Using optical sensors for indoor monitoring has been widely adopted in many smart building applications. An important design problem in this space is to explore the tradeoff between energy consumption and coverage quality. While it is important that the sensors achieve full coverage (i.e., every interesting target point can be monitored by at least one sensors), it is often a waste of energy to keep sensors on all the time as events are typically stochastic and rare and most of the time the sensors are on idle monitoring. In this paper we design efficient sensor duty cycles to ensure that any target point of interest is still covered sufficiently frequently while only a subset of sensors are kept on at any time slot. We denote by the maximum dark length for each target point p as the maximum duration in which p is covered at least once. We formulate two optimization problems: the min max dark length scheduling and the min average dark length scheduling. For both versions we provide efficient, practical algorithms with provable approximation guarantee. The two algorithms have been tested on two real testbed scenarios to evaluate its efficiency and coverage quality.

In this paper we investigate sensor deployment and coverage algorithms for using infrared signals in indoor applications. Infrared signals are directional and reliable signals that have little interference with other electromagnetic signals that are commonly found in the deployment domain such as visible light and wireless radio waves. Since the angle of arrival is used, and line of sight is the main constraint for IR signals, we investigate the problem called robust guarding, i.e., placing emitters to ensure that all points of the domain are robustly covered by two emitters that are from sufficiently different directions. We prove combinatorial upper and lower bounds for the number of emitters needed and prove that finding the minimum number of guards is NP-hard. We show that n/2 guards are always sufficient and sometimes necessary for rectilinear polygons and we provide practical algorithms in general. We also developed a testbed with low cost off-the-shelf infrared (IR) emitters and sensors for indoor device-free localization. We tested the algorithms for using infrared sensors for indoor localization and our system achieves an average accuracy of 11.7 cm in a typical office setting.

Bulk transfer on the wide-area network (WAN) is a fundamental service to many globally-distributed applications. It is challenging to efficiently utilize expensive WAN bandwidth to achieve short transfer completion time and meet mission-critical deadlines. Advancements in software-defined networking (SDN) and optical hardware make it feasible and beneficial to quickly reconfigure optical devices in the optical layer, which brings a new opportunity for traffic management on the WAN.

Social behaviors and choices spread through interactions and may lead to a cascading behavior. Understanding how such social cascades spread in a network is crucial for many applications ranging from viral marketing to political campaigns. The behavior of cascade depends crucially on the model of cascade or social influence and the topological structure of the social network.

Motivated by mobile sensor networks as in participatory sensing applications, we are interested in developing a practical, lightweight solution for routing in a mobile network. While greedy routing is robust to mobility, it may get stuck in a local minimum, which then requires non-trivial recovery methods. We find an embedding of the network such that greedy routing using the virtual coordinates guarantees delivery, thus eliminating the necessity of any recovery methods. Our contribution is to replace the in-network computation of the embedding by a preprocessing of the domain before network deployment and encode the map of network domain to virtual coordinate space by using a small number of parameters which can be preloaded to all sensor nodes. As a result, the map is only dependent on the network domain and is independent of the network connectivity. Each node can directly compute or update its virtual coordinates by applying the locally stored map on its geographical coordinates. This represents the first practical solution for using virtual coordinates for greedy routing in a sensor network and could be easily extended to the case of a mobile network. The paper describes algorithmic innovations as well as implementations on a real testbed.

Advances in wireless communication technology have enabled information exchange opportunities between moving vehicles within proximity. Potentially through such physical contacts a piece of information can diffuse to the entire network. While there has been extensive research on information diffusion in social networks, we do not know much about the spatial patterns in vehicle motion and how such patterns can support information dissemination. To this end, in this paper, we provide a systematic study of three large-scale data sets of taxi GPS traces from three big cities. The study shows the following properties universal of the three data sets: 1) the small world property, that information can be disseminated to almost the entire set of participants, within a very small number of hops; 2) certain physical contacts can be extremely effective in exchanging messages and such effectiveness shows a power law distribution; 3) the lack of hubs, no vehicle behaves as major hubs; removing top 20% nodes that have the highest number of physical contacts does not affect the effectiveness of information dissemination. 4) the information dissemination exhibits strong spatial temporal correlation. Finally, to explain the observations in particular the small world property, we develop mathematical models of the taxi movement patterns such that on graph topologies exhibiting properties of real-world road networks a number of observations can be rigorously proved.

With the recent development of localization and tracking systems for both indoor and outdoor settings, we consider the problem of analyzing and representing the huge amount of natural trajectories from human movements that we expect to gather in the near future. In this paper we argue the topological representation, which records how a target moves around the natural obstacles in the underlying environment, can be sufficiently descriptive for many applications and efficient enough for both storing, comparing and classifying these natural human trajectories. Technically, the representation uses the homotopy type of the trajectory. By using harmonic one-forms and Hodge decomposition, we pre-process the sensor network with a purely decentralized algorithm such that the homology class of a trajectory can be obtained by a simple integration along the trajectory. This supports real-time classification of trajectories up to the homology accuracy with minimum communication cost. We test the effectiveness of our approach by showing how to classify randomly generated trajectories in a multi-level arts museum layout as well as how to distinguish real world taxi trajectories in a large city.

Analysis of Internet topologies has shown that the Internet topology has negative curvature, measured by Gromov's “thin triangle condition”, which is tightly related to core congestion and route reliability. In this work we analyze the discrete Ricci curvature of the Internet, defined by Ollivier [1], Lin et al. [2], etc. Ricci curvature measures whether local distances diverge or converge. It is a more local measure which allows us to understand the distribution of curvatures in the network. We show by various Internet data sets that the distribution of Ricci cuvature is spread out, suggesting the network topology to be non-homogenous. We also show that the Ricci curvature has interesting connections to both local measures such as node degree and clustering coefficient, global measures such as betweenness centrality and network connectivity, as well as auxilary attributes such as geographical distances. These observations add to the richness of geometric structures in complex network theory.

Complex contagions describe diffusion of behaviors in a social network in settings where spreading requires influence by two or more neighbors. In a k-complex contagion, a cluster of nodes are initially infected, and additional nodes become infected in the next round if they have at least k already infected neighbors. It has been argued that complex contagions better model behavioral changes such as adoption of new beliefs, fashion trends or expensive technology innovations. This has motivated rigorous understanding of spreading of complex contagions in social networks. Despite simple contagions (k=1) that spread fast in all small world graphs, how complex contagions spread is much less understood. Previous work [11] analyzes complex contagions in Kleinberg's small world model [14] where edges are randomly added according to a spatial distribution (with exponent γ) on top of a two dimensional grid structure. It has been shown in [11] that the speed of complex contagions differs exponentially when γ=0 compared to when γ=2.

Graph filters are a recent and powerful tool to process information in graphs. Yet despite their advantages, graph filters are limited. The limitation is exposed in a filtering task that is common, but not fully solved in sensor networks: the identification of a signal's peaks and pits. Choosing the correct filter necessitates a-priori information about the signal and the network topology. Furthermore, in sparse and irregular networks graph filters introduce distortion, effectively rendering identification inaccurate, even when signal-specific information is available. Motivated by the need for a multi-scale approach, this paper extends classical results on scale-space analysis to graphs. We derive the family of scale-space kernels (or filters) that are suitable for graphs and show how these can be used to observe a signal at all possible scales: from fine to coarse. The gathered information is then used to distributedly identify the signal's peaks and pits. Our graph scale-space approach diminishes the need for a-priori knowledge, and reduces the effects caused by noise, sparse and irregular topologies, exhibiting: (i) superior resilience to noise than the state-of-the-art, and (ii) at least 20% higher precision than the best graph filter, when evaluated on our testbed.

Recent advances in energy transfer technology is boosting the development of renewable sensor networks. To sustain such a network, a mobile robot travels from node to node to recharge each sensor before its battery runs out. Consider each node's recharge as a real-time task, the robot needs to serve these tasks by their deadlines. This represents a class of challenging mobility scheduling problems, where the nodes' deadlines and spatial distribution are often at odds with each other. In this paper, we focus on the scenario where nodes have heterogeneous energy consumption rates, and our goal is to maximize the percentage of nodes alive. We formulate this scheduling problem and prove its NP-completeness. To solve this problem, we propose a spatial dependent task scheduling algorithm, which quantifies the impact of scheduling proximate tasks on the other tasks. With extensive simulations, we reveal the trade-offs of existing solutions under a wide range of network scenarios. Our evaluation results show that our algorithms out-perform classical TSP scheduler by up to 10% and 85% in terms of coverage ratio and average tardiness, respectively.

Recent advances in energy transfer technology is boosting the development of renewable sensor networks. To sustain such a network, a mobile robot travels from node to node to recharge each sensor before its battery runs out. To solve this problem, we propose a spatial dependent task scheduling algorithm, which quantifies the impact of scheduling proximate tasks on the other tasks. Our evaluation results show that our algorithms out-perform classical TSP scheduler by up to10% and 85% in terms of coverage ratio and average tardiness, respectively.

We describe an online algorithm to simplify large volumes of location and sensor data on the source mobile device, by eliminating redundant data points and saving important ones. Our approach is to use topological persistence to identify large scale sharp features of a data stream.

We consider the problem of deployment of cameras inside a complex indoor setting for surveillance applications. We formulate the problem of the minimum guarding network that places a minimum number of cameras satisfying both visual coverage of the domain and wireless network connectivity. We prove that finding the minimum guarding network in both the geometric setting and discrete setting is NP-hard. We also give a 2-approximation algorithm to the geometric minimum guarding network. Motivated by the connection of this problem with the watchman tour problem and the art gallery problem, we develop two algorithms that generate satisfactory results in a prototype testbed and in our simulations.

Homotopic routing asks for a path going around holes according to a given “threading”. Paths of different homo-topy types can be used to improve load balancing and routing resilience. We propose the first lightweight homotopic routing scheme that generates constant bounded stretch compared to the shortest path of the same homotopy type. Our main insight is that in a sequence of triangles to traverse, a message always routed to the nearest point on the next triangle in the sequence travels at most a constant times the length of any shortest path going through the same sequence of triangles. Our routing scheme operates on two levels enabled by a coarse triangulation. The top level is used to specify and represent the requested homotopy type, while the bottom level executes the local greedy routing on a triangle sequence. After a preprocessing step that triangulates the given region and creates a minimum-size auxiliary structure, routing operates greedily at two different resolutions. We also present simulation analysis in a variety of settings and show that the paths indeed have small stretch in practice, considerably shorter than the bounds guaranteed by the theory.

The decentralized detection of event regions is a fundamental building block for monitoring and reasoning about spatial phenomena. However, so far the problem has been studied almost exclusively for static networks. This study proposes a theoretical framework with which we can analyze event detection algorithms suitable for large-scale mobile networks. Our analysis builds on the following insight: the inherent trends of spatial events are well captured by the spectral domain of the network graph. Using this framework, we propose novel local algorithms that are location-free; that work with mobile nodes and dynamic events; that operate on 3D topologies; and that are simple to implement. We are not aware of event detection algorithms possessing all these traits. Simulations based on complex oil spill traces showcase the resilience and robustness of our methods. Additionally, we demonstrate their validity for practical scenarios by evaluating them on a 105 node testbed.

Load balanced routing in a network, i.e., minimizing the maximum traffic load any node carries for unsplittable flows, is a well known NP-hard problem. Finding practical algorithms remains a long standing challenge. In this paper we propose greedy routing using virtual coordinates that achieves both small path stretch ratio (compared to shortest path) and small load balancing ratio (compared to optimal load balanced routing), in a large scale wireless sensor network deployed densely inside a geometric domain with complex shape. We first provide a greedy routing scheme on a disk with a stretch ratio of at most 2, and under which the maximum load is a factor 4√2 smaller than the maximum load under shortest path routing. This is the first simple routing scheme with a small stretch that has been proven to outperform shortest path routing in terms of load balancing. Then we transform a network of arbitrary shape to a disk by an area preserving map φ. We show that both the path length and the maximum traffic load in the original network only increases by an additional factor of d2, where d is the maximum length stretch of φ. Combined with the result on a disk we again achieve both bounded stretch and bounded load balancing ratio. Our simulation results evaluated the practical performance on both quality measures.

In this paper we propose an algorithm to construct a “space filling” curve for a sensor network with holes. Mathematically, for a given multi-hole domain R, we generate a path P that is provably aperiodic (i.e., any point is covered at most a constant number of times) and dense (i.e., any point of R is arbitrarily close to P). In a discrete setting as in a sensor network, the path visits the nodes with progressive density, which can adapt to the budget of the path length. Given a higher budget, the path covers the network with higher density. With a lower budget the path becomes proportional sparser. We show how this density-adaptive space filling curve can be useful for applications such as serial data fusion, motion planning for data mules, sensor node indexing, and double ruling type in-network data storage and retrieval. We show by simulation results the superior performance of using our algorithm vs standard space filling curves and random walks.

Motivated by mobile sensor networks as in participatory sensing applications, we are interested in developing a practical, lightweight solution for routing in a mobile network. While greedy routing is robust to mobility, location errors and link dynamics, it may get stuck in a local minimum, which then requires non-trivial recovery methods. We follow the approach taken by Sarkar et. al. [24] to find an embedding of the network such that greedy routing using the virtual coordinates guarantees delivery, thus eliminating the necessity of any recovery methods. Our new contribution is to replace the in-network computation of the embedding by a preprocessing of the domain before network deployment and encode the map of network domain to virtual coordinate space by using a small number of parameters which can be pre-loaded to all sensor nodes. As a result, the map is only dependent on the network domain and is independent of the network connectivity. Each node can directly compute or update its virtual coordinates by applying the locally stored map on its geographical coordinates. This represents the first practical solution for using virtual coordinates for greedy routing in a sensor network and could be easily extended to the case of a mobile network. Being extremely light-weight, greedy routing on the virtual coordinates is shown to be very robust to mobility, link dynamics and non-unit disk graph connectivity models.

Random walk on a graph is a Markov chain and thus is `memoryless' as the next node to visit depends only on the current node and not on the sequence of events that preceded it. With these properties, random walk and its many variations have been used in network routing to `randomize' the traffic pattern and hide the location of the data sources. In this paper we examine a myth in common understanding of the memoryless property of a random walk applied for protecting source location privacy in a wireless sensor network. In particular, if one monitors only the network boundary and records the first boundary node hit by a random walk, this distribution can be related to the location of the source node. For the scenario of a single data source, a very simple algorithm by integrating along the network boundary would reveal the location of the source. We also develop a generic algorithm to reconstruct the source locations for various sources that have simple descriptions (e.g., k source locations, sources on a line segment, sources in a disk). This represents a new type of traffic analysis attack for invading sensor data location privacy and essentially re-opens the problem for further examination.

First responder applications often require safety surveillance using wireless camera networks~\cite{breadcrums}. To ensure visual sensing coverage, it is crucial to place optical sensor nodes at proper locations. Under the scenario of energy constrained wireless camera deployment, the issue of communication cost should also be considered. Previous camera deployment research (e.g Art Gallery Problem) mainly concerned sensing coverage. One well-known solution for the art gallery problem is to triangulate the objective polygon and then select vertices to ensure full coverage. However, deploying cameras only in the vertices of polygon may induce inefficiency both in number of necessary cameras and overall communication cost. To reduce the cost, we propose two deployment algorithms: 1) connected visibility region planning algorithm for static deployment given the floor plan is known, and 2) connected visibility region tracking algorithm for the dynamic deployment during the run time. In extensive simulations with real floor plans, our algorithms outperform previous solutions significantly.

Diseases, information and rumors could spread fast in social networks exhibiting the small world property. In the diffusion of these 'simple contagions', which can spread through a single contact, a small network diameter and the existence of weak ties in the network play important roles. Recent studies by sociologists [Centola and Macy 2007] have also explored 'complex contagions' in which multiple contacts are required for the spread of contagion. [Centola and Macy 2007] and [Romero et al. 2011] have shown that complex contagions exhibit different diffusion patterns than simple ones. In this paper, we study three small world models and provide rigorous analysis on the diffusion speed of a k-complex contagion, in which a node becomes active only when at least k of its neighbors are active. Diffusion of a complex contagion starts from a constant number of initial active nodes. We provide upper and lower bounds on the number of rounds it takes for the entire network to be activated. Our results show that compared to simple contagions, weak ties are not as effective in spreading complex contagions due to the lack of simultaneous active contacts; and the diffusion speed depends heavily on the the way weak ties are distributed in a network.

Social groups often exhibit a high degree of dynamism. Some groups thrive, while many others die over time. Modeling group stability dynamics and understanding whether/when a group will remain stable or shrink over time can be important in a number of social domains. In this paper, we study two different types of social networks as exemplar platforms for modeling and predicting group stability dynamics. We build models to predict if a group is going to remain stable or is likely to shrink over a period of time. We observe that both the level of member diversity and social activities are critical in maintaining the stability of groups. We also find that certain 'prolific' members play a more important role in maintaining the group stability. Our study shows that group stability can be predicted with high accuracy, and feature diversity is critical to prediction performance.

Consider mobile targets in a plane and their movements being monitored by a network such as a field of sensors. We develop distributed algorithms for in-network tracking and range queries for aggregated data (for example, returning the number of targets within any user given region). Our scheme stores the target detection information locally in the network and answers a query by examining the perimeter of the given range. The cost of updating data about mobile targets is proportional to the target displacement. The key insight is to maintain in the sensor network a function with respect to the target detection data on the graph edges that is a differential form such that the integral of this form along any closed curve C gives the integral within the region bounded by C. The differential form has great flexibility, making it appropriate for tracking mobile targets. The basic range query can be used to find a nearby target or any given identifiable target with cost O(d), where d is the distance to the target in question. Dynamic insertion, deletion, coverage holes, and mobility of sensor nodes can be handled with only local operations, making the scheme suitable for a highly dynamic network. It is extremely robust and capable of tolerating errors in sensing and target localization. Targets do not need to be identified for the tracking, thus user privacy can be preserved. In this paper, we only elaborate the advantages of differential forms in tracking of mobile targets. Similar routines can be applied for organizing many other types of information-for example, streaming scalar sensor data (such as temperature data field)-to support efficient range queries. We demonstrate through analysis and simulations that this scheme compares favorably to existing schemes that use location services for answering aggregate range queries of target detection data.

In traditional routing, the routing tables store shortest paths to all other destinations and have size linear in the size of the network, which is not scalable for resource-constrained networks such as wireless sensor networks. In this article we show that by storing selectively a much smaller set of routing paths in the routing tables one can get low-stretch, compact routing schemes. Our routing scheme includes an approximate distance oracle with which one can obtain approximate shortest path length estimates to destinations. This distance oracle can be obtained, for example, by a landmark-based scheme, or in case of sensor networks, from the geographic distance between node locations. With an approximate distance oracle one can attempt greedy routing by forwarding to the neighbor whose estimate is closer to the destination. But there is no guarantee of delivery nor of the routing path length. We augment the distance oracle by storing, for each node u , routing paths to O (log 2 n ) strategically selected nodes that serve as intermediate destinations. These nodes are selected with probability proportional to 1/ r ρ , where r is the distance to u and ρ is a suitable constant for the network. Then we derive a set of sufficient conditions to select the next step at each stage of routing, such that these conditions can be verified locally and guarantee 1+ε stretch routing on any metric. These conditions serve as the “greedy routing” or local decision rule. On graphs of bounded growth, our scheme guarantees 1+ε stretch routing with high probability, with an average routing table size of O (√n log 2 n ). This scheme is favorable for its simplicity, generality, and blindness to any global state. It demonstrates that global routing properties could emerge from purely distributed and uncoordinated routing table design.

We present a novel area-preservation mapping/flattening method using the optimal mass transport technique, based on the Monge-Brenier theory. Our optimal transport map approach is rigorous and solid in theory, efficient and parallel in computation, yet general for various applications. By comparison with the conventional Monge-Kantorovich approach, our method reduces the number of variables from O(n2) to O(n), and converts the optimal mass transport problem to a convex optimization problem, which can now be efficiently carried out by Newton's method. Furthermore, our framework includes the area weighting strategy that enables users to completely control and adjust the size of areas everywhere in an accurate and quantitative way. Our method significantly reduces the complexity of the problem, and improves the efficiency, flexibility and scalability during visualization. Our framework, by combining conformal mapping and optimal mass transport mapping, serves as a powerful tool for a broad range of applications in visualization and graphics, especially for medical imaging. We provide a variety of experimental results to demonstrate the efficiency, robustness and efficacy of our novel framework.

Location privacy is an important concern in participatory sensing applications, where users can both contribute valuable information (data reporting) as well as retrieve (location-dependent) information (query) regarding their surroundings. K-anonymity is an important measure for privacy to prevent the disclosure of personal data. In this paper, we propose a mechanism based on locality-sensitive hashing (LSH) to partition user locations into groups each containing at least K users (called spatial cloaks). The mechanism is shown to preserve both locality and K-anonymity. We then devise an efficient algorithm to answer kNN queries for any point in the spatial cloaks of arbitrary polygonal shape. Extensive simulation study shows that both algorithms have superior performance with moderate computation complexity.

We study scalable routing for a sensor network deployed in complicated 3D settings such as underground tunnels in gas system or water system. The nodes are in general 3D space but they are very sparsely located and the network has complex topology. We propose a routing scheme by first embdding the network on a surface with possibly non-zero genus. Then we compute a canonical hyperbolic metric of the embedded surface, and use geodesics to decompose the network into canonical components called pairs of `pants' whose topology is simpler (with genus zero). The adjacency of the pants components is extracted as a high level routing map and stored at every node. With the hyperbolic metric one can use greedy routing to navigate within and across pants. Altogether this leads to a two-level routing scheme by first finding a sequence of pants and then realizing the route with greedy steps. We show by simulation that the number of pants is closely related to the true `genus' of the network and that the routing scheme is efficient and scalable.

In this paper we address the problem of scalable and load balanced routing for wireless sensor networks. Motivated by the analog of the continuous setting that geodesic routing on a sphere gives perfect load balancing, we embed sensor nodes on a convex polyhedron in 3D and use greedy routing to deliver messages between any pair of nodes with guaranteed success. This embedding is known to exist by the Koebe-Andreev-Thurston Theorem for any 3-connected planar graphs. In our paper we use discrete Ricci flow to develop a distributed algorithm to compute this embedding. Further, such an embedding is not unique and differs from one another by a Möbius transformation. We employ an optimization routine to look for the Möbius transformation such that the nodes are spread on the polyhedron as uniformly as possible. We evaluated the load balancing property of this greedy routing scheme and showed favorable comparison with previous schemes.

A wormhole attack places two radio transceivers connected by a high capacity link and retransmits wireless signals from one antenna at the other. This creates a set of shortcut paths in the network, and may attract a lot of traffic to the wormhole link. The link thus gains control of a large fraction of network traffic which opens the door for more dangerous attacks afterwards. In this paper we introduce a wormhole detection and removal algorithm based on local connectivity tests.

In this article we propose a lightweight algorithm for constructing multiresolution data representations for sensor networks. At each sensor node u , we compute O (log n ) aggregates about exponentially enlarging neighborhoods centered at u . The i th aggregate is the aggregated data from nodes approximately within 2 i hops of u . We present a scheme, named the hierarchical spatial gossip algorithm , to extract and construct these aggregates, for all sensors simultaneously, with a total communication cost of O ( n polylog n ). The hierarchical gossip algorithm adopts atomic communication steps with each node choosing to exchange information with a node distance d away with probability ∼ 1/ d 3 . The attractiveness of the algorithm can be attributed to its simplicity, low communication cost, distributed nature, and robustness to node failures and link failures. We show in addition that computing multiresolution aggregates precisely (i.e., each aggregate uses all and only the nodes within 2 i hops) requires a communication cost of Ω( n √ n ), which does not scale well with network size. An approximate range in aggregate computation like that introduced by the gossip mechanism is therefore necessary in a scalable efficient algorithm. Besides the natural applications of multiresolution data summaries in data validation and information mining, we also demonstrate the application of the precomputed multiresolution data summaries in answering range queries efficiently.

Small-world experiments in which packages reach addressees unknown to the original sender through a forwarding chain confirm that acquaintance networks have short paths, a property that was later also discovered in many other networks. They further show that people can find these paths by passing the package on to the acquaintance most socially proximate to the target. This has led researchers to conjecture that perhaps also in many other networks some proximity-based algorithm can be used to find short paths, provided that nodes are given appropriate coordinates. Although potential applications are numerous, ranging from decentralized search to recommendation-based trust to disease control, this conjecture has remained largely unverified. In this paper we apply algorithmic methods to embed nodes in some latent space and employ greedy routing to deliver packages. Using these methods we empirically investigate the navigability of five real-world complex networks from diverse contexts and of varying topology. In each network, we deliver a majority of packages in fewer than six hops.

The best known upper bound on the number of topological changes in the Delaunay triangulation of a set of moving points in ℜ2 is (nearly) cubic, even if each point is moving with a fixed velocity. We introduce the notion of a stable Delaunay graph (SDG in short), a dynamic subgraph of the Delaunay triangulation, that is less volatile in the sense that it undergoes fewer topological changes and yet retains many useful properties of the full Delaunay triangulation. SDG is defined in terms of a parameter ± > 0, and consists of Delaunay edges pq for which the (equal) angles at which p and q see the corresponding Voronoi edge epq are at least ±. We prove several interesting properties of SDG and describe two kinetic data structures for maintaining it. Both structures use O*(n) storage. They process O*(n2) events during the motion, each in O*(1) time, provided that the points of P move along algebraic trajectories of bounded degree; the O*(·) notation hides multiplicative factors that are polynomial in 1/± and polylogarithmic in n. The first structure is simpler but the dependency on 1/± in its performance is higher.

We study how to characterize the families of paths between any two nodes s, t in a sensor network with holes. Two paths that can be deformed to one another through local changes are called homotopy equivalent. Two paths that pass around holes in different ways have different homotopy types. With a distributed algorithm we compute an embedding of the network in hyperbolic space by using Ricci flow such that paths of different homotopy types are mapped naturally to paths connecting s with different images of t. Greedy routing to a particular image is guaranteed with success to find a path with a given homotopy type. This leads to simple greedy routing algorithms that are resilient to both local link dynamics and large scale jamming attacks and improve load balancing over previous greedy routing algorithms.

We study the problem of maintaining group communication between m mobile agents, tracked and helped by n static networked sensors. We develop algorithms to maintain a O(lg n)-approximation to the minimum Sterner tree of the mobile agents such that the maintenance message cost is on average O(lg n) per each hop an agent moves. The key idea is to extract a 'hierarchical well-separated tree (HST)' on the sensor nodes such that the tree distance approximates the sensor network hop distance by a factor of O(lg n). We then prove that maintaining the subtree of the mobile agents on the HST uses logarithmic messages per hop movement. With the HST we can also maintain O(lg n) approximate k-center for the mobile agents with the same message cost. Both the minimum Steiner tree and the k-center problems are NP-hard and our algorithms are the first efficient algorithms for maintaining approximate solutions in a distributed setting.

For in-network storage schemes, one maps data, indexed in a logical space, to the distributed sensor locations. When the physical sensor network has an irregular shape and possibly holes, the mapping of data to sensors often creates unbalanced storage load with high data concentration on nodes near network boundaries. In this paper we propose to map data to a covering space, which is a tiling of the plane with copies of the sensor network, such that the sensors receive uniform storage load and traffic. We propose distributed algorithms to construct the covering space with Ricci flow and Möbius transforms. The use of the covering space improves the performance of many in-network storage and retrieval schemes such as geographical hash tables (GHTs) or the double rulings (quorum based schemes), and provides better load balanced routing.

Consider mobile targets moving in a plane and their movements being monitored by a network such as a field of sensors. We develop distributed algorithms for in-network tracking and range queries for aggregated data (for example returning the number of targets within any user given region). Our scheme stores the target detection information locally in the network, and answers a query by examining the perimeter of the given range. The cost of updating data about mobile targets is proportional to the target displacement. The key insight is to maintain in the sensor network a function with respect to the target detection data on the graph edges that is a differential one-form such that the integral of this one-form along any closed curve C gives the integral within the region bounded by C.

In this paper, we address the problem of preserving generated data in a sensor network in case of node failures. We focus on the type of node failures that have explicit spatial shapes such as circles or rectangles (e.g., modeling a bomb attack or a river overflow). We consider two different schemes for introducing redundancy in the network, by simply replicating data or by using erasure codes, with the objective to minimize the communication cost incurred to build such data redundancy. We prove that the problem is NP-hard using either replication or coding. We design Oα-approximation centralized and distributed algorithms for the two redundancy schemes, where α is the "fatness" of the potential node failure events. Using erasure codes, data distribution can be handled in an efficient distributed manner. Simulation results show that by exploiting the spatial properties of the node failure patterns, one can substantially reduce the communication cost compared to the resilient data storage schemes in the prior literature.

A set of k balls B 1 , …, B k in a Euclidean space is said to cover a collection of lines if every line intersects some ball. We consider the k - center problem for lines in high-dimensional space: Given a set of n lines l = { l 1 ,…, l n in R d , find k balls of minimum radius which cover l . We present a 2-approximation algorithm for the cases k = 2, 3 of this problem, having running time quasi-linear in the number of lines and the dimension of the ambient space. Our result for 3-clustering is strongly based on a new result in discrete geometry that may be of independent interest: a Helly-type theorem for collections of axis-parallel “crosses” in the plane. The family of crosses does not have finite Helly number in the usual sense. Our Helly theorem is of a new type: it depends on ε-contracting the sets. In statistical practice, data is often incompletely specified; we consider lines as the most elementary case of incompletely specified data points. Clustering of data is a key primitive in nonparametric statistics. Our results provide a way of performing this primitive on incomplete data, as well as imputing the missing values.

We introduce a new feature size for bounded domains in the plane endowed with an intrinsic metric. Given a point x in a domain X , the systolic feature size of X at x measures half the length of the shortest loop through x that is not null-homotopic in X . The resort to an intrinsic metric makes the systolic feature size rather insensitive to the local geometry of the domain, in contrast with its predecessors (local feature size, weak feature size, homology feature size). This reduces the number of samples required to capture the topology of X , provided that a reliable approximation to the intrinsic metric of X is available. Under sufficient sampling conditions involving the systolic feature size, we show that the geodesic Delaunay triangulation D x ( L ) of a finite sampling L is homotopy equivalent to X . Under similar conditions, D x ( L ) is sandwiched between the geodesic witness complex C W X ( L ) and a relaxed version C W X,ν ( L ). In the conference version of the article, we took advantage of this fact and proved that the homology of D x ( L ) (and hence the one of X ) can be retrieved by computing the persistent homology between C W X ( L ) and C W X,ν ( L ). Here, we investigate further and show that the homology of X can also be recovered from the persistent homology associated with inclusions of type C W X,ν ( L )↪ C W X,ν′ ( L ), under some conditions on the parameters ν≤ν′. Similar results are obtained for Vietoris-Rips complexes in the intrinsic metric. The proofs draw some connections with recent advances on the front of homology inference from point cloud data, but also with several well-known concepts of Riemannian (and even metric) geometry. On the algorithmic front, we propose algorithms for estimating the systolic feature size of a bounded planar domain X , selecting a landmark set of sufficient density, and computing the homology of X using geodesic witness complexes or Rips complexes.

Location information is of essential importance in sensor networks deployed for generating location-specific event reports. When such networks operate in hostile environments, it becomes imperative to guarantee the correctness of event location claims. In this article we address the problem of assessing location claims of untrusted (potentially compromised) nodes. The mechanisms introduced here prevent a compromised node from generating illicit event reports for locations other than its own. This is important because by compromising “easy target” sensors (say, sensors on the perimeter of the field that's easier to access), the adversary should not be able to impact data flows associated with other (“premium target”) regions of the network. To achieve this goal, in a process we call location certification , data routed through the network is “tagged” by participating nodes with “belief” ratings, collaboratively assessing the probability that the claimed source location is indeed correct. The effectiveness of our solution relies on the joint knowledge of participating nodes to assess the truthfulness of claimed locations. By collaboratively generating and propagating a set of “belief” ratings with transmitted data and event reports, the network allows authorized parties (e.g., final data sinks) to evaluate a metric of trust for the claimed location of such reports. Belief ratings are derived from a data model of observed past routing activity. The solution is shown to feature a strong ability to detect false location claims and compromised nodes. For example, incorrect claims as small as 2 hops (from the actual location) are detected with over 90% accuracy. Finally, these new location certification mechanisms can be deployed in tandem with traditional secure localization, yet do not require it, and, in a sense, can serve to minimize the need thereof.

We study the localization problem in sensor networks by using local angle measurement. Localization using local angle information was recently proposed as an effective localization technique, which can be used for geographical routing with guaranteed delivery. However, the existing approach is based on linear programming (LP) and can not be implemented distributedly. We propose, design, and evaluate DAL: a purely distributed localization protocol in sensor networks using local angle measurement. Localization with local angle poses unique challenge in sensor networks due to information uncertainties identified in this paper. DAL specifically addresses these challenges. Via extensive simulations using ns2 and our own simulator, we show that the performance of DAL is comparable with that of the centralized LP approach in most cases. Our preliminary results with noisy angle measurement show that DAL keeps the global geometry of the sensor network fairly well.

Replicating content across a geographically distributed set of servers and redirecting clients to the closest server in terms of latency has emerged as a common paradigm for improving client performance. In this paper, we analyze latencies measured from servers in Google's content distribution network (CDN) to clients all across the Internet to study the effectiveness of latency-based server selection. Our main result is that redirecting every client to the server with least latency does not suffice to optimize client latencies. First, even though most clients are served by a geographically nearby CDN node, a sizeable fraction of experience latencies several tens of milliseconds higher than other in the same region. Second, we find that queueing delays often override the benefits of a client interacting with a nearby server.

We propose a generic routing table design principle for scalable routing on networks with bounded geometric growth. Given an inaccurate distance oracle that estimates the graph distance of any two nodes with constant factor upper and lower bounds, we augment it by storing the routing paths of pairs of nodes, selected in a spatial distribution, and show that the routing table enables 1 + epsiv stretch routing. In the wireless ad hoc and sensor network scenario, the geographic locations of the nodes serve as such an inaccurate distance oracle. Each node p selects O (log n loglog n) other nodes from a distribution proportional to 1/r2where r is the distance to p and the routing paths to these nodes are stored on the nodes along these paths in the network. The routing algorithm selects links conforming to a set of sufficient conditions and guarantees with high probability 1 + epsiv stretch routing with routing table size O(radicn log n loglog n) on average for each node. This scheme is favorable for its simplicity, generality and blindness to any global state. It is a good example that global routing properties emerge from purely distributed and uncoordinated routing table design.

We study the anchor-free localization problem for a large-scale sensor network with a complex shape, knowing network connectivity information only. The main idea follows from our previous work in which a subset of the nodes are selected as landmarks and the sensor field is partitioned into Voronoi cells with all the nodes closest to the same landmark grouped into the same cell. We extract the combinatorial Delaunay complex as the dual complex of the landmark Voronoi diagram and embed the combinatorial Delaunay complex as a structural skeleton. In this paper we develop a new landmark selection algorithm with incremental Delaunay refinement method. This algorithm does not assume any knowledge of the network boundary and runs in a distributed manner to select landmarks incrementally until both the global rigidity property (the Delaunay complex is globally rigid and thus can be embedded uniquely) and the coverage property (every node is not far from the embedded Delaunay complex) are met. The new algorithm substantially improves the robustness and applicability of the original localization algorithm, especially in networks with very low average degree (even non- rigid networks) and complex shapes.

We study the problem of in-network processing and queries of trajectories of moving targets in a sensor network. The main idea is to exploit the spatial coherence of target trajectories for opportunistic information dissemination with no or small extra communication cost, as well as for efficient probabilistic queries searching for a given target signature in a real-time manner. Sensors near a moving target are waken up to record information about this target and take the communication opportunities to exchange their knowledge with preceding and descending sensor nodes along the trajectory. Thus a moving target's information is naturally detected, recorded, and disseminated along its trajectory, as well as the motion trajectories that enter the sensor field afterwards. We analyzed and through simulations tested the dissemination cost and query success rate for randomly generated data sets. Trajectories of reasonable length can be discovered by probabilistic in-network queries with high probability. Compared with the scheme without opportunistic dissemination, the in-network processing of trajectories, with modest cost on dissemination, allows substantially reduced query cost and delay.

We are interested in topological analysis and processing of the large-scale distributed data generated by sensor networks. Naturally, a large-scale sensor network is deployed in a geometric region with possibly holes and complex shape, and is used to sample some smooth physical signal field. We are interested in both the topology of the discrete sensor field in terms of the sensing holes (voids without sufficient sensors deployed), as well as the topology of the signal field in terms of its critical points (local maxima, minima and saddles). Towards this end, we develop distributed algorithms to construct the Morse-Smale decomposition, and study the performance benefits obtained by this approach. The sensor field is decomposed into simply-connected pieces, inside each of which the sensor signal is homogeneous, i.e., the data flows uniformly from a local maximum to a local minimum. The Morse-Smale decomposition can be efficiently constructed in the network locally, after which applications such as iso-contour queries, data-guided navigation and routing, data aggregation, and topologically faithful signal reconstructions benefit tremendously from it.

Location information is useful both for network organization and for sensor data integrity. In this article, we study the anchor-free 2D localization problem by using local angle measurements. We prove that given a unit disk graph and the angles between adjacent edges, it is NP-hard to find a valid embedding in the plane such that neighboring nodes are within distance 1 from each other and non-neighboring nodes are at least distance √2/2 away. Despite the negative results, however, we can find a planar spanner of a unit disk graph by using only local angles. The planar spanner can be used to generate a set of virtual coordinates that enable efficient and local routing schemes such as geographical routing or approximate shortest path routing. We also proposed a practical anchor-free embedding scheme by solving a linear program. We show by simulation that it gives both a good local embedding, with neighboring nodes embedded close and non-neighboring nodes far away, and a satisfactory global view such that geographical routing and approximate shortest path routing on the embedded graph are almost identical to those on the original (true) embedding.

We study the problem of localizing a large sensor network having a complex shape, possibly with holes. A major challenge with respect to such networks is to figure out the correct network layout, that is, avoid global flips where a part of the network folds on top of another. Our algorithm first selects landmarks on network boundaries with sufficient density, then constructs the landmark Voronoi diagram and its dual combinatorial Delaunay complex on these landmarks. The key insight is that the combinatorial Delaunay complex is provably globally rigid and has a unique realization in the plane. Thus an embedding of the landmarks by simply gluing the Delaunay triangles properly recovers the faithful network layout. With the landmarks nicely localized, the rest of the nodes can easily localize themselves by trilateration to nearby landmark nodes. This leads to a practical and accurate localization algorithm for large networks using only network connectivity. Simulations on various network topologies show surprisingly good results. In comparison, previous connectivity-based localization algorithms such as multidimensional scaling and rubberband representation generate globally flipped or distorted localization results.

The diversity of the deployment settings of sensor networks is naturally inherited from the diversity of geographical features of the embedded environment, and greatly influences network design. Many sensor network protocols in the literature implicitly assume that sensor nodes are deployed inside a simple geometric region, without considering possible obstacles and holes in the deployment environment. When the real deployment setting deviates from that, we often observe degraded performance. Thus, it is highly desirable to have a generic approach to handle sensor fields with complex shapes. In this article, we propose a segmentation algorithm that partitions an irregular sensor field into nicely shaped pieces such that algorithms and protocols that assume a nice sensor field can be applied inside each piece. Across the segments, problem dependent structures specify how the segments and data collected in these segments are integrated. Our segmentation algorithm does not require any extra knowledge (e.g., sensor locations) and only uses network connectivity information. This unified spatial-partitioning approach makes the protocol design become flexible and independent of deployment specifics. Existing protocols are still reusable with segmentation, and the development of new topology-adaptive protocols becomes much easier. We verified the correctness of the algorithm on various topologies and evaluated the performance improvements by integrating shape segmentation with several fundamental problems in network design.

An unweighted graph has density rho and growth rate k if the number of nodes in every ball with radius r is bounded by rhork. The communication graphs of wireless networks and peer-to-peer networks often have constant bounded density and small growth rate. In this paper, we study the trade-off between two quality measures for routing in growth-restricted graphs. The two measures we consider are the stretch factor, which measures the lengths of the routing paths, and the load-balancing ratio, which measures the evenness of the traffic distribution. We show that if the routing algorithm is required to use paths with stretch factor c, then its load-balancing ratio is bounded by O(rho1/k(n/c)1-1/k), and the bound is tight in the worst case. We show the application and extension of the trade-off to the wireless network routing and VLSI layout design. We also present a load-balanced routing algorithm with the stretch factor constraint in an online setting, in which the routing requests come one by one.

We study the problem of localizing a large sensor network having a complex shape, possibly with holes. A major challenge with respect to such networks is to figure out the correct network layout, i.e., avoid global flips where a part of the network folds on top of another. Our algorithm first selects landmarks on network boundaries with sufficient density, then constructs the landmark Voronoi diagram and its dual combinatorial Delaunay complex on these landmarks. The key insight is that the combinatorial Delaunay complex is provably globally rigid and has a unique realization in the plane. Thus an embedding of the landmarks by simply gluing the Delaunay triangles properly recovers the faithful network layout. With the landmarks nicely localized, the rest of the nodes can easily localize themselves by trilateration to nearby landmark nodes. This leads to a practical and accurate localization algorithm for large networks using only network connectivity. Simulations on various network topologies show surprisingly good results. In comparison, previous connectivity-based localization algorithms such as multi-dimensional scaling and rubberband representation generate globally flipped or distorted localization results.

Abstract—We study the problem of data-driven routing and navigation in a distributed sensor network over a continuous scalar field. Specifically, we address the problem of searching for the collection of sensors with readings within a specified range. This is named the iso-contour query problem. We develop a gradient based routing scheme such that from any query node, the query message follows the signal field gradient or derived quantities and successfully discovers all iso-contours of interest. Due to the existence of local maxima and minima, the guaranteed delivery requires preprocessing of the signal field and the construction of a contour tree in a distributed fashion. Our approach has the following properties: (i) the gradient routing uses only local node information and its message complexity is close to optimal, as shown by simulations; (ii) the preprocessing message complexity is linear in the number of nodes and the storage requirement for each node is a small constant. The same preprocessing also facilitates route computation between any pair of nodes where the the route lies within any user supplied range of values. I.

We study the problem of contour tracking with binary sensors, an important problem for monitoring spatial signals and tracking group targets. In particular, we track the boundaries of the blobs of interest and capture the topological changes as the blobs merge or split. Only the nodes on the boundaries of these deformable blobs stay active and the repair cost is proportional to the size of the contour changes. Our algorithm is completely distributed, requires only local information, and yet captures the global topological properties. The algorithm performs a fundamental monitoring function and is a foundation for further information processing of spatial sensor data.

In sensor networks we aim to achieve global objectives through local decisions at each node, based only on data available in the node's neighborhood. In this paper, we diffuse information away from source nodes holding desired data, so as to establish information potentials that allow network queries to navigate towards and reach these sources through local greedy decisions, following information gradients. We compute these information potentials by solving for a discrete approximation to a partial differential equation over appropriate network neighborhoods, through a simple local iteration that can be executed in a distributed manner and can be re-invoked to repair the information field locally when links fail, sources move, etc. The solutions to this equation are classical harmonic functions, which have a rich algebraic structure and many useful properties, including the absence of local extrema, providing a guarantee that our local greedy navigation will not get stuck.Unlike shortest path trees, which can also be used to guide queries to sources, information potentials are robust to low-level link volatility as they reflect more global properties of the underlying connectivity. By exploiting the algebraic structure of harmonic functions such potentials can be combined in interesting ways to enable far greater path diversity and thus provide better load balancing than is possible with fixed tree structures, or they can be used to answer range queries about the number of sources in a certain regions by simply traversing the boundary of the region. Potentials for multiple information types can be aggregated and compressed using a variant of the q-digest data structure. The paper provides both analytic results and detailed simulations supporting these claims.

We study the problem of landmark selection for landmark-based routing in a network of fixed wireless communication nodes. We present a distributed landmark selection algorithm that does not rely on global clock synchronization, and a companion local greedy landmark-based routing scheme. We assume no node location information, and that each node can communicate with some of its geographic neighbors. Each node is named by its hop count distances to a small number of nearby landmarks. Greedy routing at a node is performed to equalize its vector of landmark distances to that of the destination. This is done by following the shortest path to the landmark that maximizes the ratio of its distances to the source and the destination. In addition, we propose a method to alleviate the difficulty in routing to destinations near the boundaries by virtually expanding the network boundaries. The greedy routing, when combined with our landmark selection scheme, has a provable bounded path stretch relative to the best path possible, and guarantees packet delivery in the continuous domain. In the discrete domain, our simulations show that the landmark selection scheme is effective, and the companion routing scheme performs well under realistic settings. Both the landmark selection and greedy routing assumes no specific communication model and works with asymmetric links. Although some of the analysis are non-trivial, the algorithms are simple, flexible and cost-effective enough to warrant a real-world deployment.

Many sensor network protocols in the literature implicitly assume that sensor nodes are deployed uniformly inside a simple geometric region. When the real deployment deviates from that, we often observe degraded performance. It is desirable to have a generic approach to handle a sensor field with complex shape. In this paper, we propose a segmentation algorithm that partitions an irregular sensor field into nicely shaped pieces such that algorithms and protocols that assume a nice sensor field can be applied inside each piece. Across the segments, problem dependent structures specify how the segments and data collected in these segments are integrated. This unified topology-adaptive spatial partitioning would benefit many settings that currently assume a nicely shaped sensor field. Our segmentation algorithm does not require sensor locations and only uses network connectivity information. Each node is given a 'flow direction' that directs away from the network boundary. A node with no flow direction becomes a sink, and attracts other nodes in the same segment. We evaluate the performance improvements by integrating shape segmentation with applications such as distributed indices and random sampling.

We study the problem of aggregating data from a sparse set of nodes in a wireless sensor network. This is a common situation when a sensor network is deployed to detect relatively rare events. In such situations, each node that should participate in the aggregation knows this fact based on its own sensor readings, but there is no global knowledge in the network of where all these interesting nodes are located. Instead of blindly querying all nodes in the network, we show how the interesting nodes can autonomously discover each other in a distributed fashion and form an ad hoc aggregation structure that can be used to compute cumulants, moments, or other statistical summaries. Key to our approach is the capability for two nodes that wish to communicate at roughly the same time to discover each other at a cost that is proportional to their network distance. We show how to build nearly optimal aggregation structures that can further deal with network volatility and compensate for the loss or duplication of data by exploiting probabilistic techniques.

In this paper we propose a lightweight algorithm for constructing multi-resolution data representations for sensor networks. We compute, at each sensor node u, O(log n) aggregates about exponentially enlarging neighborhoods centered at u. The ith aggregate is the aggregated data among nodes approximately within 2i hops of u. We present a scheme, named the hierarchical spatial gossip algorithm, to extract and construct these aggregates, for all sensors simultaneously, with a total communication cost of O(n polylog n). The hierarchical gossip algorithm adopts atomic communication steps with each node choosing to exchange information with a node distance d away with probability 1 /d3. The attractiveness of the algorithm attributes to its simplicity, low communication cost, distributed nature and robustness to node failures and link failures. Besides the natural applications of multi-resolution data summaries in data validation and information mining, we also demonstrate the application of the pre-computed spatial multi-resolution data summaries in answering range queries efficiently.

Abstract — For a wide variety of sensor network environments, location information is unavailable or expensive to obtain. We propose a location-free, lightweight, distributed, and data-centric storage/retrieval scheme for information producers and information consumers in sensor networks. Our scheme is built upon the Gradient Landmark-Based Distributed Routing protocol (GLIDER) [8], a two-level routing scheme where sensor nodes are partitioned into tiles by their graph distances to a small set of local landmarks so that localized and efficient routing can be achieved inside and across tiles. Our information storage and retrieval scheme uses two ideas on top of the GLIDER hierarchy — a distributed hash table on the combinatorial tile adjacency graph and a double-ruling scheme within each tile. Queries follow a path that will provably reach the data replicated by the producer(s). We show that this scheme compares favorably with previously proposed schemes, such as Geographic Hash Tables (GHT), providing comparable data storage performance and better locality-aware data retrieval performance. More importantly, this scheme uses no geographic information, makes few assumptions on the network model, and achieves better load balancing and structured data processing and aggregation even for sensor fields with complex geometric shapes and non-trivial topology. I.

A Bloom filter is a simple randomized data structure that answers membership query with no false negative and a small false positive probability. It is an elegant data compression technique for membership information and has broad applications. In this paper, we generalize the traditional Bloom filter to weighted Bloom filter, which incorporates the information on the query frequencies and the membership likelihood of the elements into its optimal design. It has been widely observed that in many applications, some popular elements are queried much more often than the others. The traditional Bloom filter for data sets with irregular query patterns and non-uniform membership likelihood can be further optimized. We derive the optimal configuration of the Bloom filter with query-frequency and membership-likelihood information, and show that the adapted Bloom filter always outperforms the traditional Bloom filter. Under reasonable frequency models such as the step distribution or the Zipf's distribution, the improvement of the false positive probability of the weighted Bloom filter over that of the traditional Bloom filter has been evaluated by simulations

We study the problem of information brokerage in sensor networks, where information consumers (sinks,users)search for data acquired by information producers (sources). In-network storage such as geographical hash table (GHTs) has been proposed to store data at rendezvous nodes for consumers to retrieve. In this paper, we propose a double rulings scheme which stores data replica at a curve instead of one or multiple isolated sensors. The consumer travels along another curve which guarantees to intersect with the producer curve. The double rulings is a natural extension of the flat hashing scheme such as GHTs with improved query locality, i.e., consumers close to producers find the data quickly, and structured aggregate queries, i.e., a consumer following a curve is able to retrieve all the data. Further, by the flexibility of retrieval mechanisms we have better routing robustness and data robustness. We show by simulation that the double rulings scheme provide reduced communication costs and more balanced traffic load on the sensors.

Wireless sensor networks are tightly associated with the underlying environment in which the sensors are deployed. The global topology of the network is of great importance to both sensor network applications and the implementation of networking functionalities. In this paper we study the problem of topology discovery, in particular, identifying boundaries in a sensor network. Suppose a large number of sensor nodes are scattered in a geometric region, with nearby nodes communicating with each other directly. Our goal is to find the boundary nodes by using only connectivity information. We do not assume any knowledge of the node locations or inter-distances, nor do we enforce that the communication graph follows the unit disk graph model. We propose a simple, distributed algorithm that correctly detects nodes on the boundaries and connects them into meaningful boundary cycles. We obtain as a byproduct the medial axis of the sensor field, which has applications in creating virtual coordinates for routing. We show by extensive simulation that the algorithm gives good results even for networks with low density. We also prove rigorously the correctness of the algorithm for continuous geometric domains.

Localization is an important and extensively studied problem in ad-hoc wireless sensor networks. Given the connectivity graph of the sensor nodes,along with additional local information (e.g. distances, angles, orientations etc.), the goal is to reconstruct the global geometry of the network. In this paper, we study the problem of localization with noisy distance and angle information. With no noise at all, the localization problem with both angle (with orientation) and distance information is trivial. However, in the presence of even a small amount of noise, we prove that the localization problem is NP hard.Localization with accurate distance information and relative angle information is also hard. These hardness results motivate our study of approximation schemes. We relax the non-convex constraints to approximating convex constraints and propose linear programs (LP) for two formulations of the resulting localization problem, which we call the weak deployment and strong deployment problems.These two formulations give upper and lower bounds on the location uncertainty respectively: No sensor is located outside its weak deployment region, and each sensor can be anywhere in its strong deployment region without violating the approximate distance and angle constraints. Though LP-based algorithms are usually solved by centralized methods, we propose distributed, iterative methods, which are provably convergent to the centralized algorithm solutions. We give simulation results for the distributed algorithms, evaluating the convergence rate, dependence on measurement noises,and robustness to link dynamics.

We study routing algorithms on wireless networks that use only short paths, for minimizing latency, and achieve good load balance, for balancing the energy use. We consider the special case when all the nodes are located in a narrow strip with width at most /spl radic/3/2 /spl ap/ 0.86 times the communication radius. We present algorithms that achieve good performance in terms of both measures simultaneously. In particular, the routing path is at most four times the shortest path length and the maximum load on any node is at most three times that of the most load-balanced algorithm without path-length constraint. In addition, our routing algorithms make routing decisions by only local information and, as a consequence, are more adaptive to topology changes due to dynamic node insertions/deletions or due to mobility.

We present gradient landmark-based distributed routing (GLIDER), a novel naming/addressing scheme and associated routing algorithm, for a network of wireless communicating nodes. We assume that the nodes are fixed (though their geographic locations are not necessarily known), and that each node can communicate wirelessly with some of its geographic neighbors - a common scenario in sensor networks. We develop a protocol which in a preprocessing phase discovers the global topology of the sensor field and, as a byproduct, partitions the nodes into routable tiles - regions where the node placement is sufficiently dense and regular that local greedy methods can work well. Such global topology includes not just connectivity but also higher order topological features, such as the presence of holes. We address each node by the name of the tile containing it and a set of local coordinates derived from connectivity graph distances between the node and certain landmark nodes associated with its own and neighboring tiles. We use the tile adjacency graph for global route planning and the local coordinates for realizing actual inter- and intra-tile routes. We show that efficient load-balanced global routing can be implemented quite simply using such a scheme.

One of the challenging tasks in the deployment of dense wireless networks (like sensor networks) is in devising a routing scheme for node to node communication. Important consideration includes scalability, routing complexity, the length of the communication paths and the load sharing of the routes. In this paper, we show that a compact and expressive abstraction of network connectivity by the medial axis enables efficient and localized routing. We propose MAP, a Medial Axis based naming and routing Protocol that does not require locations, makes routing decisions locally, and achieves good load balancing. In its preprocessing phase, MAP constructs the medial axis of the sensor field, defined as the set of nodes with at least two closest boundary nodes. The medial axis of the network captures both the complex geometry and non-trivial topology of the sensor field. It can be represented compactly by a graph whose size is comparable with the complexity of the geometric features (e.g., the number of holes). Each node is then given a name related to its position with respect to the medial axis. The routing scheme is derived through local decisions based on the names of the source and destination nodes and guarantees delivery with reasonable and natural routes. We show by both theoretical analysis and simulations that our medial axis based geometric routing scheme is scalable, produces short routes, achieves excellent load balancing, and is very robust to variations in the network model.

Location information is very useful in the design of sensor network infrastructures. In this paper, we study the anchor-free 2D localization problem by using local angle measurements in a sensor network. We prove that given a unit disk graph and the angles between adjacent edges, it is NP-hard to find a valid embedding in the plane such that neighboring nodes are within distance 1 from each other and non-neighboring nodes are at least distance 1 away. Despite the negative results, however, one can find a planar spanner of a unit disk graph by using only local angles. The planar spanner can be used to generate a set of virtual coordinates that enable efficient and local routing schemes such as geographical routing or approximate shortest path routing. We also proposed a practical anchor-free embedding scheme by solving a linear program. We show by simulation that not only does it give very good local embedding, i.e., neighboring nodes are close and non-neighboring nodes are far away, but it also gives a quite accurate global view such that geographical routing and approximate shortest path routing on the embedded graph are almost identical to those on the original (true) embedding. The embedding algorithm can be adapted to other models of wireless sensor networks and is robust to measurement noise.

We propose a new routing graph, the restricted Delaunay graph (RDG), for mobile ad hoc networks. Combined with a node clustering algorithm, the RDG can be used as an underlying graph for geographic routing protocols. This graph has the following attractive properties: 1) it is planar; 2) between any two graph nodes there exists a path whose length, whether measured in terms of topological or Euclidean distance, is only a constant times the minimum length possible; and 3) the graph can be maintained efficiently in a distributed manner when the nodes move around. Furthermore, each node only needs constant time to make routing decisions. We show by simulation that the RDG outperforms previously proposed routing graphs in the context of the Greedy perimeter stateless routing (GPSR) protocol. Finally, we investigate theoretical bounds on the quality of paths discovered using GPSR.

We extend the classic notion of well-separated pair decomposition [P. B. Callahan and S. R. Kosaraju, J. ACM, 42 (1975), pp. 67--90] to theunit-disk graph metric: the shortest path distance metric induced by the intersection graph of unit disks. We show that for the unit-disk graph metric of n points in the plane and for any constant $c\geq 1$, there exists a c-well-separated pair decomposition with O(n log n) pairs, and the decomposition can be computed in O(n log n) time. We also show that for the unit-ball graph metric in k dimensions where $k\geq 3$, there exists a c-well-separated pair decomposition with O(n 2-2/k ) pairs, and the bound is tight in the worst case. We present the application of the well-separated pair decomposition in obtaining efficient algorithms for approximating the diameter, closest pair, nearest neighbor, center, median, and stretch factor, all under the unit-disk graph metric.

For a set S of points in R d,ans-spanner is a graph on S such that any pair of points is connected via some path in the spanner whose total length is at most s times the Euclidean distance between the points. In this paper we propose a new sparse (1 + ε)-spanner with O(n/ε d) edges, where ε is a specified parameter. The key property of this spanner is that it can be efficiently maintained under dynamic insertion or deletion of points, as well as under continuous motion of the points in both the kinetic data structures setting and in the more realistic blackbox displacement model we introduce. Our deformable spanner succinctly encodes all proximity information in a deforming point cloud, giving us efficient kinetic algorithms for problems such as the closest pair, the near neighbors of all points, approximate nearest neighbor search (aka approximate Voronoi diagram), well-separated pair decomposition, and approximate k-centers. 1

Many algorithms for routing in sensor networks exploit greedy forwarding strategies to get packets to their destinations. We study a fundamental difficulty such strategies face: the "local minimum phenomena" that can cause packets to get stuck. We give a definition of stuck nodes where packets may get stuck in greedy multi-hop forwarding, and develop a local rule, the TENT rule, for each node in the network to test whether a packet can get stuck at that node. To help the packets get out of stuck nodes, we describe a distributed algorithm, BOUNDHOLE, to build routes around holes, which are connected regions of the network with boundaries consisting of all the stuck nodes. We show that these hole-surrounding routes can be used in many applications such as geographic routing, path migration, information storage mechanisms and identification of regions of interest.

We study wireless network routing algorithms that use only short paths, for minimizing latency, and achieve good load balance, for balancing the energy use. We consider the special case when all the nodes are located in a narrow strip with width at most /spl radic/3/2 /spl ap/ 0.86 times the communication radius. We present algorithms that achieve good performance in terms of both measures simultaneously. In addition, our algorithms only use local information and can deal with dynamic change and mobility efficiently.

We address the problem of distributed information aggregation and storage in a sensor network, where queries can be injected anywhere in the network. The principle we propose is that a sensor should know a "fraction" of the information from distant parts of the network, in an exponentially decaying fashion by distance. We show how a sampled scalar field can be stored in this distributed fashion, with only a modest amount of additional storage and network traffic. Our storage scheme makes neighboring sensors have highly correlated world views; this allows smooth information gradients and enables local search algorithms to work well. We study in particular how this principle of fractionally cascaded information can be exploited to answer range queries about the sampled field efficiently. Using local decisions only we are able to route the query to exactly the portions of the field where the sought information is stored. We provide a rigorous theoretical analysis showing that our scheme is close to optimal.

Typical distributed transaction environments are a heterogeneous collection of hardware and software resources. An example of such an environment is an electronic store front where users can launch a number of different transactions to complete one or more interactions with the system. One of the challenges in managing such an environment is to figure out the root cause of a performance or throughput problem that manifests itself at a user access point, and to take appropriate action, preferably in an automated way. Our paper addresses this problem by analyzing the dependency relationship among various software components. We also provide a theoretical insight into how a set of transactions can be generated to pinpoint the root cause of a performance problem that is manifested at the user access point.

A graph has growth rate k if the number of nodes in any subgraph with diameter r is bounded by O(rk). The communication graphs of wireless networks and peer-to-peer networks often have small growth rate. In this paper we study the tradeoff between two quality measures for routing in growth restricted graphs. The two measures we consider are the stretch factor, which measures the lengths of the routing paths, and the load balancing ratio, which measures how evenly the traffic is distributed. We show that if the routing algorithm is required to use paths with stretch factor c, then its load balancing ratio is bounded by O((n/c)1-1/k), where k is the graph's growth rate. We illustrate our results by focusing on the unit disk graph for modeling wireless networks in which two nodes have direct communication if their distance is under certain threshold. We show that if the maximum density of the nodes is bounded by ρ, there exists routing scheme such that the stretch factor of routing paths is at most c, and the maximum load on the nodes is at most O(min(√ρn/c, n/c)) times the optimum. In addition, the bound on the load balancing ratio is tight in the worst case. As a special case, when the density is bounded by a constant, the shortest path routing has a load balancing ratio of O(√n). The result extends to k-dimensional unit ball graphs and graphs with growth rate k. We also discuss algorithmic issues for load balanced short path routing and for load balanced routing in spanner graphs.

We extend the classic notion of well-separated pair decomposition [10] to the (weighted) unit-disk graph metric: the shortest path distance metric induced by the intersection graph of unit disks. We show that for the unit-disk graph metric of n points in the plane and for any constant c≥1, there exists a c-well-separated pair decomposition with O(n log n) pairs, and the decomposition can be computed in O(n log n) time. We also show that for the unit-ball graph metric in k dimensions where k≥3, there exists a c-well-separated pair decomposition with O(n2-2/k) pairs, and the bound is tight in the worst case. We present the application of the well-separated pair decomposition in obtaining efficient algorithms for approximating the diameter, closest pair, nearest neighbor, center, median, and stretch factor, all under the unit-disk graph metric.

\emph{We propose a new randomized algorithm for maintaining a set of c lusters among moving nodes in the plane. Given a specified cluster radius, our algorithm selects and maintains a variable subset of the nodes as cluster centers. This subset has the property that (1) balls of the given radius centered at the chosen nodes cover all the others and (2) the number of centers selected is a constant-factor approximation of the minimum possible. As the nodes move, an event-based kinetic data structure updates the clustering as necessary. This kinetic data structure is shown to be responsive, efficient, local, and compact. The produced cover is also smooth, in the sense that wholesale cluster re-arrangements are avoided. The algorithm can be implemented without exact knowledge of the node positions, if each node is able to sense its distance to other nodes up to the cluster radius. Such a kinetic clustering can be used in numerous applications where mobile devices must be interconnected into an ad-hoc network to collaboratively perform some task.}

We propose a new routing graph, the Restricted Delaunay Graph (RDG), for ad hoc networks. Combined with a node clustering algorithm RDG can be used as an underlying graph for geographic routing protocols. This graph has the following attractive properties: (1) it is a planar graph; (2) between any two nodes there exists a path in the RDG whose length, whether measured in terms of topological or Euclidean distance, is only a constant times the optimum length possible; and (3) the graph can be maintained efficiently in a distributed manner when the nodes move around. Furthermore, each node only needs constant time to make routing decisions. We also show by simulation that the RDG outperforms the previously proposed routing graphs under the Greedy Perimeter Stateless Routing (GPSR) protocol. In addition, we investigate theoretical bounds on the quality of paths discovered using GPSR