We present a new algorithm for computing zigzag persistent homology, an algebraic structure which encodes changes to homology groups of a simplicial complex over a sequence of simplex additions and deletions. Provided that there is an algorithm that multiplies two n×n matrices in M(n) time, our algorithm runs in O(M(n) + n2 log2 n) time for a sequence of n additions and deletions. In particular, the running time is O(n2.376), by result of Coppersmith and Winograd. The fastest previously known algorithm for this problem takes O(n3) time in the worst case.

Known algorithms for reconstructing a 2-manifold from a point sample in ℝ3 are naturally based on decisions/predicates that take the geometry of the point sample into account. Facing the always present problem of round-off errors that easily compromise the exactness of those predicate decisions, an exact and robust implementation of these algorithms is far from being trivial and typically requires the employment of advanced datatypes for exact arithmetic as provided by libraries like CORE, LEDA or GMP. In this paper we present a new reconstruction algorithm, one of whose main novelties is to throw away geometry information early on in the reconstruction process and to mainly operate combinatorially on a graph structure. As such it is less susceptible to robustness problems due to round-off errors and also benefits from not requiring expensive exact arithmetic by faster running times. A more theoretical view on our algorithm including correctness proofs under suitable sampling conditions can be found in a companion paper [3].

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 describe several new bottom-up approaches to problems in role engineering for Role-Based Access Control (RBAC). The salient problems are all NP-complete, even to approximate, yet we find that in instances that arise in practice these problems can be solved in minutes. We first consider role minimization, the process of finding a smallest collection of roles that can be used to implement a pre-existing user-to-permission relation. We introduce fast graph reductions that allow recovery of the solution from the solution to a problem on a smaller input graph. For our test cases, these reductions either solve the problem, or reduce the problem enough that we find the optimum solution with a (worst-case) exponential method. We introduce lower bounds that are sharp for seven of nine test cases and are within 3.4% on the other two. We introduce and test a new polynomial-time approximation that on average yields 2% more roles than the optimum. We next consider the related problem of minimizing the number of connections between roles and users or permissions, and we develop effective heuristic methods for this problem as well. Finally, we propose methods for several related problems.

Geographic routing protocols like GOAFR or GPSR rely on exact location information at the nodes, as when the greedy routing phase gets stuck at a local minimum, they require, as a fallback, a planar subgraph whose identification, in all existing methods, depends on exact node positions. In practice, however, location information at the network nodes is hardly precise; be it because the employed location hardware, such as GPS, exhibits an inherent measurement imprecision, or because the localization protocols which estimate positions of the network nodes cannot do so without errors. In this paper we propose a novel naming and routing scheme that can handle the uncertainty in location information. It is based on a macroscopic variant of geographic greedy routing, as well as a macroscopic planarization of the communication graph. If an upper bound on the deviation from true node locations is available, our routing protocol guarantees delivery of messages. Due to its macroscopic view, our routing scheme also produces shorter and more load-balanced paths than common geographic routing schemes, in particular in sparsely connected networks or in the presence of obstacles.

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.

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.