We consider the Euclidean bi-chromatic matching problem in the dynamic setting, where the goal is to efficiently process point insertions and deletions while maintaining a high-quality solution. Computing the minimum cost bi-chromatic matching is one of the core problems in geometric optimization that has found many applications, most notably in estimating Wasserstein distance between two distributions. In this work, we present the first fully dynamic algorithm for Euclidean bi-chromatic matching with sublinear update time. For any fixed $\varepsilon > 0$, our algorithm achieves $O(1/\varepsilon)$-approximation and handles updates in $O(n^{\varepsilon})$ time. Our experiments show that our algorithm enables effective monitoring of the distributional drift in the Wasserstein distance on real and synthetic data sets, while outperforming the runtime of baseline approximations by orders of magnitudes.

Uniquely represented (UR) data structures represent each logical state with a unique storage state. We study the problem of maintaining a dynamic set of n keys from a totally ordered universe in this context. UR structures are also called "strongly history independent" structures in the literature. We introduce a two-layer data structure called (α,ε)-Randomized Block Search Tree (RBST) that is uniquely represented and suitable for external memory (EM). Though RBSTs naturally generalize the well-known binary Treaps, several new ideas are needed to analyze the expected search, update, and storage efficiency in terms of block-reads, block-writes, and blocks stored. We prove that searches have O(ε^{-1} + log_α n) block-reads, that dynamic updates perform O(ε^{-1} + log_α(n)/α) block-writes and O(ε^{-2}+(1+(ε^{-1}+log n)/α)log_α n) block-reads, and that (α, ε)-RBSTs have an asymptotic load-factor of at least (1-ε) for every ε ∈ (0,1/2]. Thus (α, ε)-RBSTs improve on the known, uniquely represented B-Treap [Golovin; ICALP'09]. Compared with non-UR structures, the RBST is also, to the best of our knowledge, the first external memory structure that is storage-efficient and has a non-amortized, write-efficient update bound.

Representing a polygon using a set of simple shapes has numerous applications in different use-case scenarios. We consider the problem of covering the interior of a rectilinear polygon with holes by a set of area-weighted, axis-aligned rectangles such that the total weight of the rectangles in the cover is minimized. Already the unit-weight case is known to be NP-hard and the general problem has, to the best of our knowledge, not been studied experimentally before.

Given a set of $n$ sites from $\mathbb{R}^d$, each having some positive weight factor, the Multiplicatively Weighted Voronoi Diagram is a subdivision of space that associates each cell to the site whose weighted Euclidean distance is minimal for all points in the cell. We give novel approximation algorithms that output a cube-based subdivision such that the weighted distance of a point with respect to the associated site is at most $(1+\varepsilon)$ times the minimum weighted distance, for any fixed parameter $\varepsilon \in (0,1)$. The diagram size is $O_d(n \log(1/\varepsilon)/\varepsilon^{d-1})$ and the construction time is within an $O_D(\log(n)/\varepsilon^{(d+5)/2})$-factor of the size bound. We also prove a matching lower bound for the size, showing that the proposed method is the first to achieve \emph{optimal size}, up to $Θ(1)^d$-factors. In particular, the obscure $\log(1/\varepsilon)$ factor is unavoidable. As a by-product, we obtain a factor $d^{O(d)}$ improvement in size for the unweighted case and $O(d \log(n) + d^2 \log(1/\varepsilon))$ point-location time in the subdivision, improving the known query bound by one $d$-factor. The key ingredients of our approximation algorithms are the study of convex regions that we call cores, an adaptive refinement algorithm to obtain optimal size, and a novel notion of \emph{bisector coresets}, which may be of independent interest. In particular, we show that coresets with $O_d(1/\varepsilon^{(d+3)/2})$ worst-case size can be computed in near-linear time.

Algorithms with predictions is a new research direction that leverages machine learned predictions for algorithm design. So far a plethora of recent works have incorporated predictions to improve on worst-case bounds for online problems. In this paper, we initiate the study of complexity of dynamic data structures with predictions, including dynamic graph algorithms. Unlike online algorithms, the goal in dynamic data structures is to maintain the solution efficiently with every update. We investigate three natural models of prediction: (1) δ-accurate predictions where each predicted request matches the true request with probability δ, (2) list-accurate predictions where a true request comes from a list of possible requests, and (3) bounded delay predictions where the true requests are a permutation of the predicted requests. We give general reductions among the prediction models, showing that bounded delay is the strongest prediction model, followed by list-accurate, and δ-accurate. Further, we identify two broad problem classes based on lower bounds due to the Online Matrix Vector (OMv) conjecture. Specifically, we show that locally correctable dynamic problems have strong conditional lower bounds for list-accurate predictions that are equivalent to the non-prediction setting, unless list-accurate predictions are perfect. Moreover, we show that locally reducible dynamic problems have time complexity that degrades gracefully with the quality of bounded delay predictions. We categorize problems with known OMv lower bounds accordingly and give several upper bounds in the delay model that show that our lower bounds are almost tight. We note that concurrent work by v.d.Brand et al. [SODA '24] and Liu and Srinivas [arXiv:2307.08890] independently study dynamic graph algorithms with predictions, but their work is mostly focused on showing upper bounds.

Map matching is a common preprocessing step for analysing vehicle trajectories. In the theory community, the most popular approach for map matching is to compute a path on the road network that is the most spatially similar to the trajectory, where spatial similarity is measured using the Fréchet distance. A shortcoming of existing map matching algorithms under the Fréchet distance is that every time a trajectory is matched, the entire road network needs to be reprocessed from scratch. An open problem is whether one can preprocess the road network into a data structure, so that map matching queries can be answered in sublinear time. In this article, we investigate map matching queries under the Fréchet distance. We provide a negative result for geometric planar graphs. We show that, unless SETH fails, there is no data structure that can be constructed in polynomial time that answers map matching queries in O((pq) 1-δ ) query time for any δ > 0, where p and q are the complexities of the geometric planar graph and the query trajectory, respectively. We provide a positive result for realistic input graphs, which we regard as the main result of this article. We show that for c -packed graphs, one can construct a data structure of \(\tilde{O}(cp)\) size that can answer (1+ε)-approximate map matching queries in \(\tilde{O}(c^4 q \log ^4 p)\) time, where \(\tilde{O}(\cdot)\) hides lower-order factors and dependence on ε.

Map matching is a common preprocessing step for analysing vehicle trajectories. In the theory community, the most popular approach for map matching is to compute a path on the road network that is the most spatially similar to the trajectory, where spatial similarity is measured using the Fréchet distance. A shortcoming of existing map matching algorithms under the Fréchet distance is that every time a trajectory is matched, the entire road network needs to be reprocessed from scratch. An open problem is whether one can preprocess the road network into a data structure, so that map matching queries can be answered in sublinear time. In this paper, we investigate map matching queries under the Fréchet distance. We provide a negative result for geometric planar graphs. We show that, unless SETH fails, there is no data structure that can be constructed in polynomial time that answers map matching queries in O((pq)1-δ) query time for any δ > 0, where p and q are the complexities of the geometric planar graph and the query trajectory, respectively. We provide a positive result for realistic input graphs, which we regard as the main result of this paper. We show that for c-packed graphs, one can construct a data structure of Õ(cp) size that can answer (1 + ε)-approximate map matching queries in Õ(c4q log4p) time, where Õ(·) hides lower-order factors and dependence of ε. * The full version of the paper can be accessed at https://arxiv.org/abs/2211.02951

The Randomized Incremental Construction (RIC) of search DAGs for point location in planar subdivisions, nearest-neighbor search in 2D points, and extreme point search in 3D convex hulls, are well known to take (n log n) expected time for structures of (n) expected size. Moreover, searching takes w.h.p. (log n) comparisons in the first and w.h.p. (log2 n) comparisons in the latter two DAGs. However, the expected depth of the DAGs and high probability bounds for their size are unknown. Using a novel analysis technique, we show that the three DAGs have w.h.p. i) a size of (n), ii) a depth of (log n), and iii) a construction time of (n log n). One application of these new and improved results are remarkably simple Las Vegas verifiers to obtain search DAGs with optimal worst-case bounds. This positively answers the conjectured logarithmic search cost in the DAG of Delaunay triangulations [Guibas et al.; ICALP 1990] and a conjecture on the depth of the DAG of Trapezoidal subdivisions [Hemmer et al.; ESA 2012]. It also shows that history-based RIC circumvents a lower bound on runtime tail estimates of conflict-graph RICs [Sen; STACS 2019].

We study the problem of sub-trajectory nearest-neighbor queries on polygonal curves under the continuous Fréchet distance. Given a trajectory P with n vertices and a query trajectory Q, we seek to report a vertex-aligned sub-trajectory P' of P that is closest to Q, i.e. P' must start and end on contiguous vertices of P.

We study how to dynamize the Trapezoidal Search Tree - a well known randomized point location structure for planar subdivisions of kinetic line segments. Our approach naturally extends incremental leaf-level insertions to recursive methods and allows adaptation for the online setting. Moreover, the dynamization carries over to the Trapezoidal Search DAG, offering a linear sized data structure with logarithmic point location costs as a by-product. On a set $S$ of non-crossing segments, each update performs expected ${\mathcal O}(\log^2|S|)$ operations. We demonstrate the practicality of our method with an open-source implementation, based on the Computational Geometry Algorithms Library, and experiments on the update performance.

Given a set of prioritized balls with fixed centers in ℝd whose radii grow linearly over time, we want to compute the elimination order of these balls assuming that when two balls touch, the one with lower priority is ‘crushed’. A straightforward algorithm has running time O(n2 log n) which we improve to expected O(Δdn(log n + Δd)) where Δ = rmax/rmin is the ratio between largest and smallest radius amongst the balls. For a natural application of this problem, namely drawing labels on the globe, we have Δ = O(1). An efficient implementation based on a spherical Delaunay triangulation allows to compute the elimination order for millions of labels on commodity Desktop hardware. Dealing with rounding error induced robustness issues turned out to be one of the major challenges in the implementation.

Each non-zero point in Rd identifies one closest point x on the unit sphere Sd-1. We are interested in computing an ε-approximation y ∈ Qd for x, that is exactly on Sd-1 and has low bit size. We revise lower bounds on rational approximations and provide explicit, spherical instances. We prove that floating-point numbers can only provide trivial solutions to the sphere equation in R2 and R3. Moreover, we show how to construct a rational point with denominators of at most 32(d-1)2/ε2 for any given ε, improving on a previous result. The method further benefits from algorithms for simultaneous Diophantine approximation.

For a given sequence of location measurements, the goal of the geometric map matching problem is to compute a sequence of movements along edges of a spatially embedded graph which provides a ‘good explanation’ for the measurements. The problem gets challenging as real world data, like traces or graphs from the OpenStreetMap project, does not exhibit homogeneous data quality. Graph details and errors vary in areas and each trace has changing noise and precisions. Hence formalizing what a ‘good explanation’ is, becomes quite difficult. We propose a novel map matching approach which locally adapts to the data quality by constructing what we call dominance decompositions. While our approach is computationally more expensive than previous approaches, our experiments show that it allows for high quality map matching even in presence of highly variable data quality without parameter tuning.