Shortest Paths in Geometric Intersection Graphs

This thesis studies shortest paths in geometric intersection graphs, which can model, among others, ad-hoc communication and transportation networks. First, we consider two classical problems in the field of algorithms, namely Single-Source Shortest Paths (SSSP) and All-Pairs Shortest Paths (APSP). In SSSP we want to compute the shortest paths from one vertex of a graph to all other vertices, while in APSP we aim to find the shortest path between every pair of vertices. Although there is a vast literature for these problems in many graph classes, the case of geometric intersection graphs has been only partially addressed. In unweighted unit-disk graphs, we show that we can solve SSSP in linear time, after presorting the disk centers with respect to their coordinates. Furthermore, we give the first (slightly) subquadratic-time APSP algorithm by using our new SSSP result, bit tricks, and a shifted-grid-based decomposition technique. In unweighted, undirected geometric intersection graphs, we present a simple and general technique that reduces APSP to static, offline intersection detection. Consequently, we give fast APSP algorithms for intersection graphs of arbitrary disks, axis-aligned line segments, arbitrary line segments, d-dimensional axis-aligned boxes, and d-dimensional axis-aligned unit hypercubes. We also provide a near-linear-time SSSP algorithm for intersection graphs of axis-aligned line segments by a reduction to dynamic orthogonal point location. Then, we study two problems that have received considerable attention lately. The first is that of computing the diameter of a graph, i.e., the longest shortest-path distance between any two vertices. In the second, we want to preprocess a graph into a data structure, called distance oracle, such that the shortest path (or its length) between any two query vertices can be found quickly. Since these problems are often too costly to solve exactly, we study their approximate versions. Following a long line of research, we employ Voronoi diagrams to compute a (1+epsilon)-approximation of the diameter of an undirected, non-negatively-weighted planar graph in time near linear in the input size and polynomial in 1/epsilon. The previously best solution had exponential dependency on the latter. Using similar techniques, we can also construct the first (1+epsilon)-approximate distance oracles with similar preprocessing time and space and only O(log(1/\epsilon)) query time. In weighted unit-disk graphs, we present the first near-linear-time (1+epsilon)-approximation algorithm for the diameter and for other related problems, such as the radius and the bichromatic closest pair. To do so, we combine techniques from computational geometry and planar graphs, namely well-separated pair decompositions and shortest-path separators. We also show how to extend our approach to obtain O(1)-query-time (1+epsilon)-approximate distance oracles with near linear preprocessing time and space. Then, we apply these oracles, along with additional ideas, to build a data structure for the (1+epsilon)-approximate All-Pairs Bounded-Leg Shortest Paths (apBLSP) problem in truly subcubic time

Sublinear Average-Case Shortest Paths in Weighted Unit-Disk Graphs

We consider the problem of computing shortest paths in weighted unit-disk graphs in constant dimension $d$. Although the single-source and all-pairs variants of this problem are well-studied in the plane case, no non-trivial exact distance oracles for unit-disk graphs have been known to date, even for $d=2$. The classical result of Sedgewick and Vitter [Algorithmica '86] shows that for weighted unit-disk graphs in the plane the $A^*$ search has average-case performance superior to that of a standard shortest path algorithm, e.g., Dijkstra's algorithm. Specifically, if the $n$ corresponding points of a weighted unit-disk graph $G$ are picked from a unit square uniformly at random, and the connectivity radius is $r\in (0,1)$, $A^*$ finds a shortest path in $G$ in $O(n)$ expected time when $r=\Omega(\sqrt{\log n/n})$, even though $G$ has $\Theta((nr)^2)$ edges in expectation. In other words, the work done by the algorithm is in expectation proportional to the number of vertices and not the number of edges. In this paper, we break this natural barrier and show even stronger sublinear time results. We propose a new heuristic approach to computing point-to-point exact shortest paths in unit-disk graphs. We analyze the average-case behavior of our heuristic using the same random graph model as used by Sedgewick and Vitter and prove it superior to $A^*$. Specifically, we show that, if we are able to report the set of all $k$ points of $G$ from an arbitrary rectangular region of the plane in $O(k + t(n))$ time, then a shortest path between arbitrary two points of such a random graph on the plane can be found in $O(1/r^2 + t(n))$ expected time. In particular, the state-of-the-art range reporting data structures imply a sublinear expected bound for all $r=\Omega(\sqrt{\log n/n})$ and $O(\sqrt{n})$ expected bound for $r=\Omega(n^{-1/4})$ after only near-linear preprocessing of the point set.Comment: Full version of a SoCG'21 paper. Abstract truncated to meet arxiv requirement

Near-Optimal Algorithms for Shortest Paths in Weighted Unit-Disk Graphs

We revisit a classical graph-theoretic problem, the single-source shortest-path (SSSP) problem, in weighted unit-disk graphs. We first propose an exact (and deterministic) algorithm which solves the problem in O(n log^2 n) time using linear space, where n is the number of the vertices of the graph. This significantly improves the previous deterministic algorithm by Cabello and Jejcic [CGTA\u2715] which uses O(n^{1+delta}) time and O(n^{1+delta}) space (for any small constant delta>0) and the previous randomized algorithm by Kaplan et al. [SODA\u2717] which uses O(n log^{12+o(1)} n) expected time and O(n log^3 n) space. More specifically, we show that if the 2D offline insertion-only (additively-)weighted nearest-neighbor problem with k operations (i.e., insertions and queries) can be solved in f(k) time, then the SSSP problem in weighted unit-disk graphs can be solved in O(n log n+f(n)) time. Using the same framework with some new ideas, we also obtain a (1+epsilon)-approximate algorithm for the problem, using O(n log n + n log^2(1/epsilon)) time and linear space. This improves the previous (1+epsilon)-approximate algorithm by Chan and Skrepetos [SoCG\u2718] which uses O((1/epsilon)^2 n log n) time and O((1/epsilon)^2 n) space. Because of the Omega(n log n)-time lower bound of the problem (even when approximation is allowed), both of our algorithms are almost optimal

Shortest-Path Queries in Geometric Networks

A Euclidean t-spanner for a point set V ? ?^d is a graph such that, for any two points p and q in V, the distance between p and q in the graph is at most t times the Euclidean distance between p and q. Gudmundsson et al. [TALG 2008] presented a data structure for answering ?-approximate distance queries in a Euclidean spanner in constant time, but it seems unlikely that one can report the path itself using this data structure. In this paper, we present a data structure of size O(nlog n) that answers ?-approximate shortest-path queries in time linear in the size of the output

06481 Abstracts Collection -- Geometric Networks and Metric Space Embeddings

The Dagstuhl Seminar 06481 ``Geometric Networks and Metric Space Embeddings\u27\u27 was held from November~26 to December~1, 2006 in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. In this paper we describe the seminar topics, we have compiled a list of open questions that were posed during the seminar, there is a list of all talks and there are abstracts of the presentations given during the seminar. Links to extended abstracts or full papers are provided where available