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

Faster Approximate Diameter and Distance Oracles in Planar Graphs

We present an algorithm that computes a (1+varepsilon)-approximation of the diameter of a weighted, undirected planar graph of n vertices with non-negative edge lengths in O(nlog n(log n + (1/varepsilon)^5)) expected time, improving upon the O(n((1/varepsilon)^4 log^4(n) + 2^{O(1/varepsilon)}))-time algorithm of Weimann and Yuster [ICALP 2013]. Our algorithm makes two improvements over that result: first and foremost, it replaces the exponential dependency on 1/varepsilon with a polynomial one, by adapting and specializing Cabello\u27s recent abstract-Voronoi-diagram-based technique [SODA 2017] for approximation purposes; second, it shaves off two logarithmic factors by choosing a better sequence of error parameters during recursion. Moreover, using similar techniques, we improve the (1+varepsilon)-approximate distance oracle of Gu and Xu [ISAAC 2015] by first replacing the exponential dependency on 1/varepsilon on the preprocessing time and space with a polynomial one and second removing a logarithmic factor from the preprocessing time

All-Pairs Shortest Paths in Unit-Disk Graphs in Slightly Subquadratic Time

In this paper we study the all-pairs shortest paths problem in (unweighted) unit-disk graphs. The previous best solution for this problem required O(n^2 log n) time, by running the O(n log n)-time single-source shortest path algorithm of Cabello and Jejcic [Comput. Geom., 2015] from every source vertex,where n is the number of vertices. We not only manage to eliminate the logarithmic factor, but also obtain the first (slightly) subquadratic algorithm for the problem, running in O(n^2 sqrt{ frac{log log n}{log n} }) time. Our algorithm computes an implicit representation of all the shortest paths, and, in the same amount of time, can also compute the diameter of the graph

Approximate shortest paths and distance oracles in weighted unit-disk graphs

$\newcommand{\OO}[1]{O\left(#1\right)}\newcommand{\eps}{\varepsilon}$We present the first near-linear-time algorithm that computes a (1+\eps)-approximation of the diameter of a weighted unit-disk graph of $n$ vertices. Our algorithm requires \OO{n \log^2 n} time for any constant \eps>0, so we considerably improve upon the near-\OO{n^{3/2}}-time algorithm of Gao and Zhang (2005). Using similar ideas we develop (1+\eps)-approximate \emph{distance oracles} of \OO{1} query time with a likewise improvement in the preprocessing time, specifically from near \OO{n^{3/2}} to \OO{n \log^3 n}. We also obtain similar new results for a number of related problems in the weighted unit-disk graph metric such as the radius and the bichromatic closest pair.As a further application we employ our distance oracle, along with additional ideas, to solve the (1+\eps)-approximate \emph{all-pairs bounded-leg shortest paths\/} (apBLSP) problem for a set of $n$ planar points. Our data structure requires \OO{n^2 \log n} space, \OO{\log{\log n}} query time, and nearly \OO{n^{2.579}} preprocessing time for any constant \eps>0, and is the first that breaks the near-cubic preprocessing time bound given by Roditty and Segal (2011)

All-Pairs Shortest Paths in Geometric Intersection Graphs

$\newcommand{\OO}[1]{O\left(#1\right)}$We present a simple and general algorithm for the all-pairs shortest paths (APSP) problem in unweighted geometric intersection graphs.  Specifically we reduce the problem to the design of static data structures for offline intersection detection. Consequently we can solve APSP in unweighted intersection graphs of $n$ arbitrary disks in \OO{n^2 \log n} time, axis-aligned line segments in \OO{n^2 \log{\log n}} time, arbitrary line segments in \OO{n^{7/3} \log^{1/3} n} time, $d$-dimensional axis-aligned unit hypercubes in \OO{n^2 \log\log n} time for $d=3$ and \OO{n^2 \log^{d-3} n} time for $d\geq4$, and $d$-dimensional axis-aligned boxes in \OO{n^2 \log^{d-1.5} n} time for $d\geq2$.We also reduce the single-source shortest paths (SSSP) problem in unweighted geometric intersection graphs to decremental intersection detection. Thus, we obtain an \OO{n \log n}-time SSSP algorithm in unweighted intersection graphs of $n$ axis-aligned line segments