Abstract
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 Jejčič [CGTA’15] which uses \(O(n^{1+\delta })\) time and \(O(n^{1+\delta })\) space (for any constant \(\delta >0\)) and the previous randomized algorithm by Kaplan et al. [SODA’17] 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 \((1+\varepsilon )\)-approximate algorithm for the problem, using \(O(n\log n+n\log ^2(1/\varepsilon ))\) time and linear space. This improves the previous \((1+\varepsilon )\)-approximate algorithm by Chan and Skrepetos [SoCG’18] which uses \(O((1/\varepsilon )^2n\log n)\) time and \(O((1/\varepsilon )^2 n)\) space. More specifically, we show that if the 2D offline insertion-only weighted nearest-neighbor problem with \(k_1\) operations in which at most \(k_2\) operations are insertions can be solved in \(f(k_1,k_2)\) time, then the \((1+\varepsilon )\)-approximate SSSP problem in weighted unit-disk graphs can be solved in \(O(n\log n+f(n,O(\varepsilon ^{-2})))\) time. 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.
Highlights
Given a set S of n points in the plane, its unit-disk graph is an undirected graph in which the vertices are points of S and two vertices are connected by an edge iff the (Euclidean) distance between them is at most 1
Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 60:2 Near-Optimal Algorithms for Shortest Paths in Weighted Unit-Disk Graphs unit-disk graphs have been extensively studied in computational geometry
We present new exact and approximation algorithms for the problem in weighted unit-disk graphs, which significantly improve the previous results and almost match the lower bound of the problem
Summary
Given a set S of n points in the plane, its unit-disk graph is an undirected graph in which the vertices are points of S and two vertices are connected by an edge iff the (Euclidean) distance between them is at most 1. A unit-disk graph (either unweighted or weighted), though having quadratic number of edges in worst case (e.g., all the vertices are very close to each other), can be represented by only giving the locations of its vertices in the plane. This linear-complexity representation allows us to solve the SSSP problem without explicitly constructing the graph and beat the Ω(|E|)-time lower bound. We present new exact and approximation algorithms for the problem in weighted unit-disk graphs, which significantly improve the previous results and almost match the lower bound of the problem. Due to the page limit, some lemma proofs are omitted but can be found in the full version [13]
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