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

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+δ ) time and O(n 1+δ ) space (for any small constant δ > 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+ε)-approximate algorithm for the problem, using O(n log n + n log 2 (1/ε)) time and linear space. This improves the previous (1 + ε)-approximate algorithm by Chan and Skrepetos [SoCG'18] which uses O((1/ε) 2 n log n) time and O((1/ε) 2 n) space. Because of the Ω(n log n)-time lower bound of the problem (even when approximation is allowed), both of our algorithms are almost optimal.

Efficient algorithms for optimization problems involving distances in a point set

“…Consider next the case where we stop because of the second condition, namely T contains 6 We remark that, unfortunately, this is not the case with the decision procedures used in Wang and Zhao [21] and in Wang and Xue [20] for the reverse shortest-path problem in unit-disk graphs and for its weighted variant. We will modify these procedures, in Sections 3.1, 3.2, and 3.4, to fit into this setup.…”

“…The decision procedure for this problem has been studied by Wang and Xue [20], who showed that, for a given r and a source point s, all shortest paths in G(r) from s to the other points of P can be computed in O(n log 2 n) time.…”

“…Unlike the problem of Section 3.3, whose decision procedure runs in O(n) time in any fixed dimension d, the running time of both the decision procedure of [10], for the unweighted planar reverse shortest path problem, and that of [20], for its weighted variant, increases in d as we move to higher dimensions. We show below that both procedures can be adapted to three dimensions, so that the resulting running time is O * (n 4/3 ) for both variants.…”

“…We then consider the weighted variant of this problem, in which we want to find the minimum value r * such that G(r * ), defined as above, contains a path from s to t of length at most w, where each edge of G(r * ) has a weight equal to its Euclidean length, and the length of the path is the sum of the weights of its edges. The decision problem for this task, in which we specify r and seek the shortest path in G(r) from s to t, has been considered by Wang and Xue [20], who gave an algorithm that takes O(n log 2 n) time. Applying our technique and using this decision procedure, we solve the weighted variant of the reverse shortest path problem for unit-disk graphs in O * (n 6/5 ) time.…”

Dynamic Planar Voronoi Diagrams for General Distance Functions and Their Algorithmic Applications

“…Our new structures give improved bounds almost in a black-box fashion, using the respective techniques of Cabello and Jejčič [10] and of Roditty and Segal [46]. Very recently, Wang and Xue [51] presented a deterministic algorithm to find the shortest-path tree in a unit disk graph in O(n log 2 n) time. The other two applications are a bit more involved.…”