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

We revisit a classical graph-theoretic problem, the \textit{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 Jej\v{c}i\v{c} [CGTA'15] 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'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 $O((1/\varepsilon)^2 n \log n)$ 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.