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

• Published in: Discrete & computational geometry Vol. 64; no. 4; pp. 1141 - 1166
• Main Authors: Wang, Haitao, Xue, Jie
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2020; Springer Nature B.V
• Subjects: Algorithms; Apexes; Combinatorics; Computational Mathematics and Numerical Analysis; Graph theory; Graphs; Insertion; Lower bounds; Mathematics; Mathematics and Statistics; Shortest-path problems; Weighted unit-disk graphs; Geometric graph algorithms; Single-source shortest paths
• ISSN: 0179-5376; 1432-0444
• DOI: 10.1007/s00454-020-00219-7

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 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. 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 + ε ) -approximate SSSP problem in weighted unit-disk graphs can be solved in O ( n log n + f ( n , O ( ε - 2 ) ) ) time. 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.