A Combinatorial Algorithm for All-Pairs Shortest Paths in Directed Vertex-Weighted Graphs with Applications to Disc Graphs

TL;DR

This paper introduces a new combinatorial randomized algorithm for all-pairs shortest paths in directed vertex-weighted graphs, with applications to transitive closure and disk graphs, improving computational efficiency under certain conditions.

Contribution

The paper presents a novel combinatorial randomized algorithm for all-pairs shortest paths in directed vertex-weighted graphs, leveraging minimum weight spanning trees of associated graphs.

Findings

Algorithm solves all-pairs shortest paths in graphs with improved time complexity.

Transitive closure can be computed efficiently using the proposed method.

Efficient solution for uniform disk graphs with bounded density.

Abstract

We consider the problem of computing all-pairs shortest paths in a directed graph with real weights assigned to vertices. For an $n\times n$ 0-1 matrix $C,$ let $K_{C}$ be the complete weighted graph on the rows of $C$ where the weight of an edge between two rows is equal to their Hamming distance. Let $MWT(C)$ be the weight of a minimum weight spanning tree of $K_{C}.$ We show that the all-pairs shortest path problem for a directed graph $G$ on $n$ vertices with nonnegative real weights and adjacency matrix $A_G$ can be solved by a combinatorial randomized algorithm in time $$\widetilde{O}(n^{2}\sqrt {n + \min\{MWT(A_G), MWT(A_G^t)\}})$$ As a corollary, we conclude that the transitive closure of a directed graph $G$ can be computed by a combinatorial randomized algorithm in the aforementioned time. $\widetilde{O}(n^{2}\sqrt {n + \min\{MWT(A_G), MWT(A_G^t)\}})$ We also conclude that the all-pairs shortest path problem for uniform disk graphs, with nonnegative real vertex weights, induced by point sets of bounded density within a unit square can be solved in time $\widetilde{O}(n^{2.75})$.