A new approach to all-pairs shortest paths on real-weighted graphs

• Published in: Theoretical computer science Vol. 312; no. 1; pp. 47 - 74
• Main Author: Pettie, Seth
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Amsterdam Elsevier B.V 01.01.2004; Elsevier
• Subjects: Algorithmics. Computability. Computer arithmetics; All-pairs shortest paths; Applied sciences; Combinatorics; Combinatorics. Ordered structures; Comparison-addition model; Computer science; control theory; systems; Exact sciences and technology; Graph theory; Mathematics; Real weights; Sciences and techniques of general use; Theoretical computing; Real weights; All-pairs shortest paths; Comparison-addition model; Hierarchy; Computer theory; Shortest path; Comparison addition model; Real weight; Fast algorithm; Directed graph
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/S0304-3975(03)00402-X

We present a new all-pairs shortest path algorithm that works with real-weighted graphs in the traditional comparison-addition model. It runs in O(mn+n 2 log log n) time, improving on the long-standing bound of O(mn+n 2 log n) derived from an implementation of Dijkstra's algorithm with Fibonacci heaps. Here m and n are the number of edges and vertices, respectively. Our algorithm is rooted in the so-called component hierarchy approach to shortest paths invented by Thorup for integer-weighted undirected graphs, and generalized by Hagerup to integer-weighted directed graphs. The technical contributions of this paper include a method for approximating shortest path distances and a method for leveraging approximate distances in the computation of exact ones. We also provide a simple, one line characterization of the class of hierarchy-type shortest path algorithms. This characterization leads to some pessimistic lower bounds on computing single-source shortest paths with a hierarchy-type algorithm.