APSP on Dense Directed Graphs with Arbitrary Weights (All-Pairs Shortest Paths (APSP))

Description

In this case, the graph $G=(V,E)$ that we consider is dense ($m = O(n^2)$), is directed, and has arbitrary weights.

Related Problems

Generalizations: APSP

Related: APSP on Dense Undirected Graphs with Arbitrary Weights, APSP on Geometrically Weighted Graphs, APSP on Dense Undirected Graphs with Positive Integer Weights, APSP on Sparse Directed Graphs with Arbitrary Weights, APSP on Sparse Undirected Graphs with Positive Integer Weights, APSP on Sparse Undirected Graphs with Arbitrary Weights, APSP on Dense Directed Unweighted Graphs, APSP on Dense Undirected Unweighted Graphs, APSP on Sparse Directed Unweighted Graphs, APSP on Sparse Undirected Unweighted Graphs, (5/3)-approximate ap-shortest paths

Parameters

$n$: number of vertices

$m$: number of edges

Table of Algorithms

Name	Year	Time	Space	Approximation Factor	Model	Reference
Shimbel Algorithm	1953	$O(n^{4})$	$O(n^{2})$	Exact	Deterministic	Time
Williams	2014	$O(n^{3} /{2}^{(\log n)^{0.5}})$	$O(n^{2})$	Exact	Deterministic	Time
Chan	2009	$O(n^{3} \log^{3} \log n / \log^{2} n)$	$O(n^{2})$	Exact	Deterministic	Time

Time Complexity Graph