On the all-pairs shortest path algorithm of Moffat and Takaoka

Abstract

We review how to solve the all-pairs shortest-path problem in a nonnegatively Ž 2. weighted digraph with n vertices in expected time O n log n. This bound is shown to hold with high probability for a wide class of probability distributions on nonnegatively weighted Ž. digraphs. We also prove that, for a large class of probability distributions, ⍀ n log n time is necessary with high probability to compute shortest-path distances with respect to a single Ž .

Algorithmic theory of random graphs

“…Moffatt and Takaoka [160] subsequently reduced the expected running time to O(n 2 log n). Recently, Mehlhorn and Priebe [157] show this algorithm runs in time O(n 2 log n) whp and not just in expectation. They also give an Ω(n log n) lower bound for the single source problem under a class of distributions.…”

Finding Real-Valued Single-Source Shortest Paths ino(n3) Expected Time

“…We proceed with a high-level description of the Moffat- Takaoka [26]) Let G be a network of n vertices with nonnegative edge costs drawn from an endpoint-independent distribution and let S be a set of vertices of G whose shortest path distances have been computed. Then MT(G, S) solves SSSP on G in O(n log n) expected time, given that the edge lists are presorted by cost.…”

“…The fastest algorithm so far under the endpoint-independent model is due to Moffat and Takaoka [27] and runs in O(n 2 log n) expected time. Mehlhorn and Priebe [26] corrected an oversight in the analysis given by Moffat and Takaoka and provided a slightly modified version of the algorithm that runs in O(n 2 log n) time with high probability. They also showed that under weak assumptions Ω(n log n) time is required with high probability for solving SSSP on networks with the endpoint-independent distribution.…”

Negative Dependence Through the FKG Inequality

“…Theorem 14 allows, for example, a simple analysis of the following probabilistic experiment from [9]. Consider a k × n matrix A that is defined as follows.…”

“…Let In [9], Mehlhorn and Priebe consider shortest path problems on complete digraphs (with loops) with respect to simple weight functions. On a graph with n vertices, for every vertex v and every integer j ∈ [n], there is exactly one edge of length j leaving v. Among other facts, Mehlhorn and Priebe use large deviation estimates for f(A) to deduce that on random simple weight functions, any algorithm for the single source shortest path problem has complexity Ω(n log n) with high probability.…”