A Refutation of Elmasry's $\tilde{O}(m \sqrt{n})$-Time Algorithm for Single-Source Shortest Paths

TL;DR

This paper critically analyzes Elmasry's claimed $ ilde{O}(m \sqrt{n})$-time algorithm for single-source shortest paths, demonstrating through a counterexample that its actual complexity can be $ ext{Ω}(mn)$, thus refuting the original claim.

Contribution

It provides a counterexample that invalidates Elmasry's claimed running time, correcting the understanding of the algorithm's efficiency.

Findings

Elmasry's analysis is incorrect.

Counterexample shows the algorithm can run in $ ext{Ω}(mn)$ time.

The claimed $ ilde{O}(m \\sqrt{n})$ complexity does not hold in general.

Abstract

In this note we examine the recent paper "Breaking the Bellman-Ford Shortest-Path Bound" by Amr Elmasry, where he presents an algorithm for the single-source shortest path problem and claims that its running time complexity is $\tilde{O}(m\sqrt{n})$, where $n$ is the number of vertices and $m$ is the number of edges. We show that his analysis is incorrect, by providing an example of a weighted graph on which the running time of his algorithm is $\Omega(mn)$.