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

2025-08-06

AI Summary

This paper challenges the claimed $ ilde{O}(msqrt{n})$ time complexity of Elmasry’s (2017) single-source shortest paths (SSSP) algorithm. Method: We construct a family of adversarial graphs with hierarchical dense structure, rigorously proving that the algorithm incurs $Omega(mn)$ runtime in the worst case—substantially exceeding its stated bound. Our analysis identifies a fundamental flaw in the original complexity derivation: an underestimation of key loop iterations due to unaccounted degradation arising from coupling between priority-queue operations and graph topology. Contribution/Results: This work provides the first tight lower-bound counterexample, correcting the theoretical understanding of Elmasry’s algorithm and preserving the rigor of known SSSP complexity boundaries. It further serves as a critical caution for future heap-based shortest-path algorithm design, highlighting the necessity of accounting for structural interactions between data structures and input graphs.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Heuristic Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $ ilde{O}(msqrt{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 $Ω(mn)$.

Problem

• Refutes Elmasry's claimed O(m√n) shortest-path algorithm
• Demonstrates incorrect runtime analysis with Ω(mn) example
• Challenges Bellman-Ford bound breakthrough claim

Innovation

• Analyzing Elmasry's shortest-path algorithm
• Refuting O(m√n) time complexity claim
• Providing counterexample with Ω(mn) runtime