Breaking the Sorting Barrier for Directed Single-Source Shortest Paths

2025-04-23

AI Summary

This paper addresses the single-source shortest paths (SSSP) problem on directed graphs with nonnegative edge weights, breaking the classical Ω(m + n log n) time lower bound for Dijkstra’s algorithm in the comparison-addition model. Methodologically, it introduces a deterministic O(m log^{2/3} n) algorithm—first to achieve sub-logarithmic improvement for sparse-graph SSSP—by integrating hierarchical bucketing, recursive graph contraction, and a novel priority-queue substitution mechanism to enable efficient distance updates and vertex processing. The contribution establishes that Dijkstra’s algorithm is not optimal in this model and sets a new asymptotic time-complexity benchmark for SSSP. Notably, the result yields significant asymptotic speedups for sparse graphs, advancing the theoretical understanding of fundamental graph algorithms in algebraic computation models.

Technology Category

Search and Optimization: Heuristic SearchPlanning, Routing, and Scheduling: Optimization of Spatio-temporal SystemsConstraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

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

Abstract

We give a deterministic $O(mlog^{2/3}n)$-time algorithm for single-source shortest paths (SSSP) on directed graphs with real non-negative edge weights in the comparison-addition model. This is the first result to break the $O(m+nlog n)$ time bound of Dijkstra's algorithm on sparse graphs, showing that Dijkstra's algorithm is not optimal for SSSP.

Problem

• Breaks Dijkstra's algorithm time bound
• Solves SSSP on directed graphs
• Uses comparison-addition model

Innovation

• Deterministic algorithm for directed SSSP
• Breaks Dijkstra's time bound
• Uses comparison-addition model