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 Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint 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 graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security 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