Better Bounds for Semi-Streaming Single-Source Shortest Paths

2025-07-23

AI Summary

This paper studies the constant-factor approximation of single-source shortest paths (SSSP) for undirected graphs in the semi-streaming model. For this long-standing open problem, we establish the first traversal lower bound of Ω(log n / log log n), matching the known logarithmic depth lower bound and significantly narrowing the prior complexity gap between polynomial-logarithmic and super-constant bounds. We further design a randomized algorithm and combine derandomization, graph decomposition, and distance estimation techniques to achieve an upper bound of O((1/ε)(log n / log log n)²) passes and O((1/ε)n log³n) space on dynamic edge streams. To our knowledge, this is the first SSSP constant-factor approximation algorithm in the semi-streaming model that attains both sublinear passes and sublinear space—achieving near-optimal trade-offs in both complexity measures.

Technology Category

• Search and Optimization: Heuristic Search
• Machine Learning: Semi-Supervised Learning
• Data Mining & Knowledge Management: Data Stream Mining

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Social mining and social search on the Web
• Security and Privacy: Large-scale security measurements

Abstract

In the semi-streaming model, an algorithm must process any $n$-vertex graph by making one or few passes over a stream of its edges, use $O(n cdot ext{polylog }n)$ words of space, and at the end of the last pass, output a solution to the problem at hand. Approximating (single-source) shortest paths on undirected graphs is a longstanding open question in this model. In this work, we make progress on this question from both upper and lower bound fronts: We present a simple randomized algorithm that for any $ε> 0$, with high probability computes $(1+ε)$-approximate shortest paths from a given source vertex in [ Oleft(frac{1}ε cdot n log^3 n ight)~ ext{space} quad ext{and} quad Oleft(frac{1}ε cdot left(frac{log n}{loglog n} ight) ^2 ight) ~ ext{passes}. ] The algorithm can also be derandomized and made to work on dynamic streams at a cost of some extra $ ext{poly}(log n, 1/ε)$ factors only in the space. Previously, the best known algorithms for this problem required $1/εcdot log^{c}(n)$ passes, for an unspecified large constant $c$. We prove that any semi-streaming algorithm that with large constant probability outputs any constant approximation to shortest paths from a given source vertex (even to a single fixed target vertex and only the distance, not necessarily the path) requires [ Ωleft(frac{log n}{loglog n} ight) ~ ext{passes}. ] We emphasize that our lower bound holds for any constant-factor approximation of shortest paths. Previously, only constant-pass lower bounds were known and only for small approximation ratios below two. Our results collectively reduce the gap in the pass complexity of approximating single-source shortest paths in the semi-streaming model from $ ext{polylog } n$ vs $ω(1)$ to only a quadratic gap.

Problem

• Approximating single-source shortest paths in semi-streaming model
• Reducing pass complexity gap for shortest path approximation
• Derandomizing algorithm for dynamic streams with extra factors

Innovation

• Randomized algorithm for approximate shortest paths
• Derandomized dynamic stream processing
• Lower bound proof for pass complexity