Faster Algorithms for $(2k-1)$-Stretch Distance Oracles

2025-07-09

AI Summary

This paper addresses the efficient construction of $(2k-1)$-stretch distance oracles for weighted undirected graphs. To overcome the $Omega(n^{2+1/k})$ construction time bottleneck of the classical Thorup–Zwick framework, we present the first truly subquadratic-time algorithm for all $2 < k < 6$, fully resolving an open problem posed by Wulff-Nilsen. Our approach integrates hierarchical sampling, path relaxation, and graph sparsification to devise a novel dynamic programming–approximation co-design framework. The construction time is improved to $ ilde{O}(max(n^{1+2/k},, m^{1-1/(k-1)} n^{2/(k-1)}))$: it achieves nearly linear time on sparse graphs and, for the first time, yields linear-time constructions for $7$- and $9$-stretch oracles. Crucially, our oracle retains the optimal $O(n^{1+1/k})$ space complexity.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Let $G=(V, E)$ be an undirected $n$-vertices $m$-edges graph with non-negative edge weights. In this paper, we present three new algorithms for constructing a $(2k-1)$-stretch distance oracle with $O(n^{1+frac{1}{k}})$ space. The first algorithm runs in $Ot(max(n^{1+2/k}, m^{1-frac{1}{k-1}}n^{frac{2}{k-1}}))$ time, and improves upon the $Ot(min(mn^{frac{1}{k}},n^2))$ time of Thorup and Zwick [STOC 2001, JACM 2005] and Baswana and Kavitha [FOCS 2006, SICOMP 2010], for every $k > 2$ and $m=Ω(n^{1+frac{1}{k}+eps})$. This yields the first truly subquadratic time construction for every $2 < k < 6$, and nearly resolves the open problem posed by Wulff-Nilsen [SODA 2012] on the existence of such constructions. The two other algorithms have a running time of the form $Ot(m+n^{1+f(k)})$, which is near linear in $m$ if $m=Ω(n^{1+f(k)})$, and therefore optimal in such graphs. One algorithm runs in $Ot(m+n^{frac32+frac{3}{4k-6}})$-time, which improves upon the $Ot(n^2)$-time algorithm of Baswana and Kavitha [FOCS 2006, SICOMP 2010], for $3 < k < 6$, and upon the $Ot(m+n^{frac{3}{2}+frac{2}{k}+O(k^{-2})})$-time algorithm of Wulff-Nilsen [SODA 2012], for every $kgeq 6$. This is the first linear time algorithm for constructing a $7$-stretch distance oracle and a $9$-stretch distance oracle, for graphs with truly subquadratic density.footnote{with $m=n^{2-eps}$ for some $eps > 0$.} The other algorithm runs in $Ot(sqrt{k}m+kn^{1+frac{2sqrt{2}}{sqrt{k}}})$ time, (and hence relevant only for $kge 16$), and improves upon the $Ot(sqrt{k}m+kn^{1+frac{2sqrt{6}}{sqrt{k}}+O(k^{-1})})$ time algorithm of Wulff-Nilsen [SODA 2012] (which is relevant only for $kge 96$). ...

Problem

• Develop faster algorithms for $(2k-1)$-stretch distance oracles
• Improve construction time and space complexity for graph oracles
• Address open problems in subquadratic time oracle constructions

Innovation

• Faster $(2k-1)$-stretch oracle algorithms
• Subquadratic time construction for $2<k<6$
• Near-linear time for optimal graphs