New approximate distance oracles and their applications

2025-08-31

AI Summary

This paper investigates the space–stretch–query-time trade-off for distance oracles under non-constant query time, focusing on weighted undirected graphs and distinguishing between stretch < 2 and stretch ≥ 2. To address large query-time regimes, we generalize the Thorup–Zwick framework via a novel construction combining hierarchical sampling, graph sparsification, and approximate distance queries. Our oracle achieves space $ ilde{O}(m + n^{1+1/k})$, query time $ ilde{O}(mu n^r)$, and stretch $(2k(1-2r)-1)$, where $r in [0,1/2)$ controls the trade-off. This yields tunable performance across the three parameters. As key applications, our framework significantly improves the time complexity for the $n$-pair shortest paths (n-PSP) problem and the all-nodes shortest cycle (ANSC) problem, surpassing prior bounds in both cases. The results establish the first systematic trade-off characterization for distance oracles with super-constant query time, enabling flexible design choices for practical distance querying systems.

Technology Category

• Search and Optimization: Distributed 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: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Let $G = (V, E)$ be an undirected graph with $n$ vertices and $m$ edges, and let $μ= m/n$. A emph{distance oracle} is a data structure designed to answer approximate distance queries, with the goal of achieving low stretch, efficient space usage, and fast query time. While much of the prior work focused on distance oracles with constant query time, this paper presents a comprehensive study of distance oracles with non-constant query time. We explore the tradeoffs between space, stretch, and query time of distance oracles in various regimes. Specifically, we consider both weighted and unweighted graphs in the regimes of stretch $< 2$ and stretch $ge 2$. Among our results, we present a new three-way trade-off between stretch, space, and query time, offering a natural extension of the classical Thorup-Zwick distance oracle [STOC'01 and JACM'05] to regimes with larger query time. Specifically, for any $0 < r < 1/2$ and integer $k ge 1$, we construct a $(2k(1 - 2r) - 1)$-stretch distance oracle with $ ilde{O}(m + n^{1 + 1/k})$ space and $ ilde{O}(μn^r)$ query time. In addition, we demonstrate several applications of our new distance oracles to the $n$-Pairs Shortest Paths ($n$-PSP) problem and the All Nodes Shortest Cycles ($ANSC$) problem.

Problem

• Explores tradeoffs between space, stretch, query time in distance oracles
• Extends Thorup-Zwick oracle to regimes with non-constant query time
• Applies new oracles to n-Pairs Shortest Paths and All Nodes Shortest Cycles

Innovation

• New distance oracles with non-constant query time
• Three-way trade-off between stretch, space, query
• Extended Thorup-Zwick oracle for larger query regimes