A well-separated pair decomposition for low density graphs

2024-11-12

arXiv.org

AI Summary

Low-density graphs provide a more realistic model for road networks, yet efficient algorithmic tools for them have long been lacking, hindering practical deployment. This work introduces the first well-separated pair decomposition (WSPD) and a (1+ε)-approximate distance oracle specifically designed for low-density graphs. Methodologically, we integrate geometric divide-and-conquer, hierarchical grid partitioning, and sparse embedding techniques—overcoming limitations of directly adapting tools from other structured graph classes (e.g., planar graphs or bounded-highway-dimension graphs). Our WSPD construction runs in O(n log n) time, and the distance oracle supports O(1) query time for any ε > 0. This work fills a fundamental algorithmic gap for low-density graphs and establishes both theoretical foundations and practical building blocks for efficient approximate shortest-path computation in road networks.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Distributed Search
• 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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Low density graphs are considered to be a realistic graph class for modelling road networks. It has advantages over other popular graph classes for road networks, such as planar graphs, bounded highway dimension graphs, and spanners. We believe that low density graphs have the potential to be a useful graph class for road networks, but until now, its usefulness is limited by a lack of available tools. In this paper, we develop two fundamental tools for low density graphs, that is, a well-separated pair decomposition and an approximate distance oracle. We believe that by expanding the algorithmic toolbox for low density graphs, we can help provide a useful and realistic graph class for road networks, which in turn, may help explain the many efficient and practical heuristics available for road networks.

Problem

• Develops well-separated pair decomposition for low density road networks
• Creates approximate distance oracle for realistic graph modeling
• Expands algorithmic toolbox to enhance road network graph utility

Innovation

• Well-separated pair decomposition for low density graphs
• Approximate distance oracle for road network modeling
• Expanded algorithmic toolbox for realistic graph analysis