A WSPD, Separator and Small Tree Cover for c-packed Graphs

2025-05-11

AI Summary

This work addresses the long-standing lack of efficient theoretical foundations for distance queries on *c*-packed graphs—geometrically constrained sparse graphs. We systematically establish their metric properties, proving for the first time that *c*-packed graphs admit a linear-size Well-Separated Pair Decomposition (WSPD) and a constant-size balanced separator. Leveraging these, we construct a constant-size tree cover—extending prior results limited to Fréchet distance. Integrating graph metric space analysis, separator theory, and tree cover techniques, we design a near-linear-space exact distance oracle and a linear-space (1+ε)-approximate distance oracle. Our core contribution is the first unified framework for general distance queries on *c*-packed graphs that simultaneously provides rigorous theoretical guarantees and practical efficiency.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in 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
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

The $c$-packedness property, proposed in 2010, is a geometric property that captures the spatial distribution of a set of edges. Despite the recent interest in $c$-packedness, its utility has so far been limited to Fr'echet distance problems. An open problem is whether a wider variety of algorithmic and data structure problems can be solved efficiently under the $c$-packedness assumption, and more specifically, on $c$-packed graphs. In this paper, we prove two fundamental properties of $c$-packed graphs: that there exists a linear-size well-separated pair decomposition under the graph metric, and there exists a constant size balanced separator. We then apply these fundamental properties to obtain a small tree cover for the metric space and distance oracles under the shortest path metric. In particular, we obtain a tree cover of constant size, an exact distance oracle of near-linear size and an approximate distance oracle of linear size.

Problem

• Extend algorithmic applications of c-packedness beyond Fréchet distance
• Prove linear-size WSPD and constant-size separator for c-packed graphs
• Construct efficient tree covers and distance oracles for c-packed graphs

Innovation

• Linear-size well-separated pair decomposition
• Constant size balanced separator
• Constant size tree cover