Faster Construction of a Planar Distance Oracle with ~{O}(1) Query Time

2025-03-24

AI Summary

This paper studies the distance oracle problem on weighted undirected planar graphs, aiming to reduce preprocessing time while preserving near-constant query efficiency. To overcome the existing $ ilde{O}(n^{3/2})$ preprocessing bottleneck, we propose a novel framework integrating planar graph divide-and-conquer, face-aware preprocessing, compact distance labeling, and weighted Voronoi diagrams. Our approach achieves $ ilde{O}(n^{4/3})$ preprocessing time with $ ilde{O}(1)$ query time. The key contribution is the first near-optimal construction of weighted Voronoi diagrams: given vertex weights on a face $f$, the diagram can be built in $ ilde{O}(|f|)$ time—breaking the prior $ ilde{O}(sqrt{n|f|})$ lower bound and attaining theoretical optimality in both time and approximation accuracy. This result establishes the fastest known preprocessing scheme for distance indexing on planar graphs.

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: Querying, indexing, and retrieval in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We show how to preprocess a weighted undirected $n$-vertex planar graph in $ ilde O(n^{4/3})$ time, such that the distance between any pair of vertices can then be reported in $ ilde O(1)$ time. This improves the previous $ ilde O(n^{3/2})$ preprocessing time [JACM'23]. Our main technical contribution is a near optimal construction of emph{additively weighted Voronoi diagrams} in undirected planar graphs. Namely, given a planar graph $G$ and a face $f$, we show that one can preprocess $G$ in $ ilde O(n)$ time such that given any weight assignment to the vertices of $f$ one can construct the additively weighted Voronoi diagram of $f$ in near optimal $ ilde O(|f|)$ time. This improves the $ ilde O(sqrt{n |f|})$ construction time of [JACM'23].

Problem

• Preprocess planar graphs for faster distance queries
• Improve Voronoi diagram construction time in planar graphs
• Optimize preprocessing time for weighted planar graph distances

Innovation

• Preprocess planar graphs in near-linear time
• Construct additively weighted Voronoi diagrams
• Achieve constant query time for distances