A WSPD, Separator and Small Tree Cover for c-packed Graphs
Abstract
The -packedness property, proposed in 2010, is a geometric property that captures the spatial distribution of a set of edges. Despite the recent interest in -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 -packedness assumption, and more specifically, on -packed graphs. In this paper, we prove two fundamental properties of -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.
Keywords
Cite
@article{arxiv.2505.06884,
title = {A WSPD, Separator and Small Tree Cover for c-packed Graphs},
author = {Lindsey Deryckere and Joachim Gudmundsson and André van Renssen and Yuan Sha and Sampson Wong},
journal= {arXiv preprint arXiv:2505.06884},
year = {2025}
}