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arXiv · 2505.06884

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

Lindsey Deryckere·Joachim Gudmundsson·André van Renssen·Yuan Sha·Sampson Wong
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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 Fréchet 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.

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Lindsey Deryckere, Joachim Gudmundsson, André van Renssen, Yuan Sha, Sampson Wong. 2025-05-11. A WSPD, Separator and Small Tree Cover for c-packed Graphs. https://arxiv.org/abs/2505.06884

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