A WSPD, Separator and Small Tree Cover for c-packed Graphs
Lindsey Deryckere, Joachim Gudmundsson, Andr\'e van Renssen, Yuan Sha, Sampson Wong

TL;DR
This paper establishes fundamental properties of c-packed graphs, including well-separated pair decompositions and small separators, enabling efficient data structures like small tree covers and near-linear distance oracles.
Contribution
It proves that c-packed graphs admit linear-size well-separated pair decompositions and constant-size separators, leading to new efficient data structures for these graphs.
Findings
Existence of linear-size well-separated pair decomposition.
Presence of constant-size balanced separators.
Construction of small tree covers and efficient distance oracles.
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…
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