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

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 $c$-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.