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

TL;DR

The paper studies $c$-packed graphs in fixed dimension and proves two foundational properties: a linear-size well-separated pair decomposition under the graph metric and a constant-size balanced separator. Leveraging these, it develops a framework for a small tree cover and efficient distance oracles for shortest-path queries, including a near-linear exact distance oracle and a $(1+\varepsilon)$-distortion tree cover with a constant number of trees when $c$ and $d$ are fixed. The approach introduces a novel $\delta$-connected tree to mimic Euclidean WSPD constructions in the graph distance setting and uses a separator hierarchy to enable fast queries. The results yield deterministic, spread-independent structures with polynomial dependence on $c$ and $\varepsilon$, enabling practical distance querying and proximity problems on $c$-packed graphs. Overall, the work significantly broadens the algorithmic toolkit available for $c$-packed graphs beyond Fréchet distance problems and connects doubling metrics, bounded treewidth, and graph-based WSPDs in a cohesive framework.

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.

Figures (7)

• Figure 1: An illustration of iteration $j$ of the algorithm constructing the $c$-connected tree.
• Figure 2: Set of relevant edges w.r.t. $s(u)$ (green edges) and irrelevant edges (red edges).
• Figure 3: Two $c$-connected components in $s$ must be separated by a path of length at least $\frac{\radius{s}}{\sqrt{d}}$.
• Figure 5: The dumbbell $D(a,b)$ for the pair $\{A_i=C_a,B_i=C_b\}$. $head(a)$ ($head(b)$) is the ball that is centered at $rep_a$ ($rep_b$), has radius $\gdub{C_a}$ ($\gdub{C_b}$) and only includes points in $C_a$ ($C_b$).
• Figure 6: (a) $s(b_j')$ is the cell containing the red point. The two blue points lie in two half spaces which are (respectively) delimited by (one of) the hyperplanes through two opposite faces of $s(b_j')$ and do not contain $s(b_j')$. (b) $s(b_j')$ is the cell containing the red point. The blue point is separated from $s(b_j')$ by a slab of width at least $L(s(b_j'))$.

Theorems & Definitions (37)

• Definition 1: Geometric Well-Separated Pair
• Definition 2: Geometric Well-Separated Pair Decomposition (WSPD)
• Theorem 3
• Theorem 4
• Corollary 4
• Definition 5: $\delta$-Connected Set
• Definition 6: Partition into $\delta$-Connected Sets
• Definition 7: $\delta$-Connected Tree ($\delta$-CT)
• Lemma 7
• Lemma 7