A well-separated pair decomposition for low density graphs

TL;DR

This work addresses efficient proximity queries on realistic road-network models by focusing on $\lambda$-low-density graphs. The authors develop a near-linear size $(1/\varepsilon)$-well-separated pair decomposition (WSPD) in the graph metric, and, building on this, construct a $(1+\varepsilon)$-approximate distance oracle with near-linear size and constant query time. Key innovations include the semi-compressed quadtree and the net-tree-based clustering to bridge Euclidean and graph metrics, plus a semi-weight framework to handle unbounded spread. The results unify geometric proximity tools with graph-based distance data structures, enabling scalable routing and proximity queries in large road networks and offering a path toward broader applicability to realistic graph classes. The practical impact lies in providing compact, fast-query data structures for real-world networks where traditional planar or minor-graph assumptions fail, while also posing open questions about optimal sizes and extensions to more general graph families.

Abstract

Low density graphs are considered to be a realistic graph class for modelling road networks. It has advantages over other popular graph classes for road networks, such as planar graphs, bounded highway dimension graphs, and spanners. We believe that low density graphs have the potential to be a useful graph class for road networks, but until now, its usefulness is limited by a lack of available tools. In this paper, we develop two fundamental tools for low density graphs, that is, a well-separated pair decomposition and an approximate distance oracle. We believe that by expanding the algorithmic toolbox for low density graphs, we can help provide a useful and realistic graph class for road networks, which in turn, may help explain the many efficient and practical heuristics available for road networks.

Figures (13)

• Figure 1: Left: A $\sqrt n \times \! \sqrt n$ grid has a highway dimension, skeleton dimension, and treewidth of $\Theta(\sqrt n)$. Right: A $\sqrt n \times \! \sqrt n$ comb has a spanning ratio, doubling constant, and highway dimension of $\Theta(\sqrt n)$.
• Figure 2: Left: The solid edges count towards the density of the blue disc, but the dashed edges do not. Right: A unit grid graph is a low density graph, since balls of radius $r$ have constant density for all $r \leq 1$, and have zero density for all $r > 1$.
• Figure 3: A well-separated pair under the Euclidean metric (left) and under a graph metric (right).
• Figure 4: Left: A pair of sets that are well-separated in $\mathbb R^2$ may not be well-separated in the graph metric. Right: Clustering the sets in the Euclidean WSPD to reduce their graph diameter.
• Figure 5: The length bound relates the sum of the lengths of edges in $S$ with the number of points in $3S$.

Theorems & Definitions (48)

• Definition 4: SELG
• Definition 5: Low density graph
• Definition 6: Lanky graph DBLP:conf/soda/LeT22
• Definition 7: Well-separated pair
• Definition 8: Pair decomposition
• Definition 9: Well-separated pair decomposition, WSPD
• Definition 10: Approximate distance oracle
• Definition 11: Unit-cost floating-point word RAM DBLP:journals/siamcomp/Har-PeledM06
• Theorem 12: restate of Theorem \ref{['theorem:unbounded_spread_wspd']}
• Theorem 13: restate of Theorem \ref{['theorem:wspd_construction']}