On the Edge Crossings of the Greedy Spanner

Abstract

t-spanners are used to approximate the pairwise distances between a set of points in a metric space. They have only a few edges compared to the total number of pairs and they provide a t-approximation on the distance of any two arbitrary points. There are many ways to construct such graphs and one of the most efficient ones, in terms of weight and the number of edges of the resulting graph, is the greedy spanner. In this paper, we study the edge crossings of the greedy spanner for points in the Euclidean plane. We prove a constant upper bound for the number of intersections with larger edges that only depends on the stretch factor of the spanner, t, and we show there can be more than a bounded number of intersections with smaller edges. Our results imply that greedy spanners for points in the plane have separators of size O( √ n), that their planarizations have linear size, and that a separator hierarchy for these graphs can be constructed from their planarizations in linear time.

Optimal Spanners for Unit Ball Graphs in Doubling Metrics

“…In a recent work, the authors of this paper showed that the number of edge crossings of the greedy spanner in the two dimensional Euclidean plane is linear in the number of vertices [8]. Moreover, they proved that the crossing graph of the greedy spanner has bounded degeneracy, implying the existence of sub-linear separators for these graphs [7].…”

“…Furthermore, we study the problem in the case of the two dimensional Euclidean plane, where the greedy spanner on a complete weighted graph is known to have constant upper bounds on its lightness [10], maximum degree, and average number of edge intersections per node [8]. We observe that a simple change on the this algorithm can extend these results for unit disk graphs as well.…”

“…▶ Lemma 20 (Theorem 17 from [8]). Given an arbitrary segment ∥AB∥ ≤ 1 in the plane, the number of edges e ∈ S that intersect AB and |e| ≥ ∥AB∥ is bounded by a constant.…”

“…And since the spanner is a sub-graph of the greedy spanner on the complete weighted graph on V , we can deduce that its lightness, its maximum degree, and its average number of edge intersections per node are bounded by the lightness, maximum degree, and the average number of edge intersections per node of the greedy spanner, respectively. And these are all bounded by constants according to [10] and [8]. ◀…”

“…Proof. We use the result of Eppstein and Khodabandeh [8] on the edge crossings of the greedy spanner, which states that an arbitrary edge AB of the greedy spanner intersects with at most a constant number of longer edges. It is easy to see that the same statement is true when AB is an arbitrary segment on the plane, and the assumption of AB being an edge of the spanner is not necessary in the proof of the theorem.…”