Constructing Plane Spanners of Bounded Degree and Low Weight

Abstract

Given a set S of n points in the plane, we give an O(n log n)-time algorithm that constructs a plane t-spanner for S, with t ≈ 10, such that the degree of each point of S is bounded from above by 27, and the total edge length is proportional to the weight of a minimum spanning tree of S. Previously, no algorithms were known for constructing plane t-spanners of bounded degree. Introduction.Givena set S of n points in the plane and a real number t > 1, a t-spanner for S is a graph G with vertex set S such that any two vertices u and v are connected by a path in G whose length is at most t · |uv|, where |uv| is the Euclidean distance between u and v. If this graph has O(n) edges, then it is a sparse approximation of the (dense) complete Euclidean graph on S.Many algorithms are known that compute t-spanners with O(n) edges [1], [2], [8], [20], [22],[24], [26] that have additional properties such as bounded degree [4], [6], small spanner diameter [4] (i.e., any two points are connected by a t-spanner path consisting of only a small number of edges), low weight [11], [13], [19] (i.e., the total length of all edges is proportional to the weight of a minimum spanning tree of S), planarity [3], [21], and fault-tolerance [10], [23]; see also the surveys [18] and [25]. All these algorithms compute t-spanners for any given constant t > 1. Observe that some properties are conflicting. For instance a spanner with constant degree cannot have spanner diameter o(log n).In this paper we consider the construction of plane t-spanners. A graph, whose vertices are points in R 2 with straight-line edges, is said to be plane, if no two edges intersect, except possibly at their endpoints. Obviously, in order for a t-spanner to be plane, t must be at least √ 2. It is known that the Delaunay triangulation is a t-spanner for t = 2π/(3 cos(π/6)), see [21]. Furthermore, Das and Joseph [12] showed that other plane graphs such as the minimum weight triangulation and the greedy triangulation are t-spanners, for some constant t. In 1992 Levcopoulos and Lingas [22] showed that, for

Geometric dilation of closed planar curves: New lower bounds

“…Hence, we can apply Lemma 12(1) and get l C = |C * | 2 min v∈S 1 h C * (v) (4), (5) |C| 2 min v∈S 1 h C (v) Lem. 11 = δ(C).…”

“…One should note that the geometric dilation is a concept substantially different from the standard graph-theoretic dilation usually studied in the context of spanners [5,6,12], where only the point set of all vertices is considered. With the geometric dilation defined above, two natural problems arise.…”

The Geometric Dilation of Finite Point Sets

“…Transportation networks like waterways, railroad systems, or urban street systems can be modeled by a graph G in the plane whose edges are piecewise smooth curves that do not intersect, except at vertices of G. 3 The quality of G as a means of transport can be measured in the following way. For any two points, p and q, of G, let ξ G ( p, q) denote a shortest path in G from p to q.…”

“…{ebbers,gruene,klein}@cs.uni-bonn.de. 3 That is, we do not allow bridges at this stage, but it would, in principle, be possible to enlarge our model.…”

“…One can efficiently construct spanners of bounded dilation and degree, whose weight is close to that of the minimum spanning tree, see [3]. Also, lower time bounds are known, see [4].…”