Optimally sparse spanners in 3-dimensional Euclidean space

Abstract

Let V be a set of n points in 3-dimensional Euclidean space. A subgraph of the complete Euclidean graph is a t-spanner if for any u and v in V, the length of the shortest path from u to v in the spanner is at most t times d (u, v). We show that for any t > 1, a greedy algorithm produces a t-spanner with O(n) edges, and total edge weight O(1). tot(it4ST), whereMST is a minimum spanning tree of V.

Constructing degree-3 spanners with other sparseness properties

“…Thus asymptotic optimality has been achieved up to the third dimension. But the spanners in [2,4] do not have the degree and edges bounds presented here.…”

“…6, the spanner in this paper has a weight bound that match the spanners in [2,4]. However, the general problem of constructing spanners with low weight has not yet been satisfactorily solved.…”

“…A lower bound on the number of edges is n -1 because a spanning tree is a connected graph with the fewest edges. Spanners with O(n) edges and constant stretch factors have been designed in several papers, but in most of these spanners the c~nstant implicit in the O-notation depends on the dimension [1,2,3,4,7,9,10,14]. The degree-4 spanner constructed in [11] is an improvement, because it has at most 2. n edges.…”

“…The length of the shortest path between two vertices u and v of a graph G is da(u, v). In [2,4] a greedy algorithm is developed for constructing spanners of any set V of n points in k-dimensional space. Given any ~1 > 1, this algorithm constructs a degree-s ~'-spanner, where s depends on k and ~'.…”

“…Construct a degrees greedy spanner of V z+l using the greedy algorithm in [2,4]. Let G be the graph formed by the union of the greedy spanner and the transformed forest.…”