A new way to weigh Malnourished Euclidean graphs

Abstract

In this paper, we show that any Euclidean graph over a set V of n points in k-dimensional space that satisfies either the leapfrog property or the isolation property has small weight, i.e., has weight O(1) . wt(SMT), where SMT is a Steiner minimal tree of V. Both the leapfrog property as well as the isolation property constrain the way the edges of the graph are configured in space. Our main application is to prove that certain Euclidean graphs known as t-spanners can be constructed with optimal weight of O(1) + wt(SMT), an intriguing open problem that has attracted much attention recently. The main tool in obtaining the above weight bounds is a theorem that proves the existence of long edges in a Steiner minimal tree on a restricted set of points in k-dimensional space. We also generalize this theorem for Steiner minimal trees on arbitrary point sets. Since very little is known about high-dimensional Steiner minimal trees, these results are of independent interest.