δ-Greedy t-spanner

Abstract

We introduce a new geometric spanner, δ-Greedy, whose construction is based on a generalization of the known Path-Greedy and Gap-Greedy spanners. The δ-Greedy spanner combines the most desirable properties of geometric spanners both in theory and in practice. More specifically, it has the same theoretical and practical properties as the Path-Greedy spanner: a natural definition, small degree, linear number of edges, low weight, and strong (1+ε)-spanner, for every ε>0. The δ-Greedy algorithm is an improvement over the Path-Greedy algorithm with respect to the number of shortest path queries and hence with respect to its construction time. We show how to construct such a spanner for a set of n points in the plane in O(n2log⁡n) time.The δ-Greedy spanner has an additional parameter δ, which indicates how close it is to the Path-Greedy spanner on the account of the number of shortest path queries. For δ=t, the output spanner is identical to the Path-Greedy spanner, while the number of shortest path queries is, in practice, linear.Finally, we show that, for a set of n points placed independently at random in a unit square, the expected construction time of the δ-Greedy algorithm is O(nlog⁡n). Our analysis indicates that the δ-Greedy spanner gives the best results among the known spanners of expected O(nlog⁡n) time for random point sets. Moreover, analysis implies that by setting δ=t, the δ-Greedy algorithm provides a spanner identical to the Path-Greedy spanner in expected O(nlog⁡n) time.