\(\delta \)-Greedy t-spanner
We introduce a new geometric spanner, \(\delta \)-Greedy, whose construction is based on a generalization of the known Path-Greedy and Gap-Greedy spanners. The \(\delta \)-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 \((1+\varepsilon )\)-spanner for every \(\varepsilon >0\). The \(\delta \)-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 \(O(n^2 \log n)\) time.
The \(\delta \)-Greedy spanner has an additional parameter, \(\delta \), which indicates how close it is to the Path-Greedy spanner on the account of the number of shortest path queries. For \(\delta = t\) 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 \(\delta \)-Greedy algorithm is \(O(n \log n)\). Our analysis indicates that the \(\delta \)-Greedy spanner gives the best results among the known spanners of expected \(O(n \log n)\) time for random point sets. Moreover, analysis implies that by setting \(\delta = t\), the \(\delta \)-Greedy algorithm provides a spanner identical to the Path-Greedy spanner in expected \(O(n \log n)\) time.
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