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 $(1+\epsilon )$-spanner, for every $\epsilon >0$. 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 $O(n^2\log n)$ 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 $\delta =t$, 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 δ-Greedy algorithm is $O(n\log n)$. Our analysis indicates that the δ-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 δ-Greedy algorithm provides a spanner identical to the Path-Greedy spanner in expected $O(n\log n)$ time.

我们引入了一种新的几何扳手δ - Greedy，其构造基于已知的Path-Greedy和Gap-Greedy扳手的泛化。该δ -Greedy扳手结合了在理论和实践上的几何扳手最需要的特性。更具体地说，它具有与 Path-Greedy 扳手相同的理论和实践特性：自然定义、小度数、线性边数、低权重和强$(1+\epsilon )$-扳手，对于每个 $\epsilon >0$. 该δ -Greedy算法是在路径贪婪算法的改进相对于最短路径查询的数量，因此对于它的施工时间。我们展示了如何为平面中的一组n个点构造这样的扳手$哦(n^2\mathrm{日志}n)$ 时间。

该δ -Greedy扳手具有附加参数δ，这表明它是对帐户最短路径查询的数量的路径贪婪扳手多么接近。为了$\delta =吨$，输出生成器与路径贪婪生成器相同，而最短路径查询的数量实际上是线性的。

最后，我们证明，对于在单位正方形中随机独立放置的一组n个点，δ -Greedy 算法的预期构建时间为$哦(n\mathrm{日志}n)$. 我们的分析表明，δ -Greedy 扳手在预期的已知扳手中给出了最好的结果$哦(n\mathrm{日志}n)$随机点集的时间。此外，分析意味着通过设置$\delta =吨$, δ -Greedy 算法提供了一个与 Path-Greedy 生成器相同的生成器$哦(n\mathrm{日志}n)$ 时间。