The Greedy Spanner is Existentially Optimal
The greedy spanner is arguably the simplest and most well-studied spanner construction. Experimental results demonstrate that it is at least as good as any other spanner construction, in terms of both the size and weight parameters. However, a rigorous proof for this statement has remained elusive. In this work we fill in the theoretical gap via a surprisingly simple observation: The greedy spanner is existentially optimal (or existentially near-optimal) for several important graph families, in
... terms of both the size and weight. Roughly speaking, the greedy spanner is said to be existentially optimal (or near-optimal) for a graph family G if the worst performance of the greedy spanner over all graphs in G is just as good (or nearly as good) as the worst performance of an optimal spanner over all graphs in G. Focusing on the weight parameter, the state-of-the-art spanner constructions for both general graphs (due to Chechik and Wulff-Nilsen [SODA'16]) and doubling metrics (due to Gottlieb [FOCS'15]) are complex. Plugging our observation on these results, we conclude that the greedy spanner achieves near-optimal weight guarantees for both general graphs and doubling metrics, thus resolving two longstanding conjectures in the area. Further, we observe that approximate-greedy spanners are existentially near-optimal as well. Consequently, we provide an O(n log n)-time construction of (1+ϵ)-spanners for doubling metrics with constant lightness and degree. Our construction improves Gottlieb's construction, whose runtime is O(n log^2 n) and whose number of edges and degree are unbounded, and remarkably, it matches the state-of-the-art Euclidean result (due to Gudmundsson et al. [SICOMP'02]) in all the involved parameters (up to dependencies on ϵ and the dimension).