The greedy spanner is existentially optimal: [Extended Abstract]

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. Focusing on the weight parameter, the state-of-the-art spanner constructions for both general graphs (due to Chechik and Wulff-Nilsen [SODA16]) and doubling metrics (due to Gottlieb [FOCS15]) 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 algorithms 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 Gottliebs construction, whose runtime is O(n log² 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. [SICOMP02]) in all the involved parameters (up to dependencies on ϵ and the dimension).