The greedy spanner in a low dimensional Euclidean space is fundamental geometric construction that has been extensively studied over three decades as it possesses the two most basic properties of good spanner: constant maximum degree and lightness. Recently, Eppstein Khodabandeh [28] showed \(\mathbb {R}^2 \) admits sublinear separator strong sense: any subgraph k vertices size \(O(\sqrt {k}) ....

Problem definition: Input: A graph represented (say) as the adjacency list for each vertex (or even just the degree of each vertex) Goal: Compute the average degree (equiv. number of edges) Concern: Seems to be impossible e.g. if all degrees ≤ 1, except possibly for a few vertices whose degree is about n. Theorem 1 [Feige, 2004]: There is an algorithm that estimates the average degree d of a co...