New constructions of (α, β)-spanners and purely additive spanners

An (α, β)-spanner of an unweighted graph G is a subgraph H that approximates distances in G in the following sense. For any two vertices u, v: δ_H (u, v) ≤ αδG(u, v) + β, where δG is the distance w.r.t. G. It is well known that there exist (multiplicative) (2k - 1, 0)-spanners of size O(n^1+1/k) and that there exist (purely additive) (1, 2)-spanners of size O(n^3/2). However no other (1, O(1))-spanners are known to exist.In this paper we develop a couple new techniques for constructing (α, β)-spanners. The first result is a purely additive (1, 6)-spanner of size O(n^4/3). Our construction algorithm can be understood as an economical agent that assigns costs and values to paths in the graph, purchasing affordable paths and ignoring expensive ones, which are intuitively well-approximated by paths already purchased. This general approach should lead to new spanner constructions.The second result is a truly simple linear time construction of (k, k - 1)-spanners with size O(n¹⁺¹/k). In a distributed network the algorithm terminates in a constant number of rounds and has expected size O(n¹⁺¹/k). The new idea here is primarily in the analysis of the construction. We show that a few simple and local rules for picking spanner edges induce seemingly coordinated global behavior.