Efficient algorithms for constructing (1+ε,β)-spanners in the distributed and streaming models

For an unweighted undirected graph G = (V, E), and a pair of positive integers α ≥ 1, β ≥ 0, a subgraph G′ = (V, H), H ⊆ E, is called an (α. β)-spanner of G if for every pair of vertices u, v ε V, dist_G′(u, v) ≤ α · dist _G(u, v) + β. It was shown in [21] that for any ε > 0, κ = 1, 2,..., there exists an integer β= β(ε, κ) such that for every n-vertex graph G there exists a (1 + ε, β)-spanner G′ with O(n^1+1/κ) edges. An efficient distributed protocol for constructing (1 + ε, β)-spanners was devised in [19]. The running time and the communication complexity of that protocol are O(n ^1+ρ) and O(|E|n^ρ), respectively, where ρ is an additional control parameter of the protocol that affects only the additive term β. In this paper we devise a protocol with a drastically improved running time (O(n^ρ) as opposed to O(n^1+ρ)) for constructing (1 + ε, β)-spanners. Our protocol has the same communication complexity as the protocol of [19], and it constructs spanners with essentially the same properties as the spanners that are constructed by the protocol of [19]. The protocol can be easily extended to a parallel implementation which runs in O(log n + (|E\ · n^ρ log n) / p) time using p processors in the EREW PRAM model. In particular, when the number of processors, p, is at least |E| · n^ρ, the running time of the algorithm is O(log n). We also show that our protocol for constructing (1 + ε, β)-spanners can be adapted to the streaming model, and devise a streaming algorithm that uses a constant number of passes and O (n^1+1/κ· log n) bits of space for computing all-pairs-almosl-shortest-paths of length at most by a multiplicative factor (1 + ε) and an additive term of β greater than the shortest paths. Our algorithm processes each edge in time O(n^ρ), for an arbitrarily small ρ > 0. The only previously known algorithm for the problem [23] constructs paths of length κ times greater than the shortest paths, has the same space requirements as our algorithm, but requires O(n ^1+1/κ) time for processing each edge of the input graph. However, the algorithm of [23] uses just one pass over the input, as opposed to the constant number of passes in our algorithm.¹ We also show that any streaming algorithm for o(n)-approximate distance computation requires Ω(n) bits of space.