Efficient algorithms for constructing (1+6, 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 ∈ V, distc_G' (u, v) ≤ α · distc_G(u,v) + β. It was shown in [20] that for any ε > 0, K = 1, 2,., there exists an integer β= β(ε, K) such that for every n-vertex graph G there exists a (1+ ε, β)-spanner G' with O(n^1+1/K) edges. An efficient distributed protocol for constructing (1 + ε, β)-spanners was devised in [18]. The running time and the communication complexity of that protocol are O(n^1+p) and O(\E\n^ρ), respectively, where p 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^p) as opposed to O(n ^1+p)) for constructing (1 + ε, β)-spanners. Our protocol has the same communication complexity as the protocol of [18], and it constructs spanners with essentially the same properties as the spanners that are constructed by the protocol of [18], 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/K · log n) bits of space for computing all-pairs-almost-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^p), for an arbitrarily small p > 0. The only previously known algorithm for the problem [21] constructs paths of length K times greater than the shortest paths, has the same space requirements as our algorithm, but requires O(n^1+l/K) time for processing each edge of the input graph. However, the algorithm of [21] uses just one pass over the input, as opposed to the constant number of passes in our algorithm. We also show that any streaming algorithm for o(n) -approximate distance computation requires Ω(n) bits of space.