Small stretch (α, β)-spanners in the streaming model

We present algorithms for computing small stretch (α, β)-spanners in the streaming model. An (α, β)-spanner of a graph G is a subgraph S ⊆ G such that for each pair of vertices the distance in S is at most α times the distance in G plus β. We assume that the graph is given as a stream of edges and vertices, and that only one pass over the data is allowed. Furthermore, the number of vertices and edges are not known in advance. We denote by m the current number of scanned edges and by n the current number of discovered vertices. In this model we show how to compute a (k, k - 1)-spanner of an unweighted undirected graph, for k = 2, 3, in O (1) amortized processing time per edge/vertex. The computed (k, k - 1)-spanners have O (n1 + 1 / k) edges and our algorithms use only O (n1 + 1 / k) words of memory space. In case only Θ (n) internal memory is available, the same spanners can be computed using O (n1 + 1 / k / B) external memory blocks, each of size B. Each edge/vertex is processed in O (1) amortized time, plus O (1 / B) amortized block transfers. © 2008 Elsevier B.V. All rights reserved.