Small stretch (alpha, beta)-spanners in the streaming model

We present algorithms for computing small stretch (?,s)-spanners in the streaming model. An (?,s)-spanner of a graph G is a subgraph S?G such that for each pair of vertices the distance in S is at most ? times the distance in G plus s. 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 ?(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.