Spanners and sparsifiers in dynamic streams

Linear sketching is a popular technique for computing in dynamic streams, where one needs to handle both insertions and deletions of elements. The underlying idea of taking randomized linear measurements of input data has been extremely successful in providing space-efficient algorithms for classical problems such as frequency moment estimation and computing heavy hitters, and was very recently shown to be a powerful technique for solving graph problems in dynamic streams [AGM'12]. Ideally, one would like to obtain algorithms that use one or a small constant number of passes over the data and a small amount of space (i.e. sketching dimension) to preserve some useful properties of the input graph presented as a sequence of edge insertions and edge deletions. In this paper, we concentrate on the problem of constructing linear sketches of graphs that (approximately) preserve the spectral information of the graph in a few passes over the stream. We do so by giving the first sketch-based algorithm for constructing multiplicative graph spanners in only two passes over the stream. Our spanners use ~O(n^1+1/k) bits of space and have stretch 2^k. While this stretch is larger than the conjectured optimal 2k-1 for this amount of space, we show for an appropriate k that it implies the first 2-pass spectral sparsifier with n^1+o(1) bits of space. Previous constructions of spectral sparsifiers in this model with a constant number of passes would require n^1+c bits of space for a constant c > 0. We also give an algorithm for constructing spanners that provides an additive approximation to the shortest path metric using a single pass over the data stream, also achieving an essentially best possible space/approximation tradeoff.