(1 + ϵ)-Approximate shortest paths in dynamic streams

Computing approximate shortest paths in the dynamic streaming setting is a fundamental challenge that has been intensively studied. Currently existing solutions for this problem either build a sparse multiplicative spanner of the input graph and compute shortest paths in the spanner offline, or compute an exact single source BFS tree. Solutions of the first type are doomed to incur a stretch-space tradeoff of 2κ - 1 versus n 1+1/κ, for an integer parameter κ. (In fact, existing solutions also incur an extra factor of 1 + ϵ in the stretch for weighted graphs, and an additional factor of log O(1) n in the space.) The only existing solution of the second type uses n 1/2-O(1/κ) passes over the stream (for space O(n 1+1/κ)), and applies only to unweighted graphs. In this paper we show that (1 + ϵ)-approximate single-source shortest paths can be computed with Õ(n 1+1/κ) space using just constantly many passes in unweighted graphs, and polylogarithmically many passes in weighted graphs. Moreover, the same result applies for multi-source shortest paths, as long as the number of sources is O(n 1/κ). We achieve these results by devising efficient dynamic streaming constructions of (1 + ϵ, β)-spanners and hopsets. On our way to these results, we also devise a new dynamic streaming algorithm for the 1-sparse recovery problem. Even though our algorithm for this task is slightly inferior to the existing algorithms of 26, 11, we believe that it is of independent interest.