Better Bounds for Semi-Streaming Single-Source Shortest Paths

In the semi-streaming model, an algorithm must process any n-vertex graph by making one or few passes over a stream of its edges, use O(n · polylog n) words of space, and at the end of the last pass, output a solution to the problem at hand. Approximating (single-source) shortest paths on undirected graphs is a longstanding open question in this model. In this work, we make progress on this question from both upper and lower bound fronts: We present a simple randomized algorithm that for any $ε> 0$, with high probability computes $(1+ε)$-approximate shortest paths from a given source vertex in \[ O({1}ε · n log^3 n )~space and O({1}ε · (log n/loglog n ) ^2) ~passes. \] The algorithm can also be derandomized and made to work on dynamic streams at a cost of some extra poly(log n, 1/ε) factors only in the space. Previously, the best known algorithms for this problem required 1/ε· logᶜ(n) passes, for an unspecified large constant c. We prove that any semi-streaming algorithm that with large constant probability outputs any constant approximation to shortest paths from a given source vertex (even to a single fixed target vertex and only the distance, not necessarily the path) requires \[ Ω(log n/loglog n) ~passes. \] We emphasize that our lower bound holds for any constant-factor approximation of shortest paths. Previously, only constant-pass lower bounds were known and only for small approximation ratios below two. Our results collectively reduce the gap in the pass complexity of approximating single-source shortest paths in the semi-streaming model from polylog n vs $ω(1)$ to only a quadratic gap.