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 $\tilde{O}(n) := O(n \cdot \mathrm{polylog}(n))$ 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: 1) We present a simple randomized algorithm that for any $\varepsilon \gt 0$, with high probability computes $(1+\varepsilon)$-approximate shortest paths from a given source vertex in $O\left( \frac{1}{\varepsilon} \cdot n \log^3 n \right)$ space and $O\left( \frac{1}{\varepsilon} \cdot \left(\frac{\log n}{\log\log n}\right)^{2} \right)$ passes. The algorithm can also be derandomized and made to work on dynamic streams at a cost of some extra $\mathrm{poly}(\log n,1/\varepsilon)$ factors only in the space. Previously, the best known algorithms for this problem required $1/\varepsilon \cdot \log^{c}n$ passes, for an unspecified large constant $c$. 2) 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 $\Omega\left( \frac{\log n}{\log\log n} \right)$ 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 $\mathrm{polylog}(n)$ vs. $\omega(1)$ to only a quadratic gap.