Semi-Streaming Reachability and Shortest Paths in $n^{1/3+o(1)}$ Passes

Abstract

In the semi-streaming model, the edges of an $n$-vertex graph arrive in a stream, and an algorithm makes multiple passes over that stream while using only $\widetilde{O}(n)$ space. The number of passes needed to compute single-source reachability in directed graphs is a long-standing open question in streaming algorithms, going back to Henzinger, Raghavan, and Rajagopalan (1998). The best known semi-streaming algorithm uses $n^{1/2+o(1)}$ passes due to Jambulapati, Liu, and Sidford (2019), while the best lower bound is $Ω(\log n/\log\log n)$ passes due to Guruswami and Onak (2016). We give streaming algorithms for computing single-source reachability and shortest paths on $n$-vertex directed graphs using $\widetilde{O}(n)$ space and $n^{1/3+o(1)}$ passes. Our algorithms are based on implicit hopsets, which are large hopsets that can be represented implicitly in $\widetilde{O}(n)$ space. We introduce this notion and give new constructions for implicit hopsets with small hopbound. Along the way, we obtain new work-depth tradeoffs for parallel shortest paths in directed graphs, improving the recent result of Ashvinkumar, Bernstein, Probst Gutenberg, and Saranurak (2026).