We study directed graph problems in the semi-streaming model where we are restricted to $\widetilde{O}(n):=O(n\cdot \text{polylog}(n))$ space for processing the edge stream of an $n$-node graph and can make one or a few passes to solve the given problem. A longstanding open problem in multipass streaming is determining the number of semi-streaming passes needed to solve directed $s,t$-reachability: there is an exponential gap---$Ω(\log n/\log \log n)$ vs $n^{1/2+o(1)}$---between the best-known lower and upper bounds. Several related fundamental digraph problems, e.g., topological sorting, acyclicity testing, and decomposition into strongly connected components (SCCs), are known to have the same semi-streaming pass lower bound. However, no $o(n/\text{polylog}(n))$-pass upper bound was known for any of them. Our main result achieves an $n^{1/2+o(1)}$-pass semi-streaming algorithm for each of these problems. As a corollary, we get algorithms with the same pass and space usage for a broad class of DAG problems that admit a certain form of dynamic program over a topological ordering. This class includes weighted single-source shortest paths, longest paths, max-outdegree+1 coloring, and constructing a lexicographically-first DFS (depth-first search) forest. Our algorithm for the last problem can be seen as a generalization of our results on topological sorting and acyclicity testing: recall that standard textbook algorithms for these problems use DFS as a subroutine. Our algorithms are optimal up to an $n^{o(1)}$ factor in passes under the reasonable conjecture that $s,t$-reachability requires $Ω(\sqrt{n})$ semi-streaming passes. However, using the fact that our algorithms can be parameterized by the number of passes taken by topological sorting or single-source reachability algorithms, we prove stronger lower bounds for some of the problems even under weaker conjectures.

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