We prove that any two-pass graph streaming algorithm for the $s$-$t$ reachability problem in $n$-vertex directed graphs requires near-quadratic space of $n^{2-o(1)}$ bits. As a corollary, we also obtain near-quadratic space lower bounds for several other fundamental problems including maximum bipartite matching and (approximate) shortest path in undirected graphs. Our results collectively imply that a wide range of graph problems admit essentially no non-trivial streaming algorithm even when two passes over the input is allowed. Prior to our work, such impossibility results were only known for single-pass streaming algorithms, and the best two-pass lower bounds only ruled out $o(n^{7/6})$ space algorithms, leaving open a large gap between (trivial) upper bounds and lower bounds.

翻译：我们证明了在n顶点有向图中，任何用于s-t可达性问题的双通图流算法都需要近二次空间，即n^{2-o(1)}比特。作为推论，我们还获得了其他几个基本问题的近二次空间下界，包括最大二分匹配和（近似）无向图最短路径。我们的结果共同表明，即使在允许对输入进行两次遍历的情况下，广泛的图问题也基本上不存在非平凡的流算法。在我们的工作之前，这种不可能性结果仅已知于单通流算法，而最佳的双通下界仅排除了o(n^{7/6})空间算法，留下了（平凡的）上界与下界之间的巨大差距。