For a directed graph $G$ with $n$ vertices and a start vertex $u_{\sf start}$, we wish to (approximately) sample an $L$-step random walk over $G$ starting from $u_{\sf start}$ with minimum space using an algorithm that only makes few passes over the edges of the graph. This problem found many applications, for instance, in approximating the PageRank of a webpage. If only a single pass is allowed, the space complexity of this problem was shown to be $\tilde{\Theta}(n \cdot L)$. Prior to our work, a better space complexity was only known with $\tilde{O}(\sqrt{L})$ passes. We settle the space complexity of this random walk simulation problem for two-pass streaming algorithms, showing that it is $\tilde{\Theta}(n \cdot \sqrt{L})$, by giving almost matching upper and lower bounds. Our lower bound argument extends to every constant number of passes $p$, and shows that any $p$-pass algorithm for this problem uses $\tilde{\Omega}(n \cdot L^{1/p})$ space. In addition, we show a similar $\tilde{\Theta}(n \cdot \sqrt{L})$ bound on the space complexity of any algorithm (with any number of passes) for the related problem of sampling an $L$-step random walk from every vertex in the graph.

中文翻译：

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在有向图上随机游走的近最优两遍流算法

对于具有$ n $个顶点和起始顶点$ u _ {\ sf start} $的有向图$ G $，我们希望（近似）对从$ u _ {\ sf start} $以最小的空间使用一种算法，该算法仅在图形的边缘进行很少的遍历。例如，该问题发现了许多应用程序，例如近似于网页的PageRank。如果仅允许一次通过，则此问题的空间复杂度显示为$ \ tilde {\ Theta}（n \ cdot L）$。在我们进行工作之前，只有通过$ \ tilde {O}（\ sqrt {L}）$通过才能知道更好的空间复杂性。我们通过两次匹配的上限和下限，解决了针对两遍流算法的此随机游走仿真问题的空间复杂度，表明它是$ \ tilde {\ Theta}（n \ cdot \ sqrt {L}）$。我们的下界参数扩展到每个不变的传递次数$ p $，并显示针对此问题的任何$ p $ -pass算法都使用$ \ tilde {\ Omega}（n \ cdot L ^ {1 / p}）$空间。另外，对于任何对$ L $采样的相关问题，我们展示了类似的$ \ tilde {\ Theta}（n \ cdot \ sqrt {L}）$约束于任何算法（具有任意通过次数）的空间复杂度图中每个顶点的随机步走。