Recently, Brand et al. [STOC 2018] gave a randomized mathcal O(4 k m ε -2 -time exponential-space algorithm to approximately compute the number of paths on k vertices in a graph G up to a multiplicative error of 1 ± ε based on exterior algebra. Prior to our work, this has been the state-of-the-art. In this article, we revisit the algorithm by Alon and Gutner [IWPEC 2009, TALG 2010], and obtain the following results: • We present a deterministic 4 k + O (√ k (log k +log 2 ε -1 )) m -time polynomial-space algorithm. This matches the running time of the best known deterministic polynomial-space algorithm for deciding whether a given graph G has a path on k vertices. • Additionally, we present a randomized 4 k +mathcal O(log k (log k +logε -1 )) m -time polynomial-space algorithm. Our algorithm is simple—we only make elementary use of the probabilistic method. Here, n and m are the number of vertices and the number of edges, respectively. Additionally, our approach extends to approximate counting of other patterns of small size (such as q -dimensional p -matchings).

中文翻译：

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k 路径的近似计数：更简单、确定性且在多项式空间中

最近，布兰德等人。[STOC 2018] 给出了一个随机的数学 O(4 ķ 米ε-2-时间指数空间算法来近似计算路径的数量ķ图中的顶点G基于外部代数，乘法误差高达 1 ± ε。在我们工作之前，这是最先进的。在本文中，我们重新审视了 Alon 和 Gutner [IWPEC 2009，TALG 2010] 的算法，并获得了以下结果： • 我们提出了一个确定性的4 ķ+○(√ķ（日志ķ+日志2ε-1)) 米-时间多项式空间算法。这火柴最著名的确定性多项式空间算法的运行时间决定是否给定图G有一条路在ķ顶点。• 此外，我们提出了一个随机的4 ķ+数学 O(logķ（日志ķ+logε-1)) 米-时间多项式空间算法。我们的算法很简单——我们只基本使用了概率方法。这里，n和米分别是顶点数和边数。此外，我们的方法扩展到对其他小尺寸模式的近似计数（例如q维p-匹配）。