Consider an unweighted, directed graph G with the diameter D. In this paper, we introduce the framework for counting cycles and walks of given length in matrix multiplication time \(\widetilde {O}(n^{\omega })\). The framework is based on the fast decomposition into Frobenius normal form and the Hankel matrix-vector multiplication. It allows us to solve the All-Nodes Shortest Cycles, All-Pairs All Walks problems efficiently and also give some improvement upon distance queries in unweighted graphs.

中文翻译：

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通过Frobenius范式改进了距离查询和周期计数。

考虑未加权，有向图G ^与直径d。在本文中，我们介绍了在矩阵乘法时间\（\ widetilde {O}（n ^ {\ omega}）\）中计算给定长度的周期和步数的框架。该框架基于快速分解为Frobenius范式和Hankel矩阵向量乘法。它使我们能够有效地解决“所有节点最短周期，所有对所有步行”的问题，并且还可以改善未加权图中的距离查询。