枚举小环
Enumerating Small Cycles
中文总结 AI 辅助
本研究推广Yuster等人的环检测枚举结果，提出k≤8时2k-环、任意固定k时≤2k的环及[i,2k]范围环的最优枚举算法，预处理与延迟均达Õ(n²)、Õ(1)界。
中文摘要 AI 辅助
Yuster和Zuster的开创性结果表明，对于任意固定的k，n顶点图中的偶环C_{2k}可在O(n²)时间内检测。对于4-环，民间算法可扩展至枚举：若存在任意t个不同的4-环，可在O(n²+t)时间内列出。近期Jin、Vassilevska-Williams和Zhou得到了6-环枚举的类似界。本研究将上述结果推广至8、10、12、14、16大小的环；证明对于所有k≤8，可在Õ(n²+t)时间内枚举t个不同的2k-环。实际上，本算法的预处理时间为Õ(n²)，延迟为Õ(1)。此外，对于任意固定k，本研究提出了所有大小不超过2k的环的最优枚举（因此也是枚举）算法。更一般地，对于任意固定k和3≤i≤4k/3，本研究提出了一种预处理时间为Õ(n²)、延迟为Õ(1)的算法，可枚举[i,2k]范围内所有大小的环。
英文摘要
In a seminal result of Yuster and Zwick, they showed that for any fixed $k$, the even cycle $C_{2k}$ can be detected in an $n$-vertex graph in time $O(n^2)$. For $4$-cycles, a folklore algorithm extends to listing: for any $t$, we can list $t$ different $4$-cycles, if such exist, in $O(n^2+t)$ time. Recently, Jin, Vassilevska-Williams, and Zhou obtained similar bounds for listing $6$-cycles. In this work, we generalize the above to cycles of sizes $8, 10, 12, 14,$ and $16$; we show that for all $k\leq 8$, we can list $t$ distinct $2k$-cycles in $\tilde{O}(n^2+t)$ time. In fact, our algorithm gives enumeration with pre-processing time $\tilde{O}(n^2)$ and delay $\tilde{O}(1)$. Additionally, for any fixed $k$, we present an optimal enumeration (and hence also listing) algorithm for all cycles of size at most $2k$. More generally, for any fixed $k$ and any $3\le i\le \frac{4k}{3}$, we present an algorithm with preprocessing time $\tilde{O}(n^2)$ and delay $\tilde{O}(1)$ that enumerates all cycles of sizes in the range $[i,2k]$.
