Let $\kappa(s,t)$ denote the maximum number of internally disjoint paths in an undirected graph $G$. We consider designing a data structure that includes a list of cuts, and answers the following query: given $s,t \in V$, determine whether $\kappa(s,t) \leq k$, and if so, return a pointer to an $st$-cut of size $\leq k$ (or to a minimum $st$-cut) in the list. A trivial data structure that includes a list of $n(n-1)/2$ cuts and requires $\Theta(kn^2)$ space can answer each query in $O(1)$ time. We obtain the following results. In the case when $G$ is $k$-connected, we show that $n$ cuts suffice, and that these cuts can be partitioned into $(2k+1)$ laminar families. Thus using space $O(kn)$ we can answers each min-cut query in $O(1)$ time, slightly improving and substantially simplifying a recent result of Pettie and Yin. We then extend this data structure to subset $k$-connectivity. In the general case we show that $(2k+1)n$ cuts suffice to return an $st$-cut of size $\leq k$,and a list of size $k(k+2)n$ contains a minimum $st$-cut for every $s,t \in V$. Combining our subset $k$-connectivity data structure with the data structure of Hsu and Lu for checking $k$-connectivity, we give an $O(k^2 n)$ space data structure that returns an $st$-cut of size $\leq k$ in $O(\log k)$ time, while $O(k^3 n)$ space enables to return a minimum $st$-cut.
翻译:Lets\ kapa (s, t) 美元, 如果是的话, 将一个指针返回到一个大小为 $\ leq k$( 或至少是 $- 美元) 的列表中。 一个包含 $n( n-1) / 美元 削减并需要 $( ta) (k) 美元 的数据结构, 包括一个包含 $ (n) (n) 美元 削减 和 $ (k) 的列表中内部脱节路径的最大数量 。 我们考虑设计一个包含 削减 列表的数据结构, 包括 $( t) 的削减, 并回答 $( t) (k) 削减, 并且回答 $ (k) 美元 。 当$( t) 美元 连接 $( t) 美元 (k) 时, 我们可以用 $ (k) 美元 的最小( k) 美元来解答 。