(1)

Structure: There is an 𝒪(m) size 2-level DAG structure that stores all minimum+1 (s,t)-cuts of G such that each minimum+1 (s,t)-cut appears as 3-transversal cut—it intersects any path in this structure at most thrice. We also show that there is an 𝒪(mn) size structure for storing and characterizing all minimum+1 (s,t)-cuts in terms of 1-transversal cuts.

(2)

Data structure: There exists an 𝒪(n²) size data structure that, given a pair of vertices {u,v} that are not separated by an (s,t)-mincut, can determine in 𝒪(1) time if there exists a minimum+1 (s,t)-cut, say (A,B), such that s,u ∊ A and v,t∊ B; the corresponding cut can be reported in 𝒪(|B|) time.

(3)

Sensitivity oracle: There exists an 𝒪(n²) size data structure that solves the dual-edge sensitivity problem for (s,t)-mincuts. It takes 𝒪(1) time to report the capacity of a resulting (s,t)-mincut (A,B) and 𝒪(|B|) time to report the cut.

(4)

Lower bounds: For the data structure problems addressed in results (2) and (3) above, we also provide a matching conditional lower bound. We establish a close relationship among three seemingly unrelated problems—all-pairs directed reachability problem, the dual-edge sensitivity problem for (s,t)-mincuts, and the problem of reporting the capacity of ({x,y}, {u,v})-mincut for any four vertices x,y,u,v in G. Assuming the Directed Reachability Hypothesis by Patrascu [SIAM J. Computing, 827–847 (2011)] and Goldstein et al. [WADS, 421–436 (2017)], this leads to \(\tilde{\Omega }(n^2)\) lower bounds on the space for the latter two problems.

(1)

结构：有一个 𝒪( m ) 大小的 2 级 DAG 结构，存储G的所有最小+1 (s,t)割，使得每个最小+1 ( s,t ) 割表现为 3 横向割 —它与该结构中的任何路径最多相交三次。我们还表明，存在一个 𝒪( mn ) 尺寸结构，用于存储和表征所有最小+1 (s,t)切割的 1 横向切割。

(2)

数据结构：存在一个 𝒪( n ² ) 大小的数据结构，给定一对没有被 ( s, t )-mincut 分隔的顶点 {u,v}，可以在 𝒪(1) 时间内确定是否存在存在最小+1 ( s,t )-割，例如 ( A,B )，使得s,u ∊ A和v,t∊ B；相应的削减可以在𝒪(| B |)时间内报告。

(3)

灵敏度预言机：存在一个 𝒪( n ^{2 ) 大小的数据结构，可以解决}(s,t) -mincuts的双边缘灵敏度问题。需要 𝒪(1) 时间来报告生成的(s,t) -mincut (A,B)的容量，并花费 𝒪(| B |) 时间来报告切割。

(4)

下界：对于上面结果（2）和（3）中解决的数据结构问题，我们还提供了匹配的条件下界。我们在三个看似无关的问题之间建立了密切的关系——所有对定向可达性问题、( s,t )-mincuts的双边缘敏感性问题以及报告 ({ x,y }, { u )的容量问题,v })- G中任意四个顶点x,y,u,v 的最小割。假设 Patrascu [SIAM J.Computing, 827–847 (2011)] 和 Goldstein 等人提出的定向可达性假设。[WADS, 421–436 (2017)]，这导致了后两个问题的空间下限 \(\tilde{\Omega }(n^2)\)。