Fault-Tolerant All-Pairs Mincuts
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Let G=(V,E) be an undirected unweighted graph on n vertices and m edges. We address the problem of fault-tolerant data structure for all-pairs mincuts in G defined as follows. Build a compact data structure that, on receiving a pair of vertices s,t∈ V and any edge (x,y) as query, can efficiently report the value of the mincut between s and t upon failure of the edge (x,y). To the best of our knowledge, there exists no data structure for this problem which takes o(mn) space and a non-trivial query time. We present two compact data structures for this problem. - Our first data structure guarantees O(1) query time. The space occupied by this data structure is O(n^2) which matches the worst-case size of a graph on n vertices. - Our second data structure takes O(m) space which matches the size of the graph. The query time is O(min(m,n c_s,t)) where c_s,t is the value of the mincut between s and t in G. The query time guaranteed by our data structure is faster by a factor of Ω(√(n)) compared to the best known algorithm to compute a (s,t)-mincut.
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