@MISC{Twigg08worst-casetime, author = {Andrew Twigg}, title = {Worst-case time decremental connectivity and k-edge witness}, year = {2008} }

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Abstract

We give a simple algorithm for decremental graph connectivity that handles edge deletions in worst-case time O(k log n) and connectivity queries in O(log k), where k is the number of edges deleted so far, and uses worst-case space O(m 2). We use this to give an algorithm for k-edge witness (“does the removal of a given set of k edges disconnect two vertices u, v?”) with worst-case time O(k 2 log n) and space O(k 2 n 2). For k = o ( √ n) these improve the worst-case O ( √ n) bound for deletion due to Eppstein et al. We also give a decremental connectivity algorithm using O(n 2 log n/log log n) space, whose time complexity depends on the toughness and independence number of the input graph. Finally, we show how to construct a distributed data structure for k-vertex witness by giving a labeling scheme. This is the first data structure for k-vertex witness that can efficiently distributed without just giving each vertex a copy of the whole structure. Its complexity depends on being able to construct a linear layout with good properties.