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**1 - 2**of**2**### Dynamic 2-Connectivity With Backtracking

, 1998

"... . We give algorithms and data structures that maintain the 2-edge and 2-vertexconnected components of a graph under insertions and deletions of edges and vertices, where deletions occur in a backtracking fashion (i.e., deletions undo the insertions in the reverse order). Our algorithms ru ..."

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.<F3.83e+05> We give algorithms and data structures that maintain the 2-edge and 2-vertexconnected components of a graph under insertions and deletions of edges and vertices, where deletions occur in a backtracking fashion (i.e., deletions undo the insertions in the reverse order). Our algorithms run in #(log<F3.054e+05><F3.83e+05> n) worst-case time per operation and use<F3.054e+05><F3.83e+05> #(n) space, where<F3.054e+05> n<F3.83e+05> is the number of vertices. Using our data structure we can answer queries, which ask whether vertices<F3.054e+05> u<F3.83e+05> and<F3.054e+05> v<F3.83e+05> belong to the same 2-connected component, in #(log<F3.054e+05><F3.83e+05> n) worst-case time.<F4.005e+05> Key words.<F3.83e+05> dynamic graph algorithms, backtracking<F4.005e+05> AMS subject classifications.<F3.83e+05> 68Q20, 68Q25<F4.005e+05> PII.<F3.83e+05> S0097539794272582<F5.251e+05> 1. Introduction.<F4.483e+05> Dynamic graph problems have been studied extensively in the last several years. Rou...

### Backtracking

"... Contents 1 Introduction 3 2 Models of computation 6 3 The Set Union Problem 9 4 The Worst--Case Time Complexity of a Single Operation 15 5 The Set Union Problem with Deunions 18 6 Split and the Set Union Problem on Intervals 22 7 The Set Union Problem with Unlimited Backtracking 26 1 Introduction A ..."

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Contents 1 Introduction 3 2 Models of computation 6 3 The Set Union Problem 9 4 The Worst--Case Time Complexity of a Single Operation 15 5 The Set Union Problem with Deunions 18 6 Split and the Set Union Problem on Intervals 22 7 The Set Union Problem with Unlimited Backtracking 26 1 Introduction An equivalence relation on a finite set S is a binary relation that is reflexive symmetric and transitive. That is, for s; t and u in S, we have that sRs, if sRt then tRs, and if sRt and tRu then sRu. Set S is partitioned by R into equivalence classes where each class cointains all and only the elements that obey R pairwise. Many computational problems involve representing, modifying and tracking the evolution of equivalenc