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Marked Ancestor Problems
, 1998
"... Consider a rooted tree whose nodes can be marked or unmarked. Given a node, we want to find its nearest marked ancestor. This generalises the wellknown predecessor problem, where the tree is a path. ..."
Abstract

Cited by 52 (7 self)
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Consider a rooted tree whose nodes can be marked or unmarked. Given a node, we want to find its nearest marked ancestor. This generalises the wellknown predecessor problem, where the tree is a path.
Worstcase and Amortised Optimality in UnionFind (Extended Abstract)
, 1999
"... We study the interplay between worstcase and amortised time bounds for the classic Disjoint Set Union problem (UnionFind). We ask whether it is possible to achieve optimal worstcase and amortised bounds simultaneously. Furthermore we would like to allow a tradeoff between the worstcase time for ..."
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Cited by 1 (0 self)
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We study the interplay between worstcase and amortised time bounds for the classic Disjoint Set Union problem (UnionFind). We ask whether it is possible to achieve optimal worstcase and amortised bounds simultaneously. Furthermore we would like to allow a tradeoff between the worstcase time for a query and for an update. We answer this question by first providing lower bounds for the possible worstcase time tradeoffs, as well as lower bounds which show where in this tradeoff range optimal amortised time is achievable. We then give an algorithm which tightly matches both lower bounds simultaneously. The lower bounds are provided in the cellprobe model as well as in the algebraic realnumber RAM, and the upper bounds hold for a RAM with logarithmic word size and a modest instruction set. Our lower bounds show that for worstcase query and update time t q and t u respectively, one must have t q = 780 n= log t u ), and only for t q (m; n) can this tradeoff be achieved simultaneou...
Backtracking
"... Contents 1 Introduction 3 2 Models of computation 6 3 The Set Union Problem 9 4 The WorstCase 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 WorstCase 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