A Parallel Solution to the Extended Set Union Problem with Unlimited Backtracking
IPPS '96 Proceedings of the 10th International Parallel Processing Symposium
Algorithms and theory of computation handbook
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An extension of the disjoint set union problem is considered, where the extra primitive backtrack($i$) can undo the last $i$ unions not yet undone. Let $n$ be the total number of elements in all the sets. A data structure is presented that supports each union and find in $O(\log n/\log \log n)$ worst-case time and each backtrack($i$) in $O(1)$ worst-case time, irrespective of $i$. The total space required by the data structure is $O(n)$. A byproduct of this construction is a partially persistent data structure for the standard set union problem, capable of supporting union, find, and find-in-the-past operations, each in $O(\log n/\log \log n)$ worst-case time. All these upper bounds are tight for the class of separable-pointer algorithms as well as in the cell probe model of computation.