The Set Union Problem With Unlimited Backtracking

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(lognj loglogn) worst-case time and each backtrack(i) in 0(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(lognj 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.