# Efficient equality-testing and updating of sets

## Abstract

This paper is concerned with data structures and algorithms for managing an arbitrary number of sets such that we can dynamically update each individual set and test whether any two sets are equal. Previous schemes can support set equality-testing in constant time and an update operation (i.e. insert or delete an element) in time O(log^{2}*m*) [7, 5] or O(log m log^{*}*m*) [2], where *m* is the number of insert operations performed. Note that *m* is an upper bound of *n*, the total size of the sets, but maybe a loose one. When we have performed a lot of delete operations, having few elements left in the sets, it is natural to expect the operations to be performed faster. Yet existing schemes are not favored when *n* is much smaller than *m*. It is desirable to have a scheme whose performance is in terms of *n* instead of *m*.

This paper presents a new scheme which is more dynamic in nature and supports each insert or delete operation in O(log *n*) time, while maintaining the constant time complexity of set equality-testing.

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