The Complexity of Identifying Large Equivalence Classes

We prove that at least (3k−4) / k(2k−3) n(n-1)/2 − O(k) equivalence tests and no more than 2/k n(n-1)/2 + O(n) equivalence tests are needed in the worst case to identify the equivalence classes with at least k members in set of n elements. The upper bound is an improvement by a factor 2 compared to known results. For k = 3 we give tighter bounds. Finally, for k > n/2 we prove that it is necessary and it suffices to make 2n − k − 1 equivalence tests which generalizes a known result for k = [(n+1)/2] .