Quicksort for Equal Keys

When sorting a multiset of N elements with n ≪ N distinct values, considerable savings can be obtained from an algorithm which sorts in time O(N log n) rather than O(N log N). In a previous paper, two Quicksort derivatives operating on linked lists were introduced which are stable, i.e., maintain the relative order of equal keys, and which achieve the previously unattainable lower bound in partition exchange sorting. Here, six more algorithms are presented which are unstable but operate on arrays. One of the algorithms is also designed for a speedup on presorted input. The algorithms are analyzed and compared to potential contenders in multiset sorting.