Sorting multisets stably in minimum space

AbstractWe consider the problem of sorting a multiset of sizen containingm distinct elements, where theith distinct element appearsni times. Under the assumption that our model of computation allows only the operations of comparing elements and moving elements in the memory,Ω(n logn − ∑i=1mni logni +n) is known to be a lower bound for the computational complexity of the sorting problem. In this paper we present aminimum space algorithm that sortsstably a multiset in asymptoticallyOptimal worst-case time. A Quicksort type approach is used, where at each recursive step the median is chosen as the partitioning element. To obtain a stable minimum space implementation, we develop linear-time in-place algorithms for the following problems, which have interest of their own: Stable unpartitioning: Assume that ann-element arrayA is stably partitioned into two subarraysA0 andA1. The problem is to recoverA from its constitutentsA0 andA1. The information available is the partitioning element used and a bit array of sizen indicating whether an element ofA0 orA1 was originally in the corresponding position ofA. Stable selection: The task is to find thekth smallest element in a multiset ofn elements such that the relative order of identical elements is retained.