Sorting multisets stably in minimum space

Abstract

In a decision tree model, Ω(n log₂ n − ∑ ⁱ⁼¹_m n _i log₂ n _i + n) is known to be a lower bound for sorting a multiset of size n containing m distinct elements, where the ith distinct element appears n _t times. We present a minimum space algorithm that sorts stably a multiset in asymptotically optimal 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 implemention, we develop linear-time in-place algorithms for the following problems, which have interest of their own: Stable unpartitioning: Assume that an n-element array A is stably partitioned into two subarrays A ₀ and A ₁. The problem is to recover A from its constituents A ₀ and A ₁. The information available is the partitioning element used and a bit array of size n indicating whether an element of A ₀ or A ₁ was originally in the corresponding position of A.

Stable selection: The task is to find the kth smallest element in a multiset of n elements such that the relative order of identical elements is retained.