Abstract
We consider the problem of sorting a multiset of sizen containingm distinct elements, where theith distinct element appearsn i 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 − ∑ m i=1 n i logn i +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 subarraysA 0 andA 1. The problem is to recoverA from its constitutentsA 0 andA 1. The information available is the partitioning element used and a bit array of sizen indicating whether an element ofA 0 orA 1 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.
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Katajainen, J., Pasanen, T. Sorting multisets stably in minimum space. Acta Informatica 31, 301–313 (1994). https://doi.org/10.1007/BF01178508
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DOI: https://doi.org/10.1007/BF01178508