Stable Minimum Space Partitioning in Linear Time

In the stable 0-1 sorting problem the task is to sort an array of n elements with two distinct values such that equal elements retain their relative input order. Recently, Munro, Raman and Salowe [BIT 1990] gave an algorithm which solves this problem in O(nlog*n)3 time and constant extra space. We show that by a modification of their method the stable 0-1 sorting is possible in O(n) time and O(1) extra space. Stable three-way partitioning can be reduced to stable 0-1 sorting. This immediately yields a stable minimum space quicksort, which sorts multisets in asymptotically optimal time with high probability. CR categories: E.5, F.2.2. The stable 0-1 sorting problem is defined as follows: Given an array of n elements and a function f mapping each element to the set {0,1}, the task is to rearrange the elements such that all elements, whose f-value is zero, become before elements, whose f-value is one. Moreover, the relative order of elements with equal f-values should be maintained. For the sake of simplicity, we hereafter refer to bits instead of the f-values of elements. Stable partitioning is a special case of stable 0-1 sorting, where the f-values are obtained by comparing every element xi to some pivot element xj (which will not take part in partitioning): 1 Department of Computer Science, University of Copenhagen, Universitetsparken 1, DK-2100 COPENHAGEN EAST, Denmark, e-mail: [email protected]. On leave from Department of Computer Science, University of Turku, Lemminkäisenkatu 14 A, SF-20520 TURKU, Finland. 2 Department of Computer Science, University of Turku, Lemminkäisenkatu 14 A, SF-20520 TURKU, Finland. 3 log*n is defined as 1+log*(log n), if n>2, and otherwise it is 1.