Selection from heaps, row-sorted matrices and X+Y using soft heaps

02/20/2018

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by   Haim Kaplan, et al.

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We use soft heaps to obtain simpler optimal algorithms for selecting the k-th smallest item, and the set of k smallest items, from a heap-ordered tree, from a collection of sorted lists, and from X+Y, where X and Y are two unsorted sets. Our results match, and in some ways extend and improve, classical results of Frederickson (1993) and Frederickson and Johnson (1982). In particular, for selecting the k-th smallest item, or the set of k smallest items, from a collection of m sorted lists we obtain a new optimal "output-sensitive" algorithm that performs only O(m+∑_i=1^m (k_i+1)) comparisons, where k_i is the number of items of the i-th list that belong to the overall set of k smallest items.