The Ultimate Heapsort

A variant of Heapsort|named Ultimate Heapsort|is presented that sorts n elements in-place in (n log 2 (n + 1)) worst-case time by performing at most n log 2 n + (n) key comparisons and n log 2 n + (n) element moves. The secret behind Ultimate Heapsort is that it occasionally transforms the heap it operates with to a two-layer heap which keeps small elements at the leaves. Basically, Ultimate Heapsort is like Bottom-Up Heapsort but, due to the two-layer heap property, an element taken from a leaf has to be moved towards the root only O(1) levels, on an average. Let a1::n] be an array of n elements each consisting of a key and some information associated with this key. This array is a (maximum) heap if, for all i 2 f2; : : :; ng, the key of element abi=2c] is larger than or equal to that of element ai]. That is, a heap is a pointer-free representation of a left complete binary tree, where the elements stored are partially ordered according to their keys. Element a1] with the largest key is stored at the root. Elements abi=2c], a2i] and a2i + 1] (if they exist) are respectively stored at the parent, the left child and the right child of the node at which element ai] is stored. Heapsort is a classical sorting method that is described in almost all algo-rithmic textbooks. Heapsort sorts the given elements in ascending order with respect to their keys as follows: Input: Array a1::n] of n elements. Output: The elements in a1::n] rearranged in sorted order, i.e., the key of ai] should be smaller than or equal to that of ai+1], for all i 2 f1; 2; : : :; n?1g. 1. Rearrange a1::n] into a heap. 2. for i n step-1 until 2 do Exchange a1] and ai]. Remake a1::i ? 1] into a heap. According to Williams 29], who invented Heapsort and the heap data structure , Heapsort is a marriage of Treesort developed by Floyd 11] and Tournament-sort described, for instance, by Iverson 16, Section 6.4] (see also 13]). Both Treesort and Tournament-sort run in (n log 2 (n+1)) worst-case time and carry out n log 2 n + (n) key comparisons and n log 2 n + (n) element moves. For example, Treesort uses 2n + (1) pointers and 2n additional locations for elements , whereas Heapsort requires …