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Instead of partitioning into two subarrays using a single pivot, multi-pivot quicksort (also multiquicksort [21]) partitions its input into some s number of subarrays using s − 1 pivots. While the dual-pivot case ( s = 3 ) was considered by Sedgewick and others already in the mid-1970s, the resulting algorithms were not faster in practice ...
For example, the worst-case occurs when pivoting on the smallest element at each step, such as applying quickselect for the maximum element to already sorted data and taking the first element as pivot each time. If one instead consistently chooses "good" pivots, this is avoided and one always gets linear performance even in the worst case.
Quickselect uses the same overall approach as quicksort, choosing one element as a pivot and partitioning the data in two based on the pivot, accordingly as less than or greater than the pivot. However, instead of recursing into both sides, as in quicksort, quickselect only recurses into one side – the side with the element it is searching for.
For example, if at each step the median is chosen as the pivot then the algorithm works in O(n log n). Finding the median, such as by the median of medians selection algorithm is however an O( n ) operation on unsorted lists and therefore exacts significant overhead with sorting.
Animation of the average-case version of quicksort, with recursive subproblems indicated by shaded arrows and with pivots (red items and blue lines) chosen as the last item in each subproblem. The quicksort algorithm for sorting a set of items can be analyzed using the harmonic numbers. The algorithm operates by choosing one item as a "pivot ...
As a baseline algorithm, selection of the th smallest value in a collection of values can be performed by the following two steps: . Sort the collection; If the output of the sorting algorithm is an array, retrieve its th element; otherwise, scan the sorted sequence to find the th element.
Tree sort can be used as a one-time sort, but it is equivalent to quicksort as both recursively partition the elements based on a pivot, and since quicksort is in-place and has lower overhead, tree sort has few advantages over quicksort. It has better worst case complexity when a self-balancing tree is used, but even more overhead.
The heapsort algorithm can be divided into two phases: heap construction, and heap extraction. The heap is an implicit data structure which takes no space beyond the array of objects to be sorted; the array is interpreted as a complete binary tree where each array element is a node and each node's parent and child links are defined by simple arithmetic on the array indexes.