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Suggested ways of improving Quickselect (#23850)

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Koustav Chowdhury
2019-02-24 09:16:00 +05:30
committed by Tom
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guide/english/algorithms/quickselection-algorithm/index.md
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@ -84,4 +84,4 @@ algorithm Hoare(A, lo, hi) is
## Time complexity
Like quicksort, the quickselect has good average performance, but is sensitive to the pivot that is chosen. If good pivots are chosen, meaning ones that consistently decrease the search set by a given fraction, then the search set decreases in size exponentially and by induction (or summing the geometric series) one sees that performance is linear, as each step is linear and the overall time is a constant times this (depending on how quickly the search set reduces). However, if bad pivots are consistently chosen, such as decreasing by only a single element each time, then worst-case performance is quadratic: O(n^2). This occurs for example in searching for the maximum element of a set, using the first element as the pivot, and having sorted data.
Like quicksort, the quickselect has good average performance, but is sensitive to the pivot that is chosen. If good pivots are chosen, meaning ones that consistently decrease the search set by a given fraction, then the search set decreases in size exponentially and by induction (or summing the geometric series) one sees that performance is linear, as each step is linear and the overall time is a constant times this (depending on how quickly the search set reduces). However, if bad pivots are consistently chosen, such as decreasing by only a single element each time, then worst-case performance is quadratic: O(n^2). This occurs for example in searching for the maximum element of a set, using the first element as the pivot, and having sorted data. There are ways of improving the time complexity. One of them is to use a random pivot. It doesn't guarantee a O(nlogn) in the worst case, but in the average case it is O(nlogn). There is also another method to guarantee O(nlogn) performance. Dividing the array in groups of five, and futher dividing the median of these groups into groups of five, and so on, unless the size of the groups of the median becomes small enough so that median is calculated in O(1) time. Select the residual median element and make it the pivot. It does guarantee a O(nlogn) and the proof of this is given in CLRS.