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In the limit as approaches infinity, the Baik-Deift-Johansson theorem says, that the length of the longest increasing subsequence of a randomly permuted sequence of items has a distribution approaching the Tracy–Widom distribution, the distribution of the largest eigenvalue of a random matrix in the Gaussian unitary ensemble.
Related problems include approximate sorting (sorting a sequence to within a certain amount of the correct order), partial sorting (sorting only the k smallest elements of a list, or finding the k smallest elements, but unordered) and selection (computing the kth smallest element). These can be solved inefficiently by a total sort, but more ...
This is a list of articles about prime numbers.A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers.
Python's standard library includes heapq.nsmallest and heapq.nlargest functions for returning the smallest or largest elements from a collection, in sorted order. The implementation maintains a binary heap , limited to holding k {\displaystyle k} elements, and initialized to the first k {\displaystyle k} elements in the collection.
The Art of Computer Programming, Volume 3: Sorting and Searching, Third Edition. Addison–Wesley, 1997. ISBN 0-201-89685-0. Pages 138–141 of Section 5.2.3: Sorting by Selection. Anany Levitin. Introduction to the Design & Analysis of Algorithms, 2nd Edition. ISBN 0-321-35828-7. Section 3.1: Selection Sort, pp 98–100. Robert Sedgewick.
A standard order is often called ascending (corresponding to the fact that the standard order of numbers is ascending, i.e. A to Z, 0 to 9), the reverse order descending (Z to A, 9 to 0). For dates and times, ascending means that earlier values precede later ones e.g. 1/1/2000 will sort ahead of 1/1/2001.
Shell sort has distinctly improved running times in practical work, with two simple variants requiring O(n 3/2) and O(n 4/3) running time. [ 5 ] [ 6 ] If the cost of comparisons exceeds the cost of swaps, as is the case for example with string keys stored by reference or with human interaction (such as choosing one of a pair displayed side-by ...
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.