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Sorted arrays are the most space-efficient data structure with the best locality of reference for sequentially stored data. [citation needed]Elements within a sorted array are found using a binary search, in O(log n); thus sorted arrays are suited for cases when one needs to be able to look up elements quickly, e.g. as a set or multiset data structure.
Complexities below assume n items to be sorted, with keys of size k, digit size d, and r the range of numbers to be sorted. Many of them are based on the assumption that the key size is large enough that all entries have unique key values, and hence that n ≪ 2 k , where ≪ means "much less than".
For example, when the items are inserted in sorted key order, the tree degenerates into a linked list with n nodes. The difference in performance between the two situations may be enormous: for example, when n = 1,000,000, the minimum height is ⌊ log 2 ( 1 , 000 , 000 ) ⌋ = 19 {\displaystyle \lfloor \log _{2}(1,000,000)\rfloor =19} .
For brevity, these words will have the specified meanings in the following tables (unless noted to be part of language syntax): funcN A function.
Output the first element of list i and remove it from its list. Re-heapify h . Searching for the next smallest element to be output (find-min) and restoring heap order can now be done in O (log k ) time (more specifically, 2⌊log k ⌋ comparisons [ 6 ] ), and the full problem can be solved in O ( n log k ) time (approximately 2 n ⌊log k ...
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Bubble sort has a worst-case and average complexity of (), where is the number of items being sorted. Most practical sorting algorithms have substantially better worst-case or average complexity, often O ( n log n ) {\displaystyle O(n\log n)} .
Used in Python 2.3 and up, and Java SE 7. Insertion sorts Insertion sort: determine where the current item belongs in the list of sorted ones, and insert it there; Library sort; Patience sorting; Shell sort: an attempt to improve insertion sort; Tree sort (binary tree sort): build binary tree, then traverse it to create sorted list