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Quicksort must store a constant amount of information for each nested recursive call. Since the best case makes at most O(log n) nested recursive calls, it uses O(log n) space. However, without Sedgewick's trick to limit the recursive calls, in the worst case quicksort could make O(n) nested recursive calls and need O(n) auxiliary space.
For example, the best case for a simple linear search on a list occurs when the desired element is the first element of the list. Development and choice of algorithms is rarely based on best-case performance: most academic and commercial enterprises are more interested in improving average-case complexity and worst-case performance. Algorithms ...
The worst-case scenario occurs when all the elements are placed in a single bucket. The overall performance would then be dominated by the algorithm used to sort each bucket, for example O ( n 2 ) {\displaystyle O(n^{2})} insertion sort or O ( n log ( n ) ) {\displaystyle O(n\log(n))} comparison sort algorithms, such as merge sort .
Together with its modest O(log n) space usage, quicksort is one of the most popular sorting algorithms and is available in many standard programming libraries. The important caveat about quicksort is that its worst-case performance is O( n 2 ); while this is rare, in naive implementations (choosing the first or last element as pivot) this ...
For example, bubble sort and timsort are both algorithms to sort a list of items from smallest to largest. Bubble sort organizes the list in time proportional to the number of elements squared ( O ( n 2 ) {\textstyle O(n^{2})} , see Big O notation ), but only requires a small amount of extra memory which is constant with respect to the length ...
Third, average-case complexity allows discriminating the most efficient algorithm in practice among algorithms of equivalent best case complexity (for instance Quicksort). Average-case analysis requires a notion of an "average" input to an algorithm, which leads to the problem of devising a probability distribution over inputs.
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.
For example, if every element is at a position that differs by at most k (k ≥ 1) from the position it is going to end up in, the complexity of cocktail shaker sort becomes (). The cocktail shaker sort is also briefly discussed in the book The Art of Computer Programming, along with similar refinements of bubble sort. In conclusion, Knuth ...