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Therefore, in the general case the convex hull of n points cannot be computed more quickly than sorting. The standard Ω( n log n ) lower bound for sorting is proven in the decision tree model of computing, in which only numerical comparisons but not arithmetic operations can be performed; however, in this model, convex hulls cannot be computed ...
Best, worst and average case behavior in terms of the size of the list. For typical serial sorting algorithms, good behavior is O(n log n), with parallel sort in O(log 2 n), and bad behavior is O(n 2). Ideal behavior for a serial sort is O(n), but this is not possible in the average case. Optimal parallel sorting is O(log n).
In fact all bounds (lower and upper) currently known for the average case are precisely matched by this lower bound. For example, this gives the new result that the Janson-Knuth upper bound is matched by the resulting lower bound for the used increment sequence, showing that three pass Shellsort for this increment sequence uses Θ ( N 23 / 15 ...
A comparison sort must have an average-case lower bound of Ω(n log n) comparison operations, [2] which is known as linearithmic time. This is a consequence of the limited information available through comparisons alone — or, to put it differently, of the vague algebraic structure of totally ordered sets.
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.
For a given list of items the number () denotes the number of bins used when algorithm is applied to list , while () denotes the optimum number for this list. The absolute worst-case performance ratio R A {\displaystyle R_{A}} for an algorithm A {\displaystyle A} is defined as
A bidirectional variant of selection sort (called double selection sort or sometimes cocktail sort due to its similarity to cocktail shaker sort) finds both the minimum and maximum values in the list in every pass. This requires three comparisons per two items (a pair of elements is compared, then the greater is compared to the maximum and the ...
If A[i] < B[j], then the algorithm copies A[i] into C[k] and increases i and k. Otherwise, the algorithm copies B[j] into C[k] and increases j and k. A special case arises if either i or j have reached the end of A or B. In this case the algorithm copies the remaining elements of B or A into C and terminates.