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Sorting a set of unlabelled weights by weight using only a balance scale requires a comparison sort algorithm. A comparison sort is a type of sorting algorithm that only reads the list elements through a single abstract comparison operation (often a "less than or equal to" operator or a three-way comparison) that determines which of two elements should occur first in the final sorted list.
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).
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 ...
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.
The next pass, 3-sorting, performs insertion sort on the three subarrays (a 1, a 4, a 7, a 10), (a 2, a 5, a 8, a 11), (a 3, a 6, a 9, a 12). The last pass, 1-sorting, is an ordinary insertion sort of the entire array (a 1,..., a 12). As the example illustrates, the subarrays that Shellsort operates on are initially short; later they are longer ...
If the elements in the problem are real numbers, the decision-tree lower bound extends to the real random-access machine model with an instruction set that includes addition, subtraction and multiplication of real numbers, as well as comparison and either division or remaindering ("floor"). [5]
Consider the example of [5, 2, 3, 1, 0], following the scheme, after the first partition the array becomes [0, 2, 1, 3, 5], the "index" returned is 2, which is the number 1, when the real pivot, the one we chose to start the partition with was the number 3. With this example, we see how it is necessary to include the returned index of the ...
A well-known lower bound for unstructured sorting, in the decision tree model, is based on the factorial number of sorted orders that an unstructured list may have. Because each comparison can at best reduce the number of possible orderings by a factor of two, sorting requires a number of comparisons at least equal to the binary logarithm of ...