16.4. Costs of Exchange Sorting¶

The running time for each of the sorts discussed so far is \(\Theta(n^2)\) in the average and worst cases. The cost summary for the Insertion Sort, Bubble Sort, and Selection Sort in terms of their required number of comparisons and swaps in the best, average, and worst cases is shown.

\[\begin{split}\begin{array}{rccc} &\textbf{Insertion}&\textbf{Bubble}&\textbf{Selection}\\ \textbf{Comparisons:}\\ \textrm{Best Case}&\Theta(n)&\Theta(n^2)&\Theta(n^2)\\ \textrm{Average Case}&\Theta(n^2)&\Theta(n^2)&\Theta(n^2)\\ \textrm{Worst Case}&\Theta(n^2)&\Theta(n^2)&\Theta(n^2)\\ \\ \textbf{Swaps:}\\ \textrm{Best Case}&0&0&\Theta(n)\\ \textrm{Average Case}&\Theta(n^2)&\Theta(n^2)&\Theta(n)\\ \textrm{Worst Case}&\Theta(n^2)&\Theta(n^2)&\Theta(n)\\ \end{array}\end{split}\]

The remaining sorting algorithms presented in this chapter are significantly better than these three under typical conditions. But before continuing on, it is instructive to investigate what makes these three sorts so slow. The crucial bottleneck is that only adjacent records are compared. Thus, comparisons and moves (for Insertion and Bubble Sort) are by single steps. Swapping adjacent records is called an exchange. Thus, these sorts are sometimes referred to as an exchange sort. The cost of any exchange sort can be at best the total number of steps that the records in the array must move to reach their “correct” location. Recall that this is at least the number of inversions for the record. An inversion occurs when a record with key value greater than the current record’s key value appears before it.

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Last 3 sections:
- Insertion Sort,
- Bubble Sort, and
- Selection Sort
Algorithm Analysis