Data Structures and Algorithms with Object-Oriented Design Patterns in C++

## Bubble Sort

The simplest and, perhaps, the best known of the exchange sorts is the bubble sort . Figure  shows the operation of bubble sort.

Figure: Bubble Sorting

To sort the sequence , bubble sort makes n-1 passes through the data. In each pass, adjacent elements are compared and swapped if necessary. First, and are compared; next, and ; and so on.

Notice that after the first pass through the data, the largest element in the sequence has bubbled up into the last array position. In general, after k passes through the data, the last k elements of the array are correct and need not be considered any longer. In this regard the bubble sort differs from the insertion sort algorithms--the sorted subsequence of k elements is never modified (by an insertion).

Figure  also shows that while n-1 passes through the data are required to guarantee that the list is sorted in the end, it is possible for the list to become sorted much earlier! When no exchanges at all are made in a given pass, then the array is sorted and no additional passes are required. A minor algorithmic modification would be to count the exchanges made in a pass, and to terminate the sort when none are made.

Program  defines the BubbleSorter<T> class template. This class simply provides an implementation for the DoSort routine. The DoSort routine takes a reference to an Array<T> instance and sorts its elements in place. The implementation makes use of the Swap routine described in Section .

Program: BubbleSorter<T> Class DoSort Member Function Definition

The outer loop (lines 11-14) is done for . That makes n-1 iterations in total. During the iteration of the outer loop, exactly i-1 iterations of the inner loop are done (lines 12-14). Therefore, the number of iterations of the inner loop, summed over all the passes of the outer loop is

Consequently, the running time of bubble sort is .

The body of the inner loop compares adjacent array elements and swaps them if necessary (lines 13-14). This takes at most a constant amount of time. Of course, the algorithm will run slightly faster when no swapping is needed. For example, this occurs if the array is already sorted to begin with. In the worst case, it is necessary to swap in every iteration of the inner loop. This occurs when the array is sorted initially in reverse order. Since only adjacent elements are swapped, bubble sort removes inversions one at time. Therefore, the average number of swaps required is . Nevertheless, the running time of bubble sort is always .