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Data Structures and Algorithms
with Object-Oriented Design Patterns in C# |
The simplest and, perhaps, the best known of the exchange sorts
is the bubble sort .
Figure 
shows the operation of bubble sort.
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 class.
The BubbleSorter class extends the AbstractSorter class
defined in Program .
It simply provides an implementation for the no-arg Sort method.
Program: BubbleSorter class Sort method.
The outer loop (lines 5-8) 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 6-8).
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 7-8).
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 .
Copyright © 2001 by Bruno R. Preiss, P.Eng. All rights reserved.
