# The Bubble Sort¶

The **bubble sort** makes multiple passes through a list. It compares
adjacent items and exchanges those that are out of order. Each pass
through the list places the next largest value in its proper place. In
essence, each item “bubbles” up to the location where it belongs.

Figure 1 shows the first pass of a bubble sort. The shaded
items are being compared to see if they are out of order. If there are
*n* items in the list, then there are \(n-1\) pairs of items that
need to be compared on the first pass. It is important to note that once
the largest value in the list is part of a pair, it will continually be
moved along until the pass is complete.

At the start of the second pass, the largest value is now in place.
There are \(n-1\) items left to sort, meaning that there will be
\(n-2\) pairs. Since each pass places the next largest value in
place, the total number of passes necessary will be \(n-1\). After
completing the \(n-1\) passes, the smallest item must be in the
correct position with no further processing required. ActiveCode 1
shows the complete `bubbleSort`

function. It takes the list as a
parameter, and modifies it by exchanging items as necessary.

The exchange operation, sometimes called a “swap,” is slightly different in Python than in most other programming languages. Typically, swapping two elements in a list requires a temporary storage location (an additional memory location). A code fragment such as

```
temp = alist[i]
alist[i] = alist[j]
alist[j] = temp
```

will exchange the ith and jth items in the list. Without the temporary storage, one of the values would be overwritten.

In Python, it is possible to perform simultaneous assignment. The
statement `a,b=b,a`

will result in two assignment statements being
done at the same time (see Figure 2). Using simultaneous
assignment, the exchange operation can be done in one statement.

Lines 5-7 in ActiveCode 1 perform the exchange of the \(i\) and \((i+1)th\) items using the three–step procedure described earlier. Note that we could also have used the simultaneous assignment to swap the items.

The following activecode example shows the complete `bubbleSort`

function working on the list
shown above.

The following animation shows `bubbleSort`

in action.

To analyze the bubble sort, we should note that regardless of how the
items are arranged in the initial list, \(n-1\) passes will be
made to sort a list of size *n*. Table 1 shows the number
of comparisons for each pass. The total number of comparisons is the sum
of the first \(n-1\) integers. Recall that the sum of the first
*n* integers is \(\frac{1}{2}n^{2} + \frac{1}{2}n\). The sum of
the first \(n-1\) integers is
\(\frac{1}{2}n^{2} + \frac{1}{2}n - n\), which is
\(\frac{1}{2}n^{2} - \frac{1}{2}n\). This is still
\(O(n^{2})\) comparisons. In the best case, if the list is already
ordered, no exchanges will be made. However, in the worst case, every
comparison will cause an exchange. On average, we exchange half of the
time.

Pass |
Comparisons |
---|---|

1 | \(n-1\) |

2 | \(n-2\) |

3 | \(n-3\) |

… | … |

\(n-1\) | \(1\) |

A bubble sort is often considered the most inefficient sorting method
since it must exchange items before the final location is known. These
“wasted” exchange operations are very costly. However, because the
bubble sort makes passes through the entire unsorted portion of the
list, it has the capability to do something most sorting algorithms
cannot. In particular, if during a pass there are no exchanges, then we
know that the list must be sorted. A bubble sort can be modified to stop
early if it finds that the list has become sorted. This means that for
lists that require just a few passes, a bubble sort may have an
advantage in that it will recognize the sorted list and stop.
ActiveCode 2 shows this modification, which is often referred
to as the **short bubble**.

Self Check