Table of Contents

1 Preliminaries

There are many sorting algorithms, some better than others. There exist strictly better algorithms than bubble sort (i.e., faster, lower-power, same-space, and simpler), the inefficiency of bubble sorts makes a proof of its correctness interesting.

2 Algorithm

Let an element in a list be considered out of order if it is followed in the list by another value strictly smaller than itself.

An element that is not out of order is said to be in order.

Observe that by this definition, the last element in the list cannot be out of order, but theoretically all other elements can. That theory can be realized in practice by a list of unique elements sorted in descending order, with the first element being the largest and the last the smallest.

Consider the following algorithm:

Given

A list x with n elements, x_1 through x_n

Do

repeat the following until x contains no out-of-order elements
for each i from 1 to n-1
if x_i is out of order then
swap x_i and x_{i+1}
end if
end for
end repeat

3 Proof of Correctness

Given any finite list x, the above algorithm will terminate after no more than n-1 iterations of the outer loop.

To prove this theorem, we first define a fixed tail of a list and then prove a lemma about the inner loop.

A fixed tail of a list is a (possibly empty) suffix of the list such that both (a) no element outside the tail is larger than any element inside the tail and (b) all elements in the tail are in order.

The maximal fixed tail of a list is the largest suffix of the list that is a fixed tail. The portion of a list that is not part of it’s maximal fixed tail is the body of the list.

Each run of the inner loop of the algorithm (lines 2–6) increases the length of the list’s maximal fixed tail by at least one.

Consider the largest element of the body of the list (if there are several elements of that value, consider the one with the largest index). Because no element before it is larger than it, the condition on line 3 will keep line 4 from moving it to an earlier index. Because it is larger than every element in the body after it, line 4 will keep pushing it to the next position until it reaches the end of the body. Because every element of the original maximal fixed tail is at least as large as any element of the body, the element will then be in order and stop moving. Because a fixed tail is in order, none of its elements will move either. Because the new element has no elements outside the tail that is larger than it, and it is in order, it becomes part of a new, larger, maximal fixed tail.

With this lemma, we can now prove the original theorem

The list x initially has a maximal fixed tail of at least 0 elements. After each iteration of the outer loop, the maximal fixed tail increases in length by at least 1 element; thus, after n-1 iterations the maximal fixed tail must include all but the first element of the list. By definition, all elements in the fixed tail are in order, so it remains to show only that the first element of the list is also in order. Because the element following the first element is in a fixed tail, the element following the first element cannot be smaller than the first element. Hence, the first element is also in order and the outer loop will terminate.