Chapter 3 - Exercises - Page 142: 12

In each iteration of the bubble sort, the bubble sort compares each element with the adjacent one, and thus filters the largest element in the unsorted part to the end of the unsorted array. Thus, in the first iteration, when the number of elements in the unsorted array is $(\mathrm{n}),$ bubble sort compares element $1$ to $2,2$ to $3,$ and so on to $(n-1)$ to $n,$ and thus does $n-1$ comparisons. Similarly, the second iterations do $(n-2)$ comparisons, and thus, the total number of comparisons is: $C=(n-1)+(n-2)+(n-3)+\ldots .+1=\frac{(n-1)(n)}{2}=\Theta\left(n^{2}\right)$

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