Exercise 7.2.4

Banks often record transactions on an account in order of the times of the transactions, but many people like to receive their bank statements with checks listed in order by check numbers. People usually write checks in order by check number, and merchants usually cash the with reasonable dispatch. The problem of converting time-of-transaction ordering to check-number ordering is therefore the problem of sorting almost-sorted input. Argue that the procedure INSERTION-SORT would tend to beat the procedure QUICKSORT on this problem.

A simple intuitive argument will suffice here.

The more sorted the array is, the less work insertion sort will do. Namely, INSERTION-SORT is $\Theta(n + d)$, where $d$ is the number of inversions in the array. In the example above the number of inversions tends to be small so insertion sort will be close to linear.

On the other hand, if PARTITION does picks a pivot that does not participate in an inversion, it will produce and empty partition. Since there is a small number of inversions, QUICKSORT is very likely to produce empty partitions.