Improvements in Bubblesort

Which distance measures improve in BUBBLESORT? If we can find one that decreases steadily, then we can prove that BUBBLESORT is correct, since it will eventually work its way down to zero (and therefore a sorted list).

Which measures decrease?

• unsortedp: That's what we're trying to find out!
• inversions: Yes! Each time a swap occurs, inversions decreases by one.

  

  Therefore, total time, because we know there's at least one swap per iteration of the ``while'' loop, and at most inversions to start.

• adjacent inversions: Not really. Each swap fixes one adjacent inversion, but might introduce 2 new ones:
  • 2 3 8-1 7
  • 2 3-1 8-7
• items out of place: Yes! After each execution of the entire ``for'' loop, the largest unsorted item moves into place. Therefore, no more than n times through the while loop (or time all together).
  • 2-3 8 1 7
  • 2 3-8 1 7
  • 2 3 1-8 7
  • 2 3 1 7-8
• insertion index: Yes. Must decrease by at least one (but perhaps more!).