1. Quicksort Lecture 4

2. Quicksort

3. Divide and Conquer

4. Partitioning Subroutine

5. Example of Partitioning

17. Running time for PARTITION The running time of PARTITION on the subarray A[p...r] is where n=r-p+1

18. Pseudo code for Quicksort

19. Analysis of Quicksort

20. Worst-case of quicksort

21. Worst-case decision tree

27. Worst-case analysis For the worst case, we can write the recurrence equation as We guess that

28. Worst-case analysis This expression achieves a maximum at either end points: q=0 or q=n-1. Using the maximum of T(n) we have Thus

29. Worst-case analysis We can also show that the recurrence equation as Has a solution of We guess that

30. Worst-case analysis Using the maximum of T(n) we have We can pick the constant c 1 large enough so that and Thus the worst case running time of quicksort is

31. Best-case analysis

32. Analysis of almost best-case

37. Best-case analysis For the best case, we can write the recurrence equation as We guess that

38. Best-case analysis

39. This expression achieves a minimum at Using the minimum of T(n) we have

40. More intuition

41. Randomized quicksort

42. Standard Problematic Algorithm :

43. Randomized quicksort R ANDOMIZED-PARTITION (A, p, r) 1i←RANDOM(p, r) 2exchange A[r]↔A[i] 3return PARTITION(A, p, r) RANDOMIZED-QUICKSORT (A,p,r) 1if p<r 2then q← R ANDOMIZED-PARTITION (A, p, r) RANDOMIZED-QUICKSORT (A, p, q-1) RANDOMIZED-QUICKSORT (A, q+1, r)

44. Randomized quicksort

45. Randomized quicksort analysis

46. Calculating Expectation

53. Substitution Method

57. Quicksort in practice