1. Quicksort Lecture 3 Prof. Dr. Aydın Öztürk

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 This expression achieves a maximum at either end points: q=0 or q=n-1.

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

43. 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. Calculating Expectation(Alternative approach) Example: Sort the following 10 distinct values: 5, 9, 3, 10, 11, 14, 8, 4, 17, 6 Pivot: 10 {5, 9, 3, 8, 4, 6}, 10, {11, 14, 17} Pivot: 6 {5, 3, 4}, 6, {9, 8}, 10, {11, 14, 17} Pivot: 4 {3}, 4, {5}, 6, {9, 8}, 10, {11, 14, 17}

58. Calculating Expectation(Alternative approach) Let X be the total number of comparisons performed in all calls to PARTITION. Let being the ith smallestelement of the array A. We define the set Let also

59. Calculating Expectation(Alternative approach) To determine the prob. that and are ever compared, we note that the values will initially in the same bracket and will remain in the same bracket if the number chosen for the first comparison is not between and. For example, if the comparison number is larger than, then all the values will go in a bracket to the left of the comparison number, and if it is smaller than, then they will all go in a bracket to the right. Thus all the values will remain in the same bracket until the first time that one of them chosen as a comparison value. At that point all the other values between i and j will be compared with this comparison value. If this comparison value is neither i or j then when compared with it, i will go into a left bracket and j will go into right bracket, consequently i and j will never be compared. If the comparison value of the set is either i or j then there will be a direct comparison between i and j.

60. Calculating Expectation(Alternative approach) The total number number of comparison performed by the algorithm

61. Calculating Expectation(Alternative approach) Taking expectation of both sides

62. Calculating Expectation(Alternative approach) Taking expectation of both sides

63. Calculating Expectation(Alternative approach)

64. Quicksort in practice