Dual-Pivot Quicksort and Beyond: Analysis of Multiway Partitioning and Its Practical Potential

1. Dual-Pivot Quicksort and Beyond Analysis of Multiway Partitioning and Its Practical Potential Sebastian Wild Vortrag zur wissenschaftlichen Aussprache im Rahmen des Promotionsverfahrens 8. Juli 2016 Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 1 / 17

2. Dual-P??? Q?????? and Beyond Analysis of Multi??? P??????? and Its Practical Potential Sebastian Wild Vortrag zur wissenschaftlichen Aussprache im Rahmen des Promotionsverfahrens 8. Juli 2016 Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 1 / 17

3. Dual-Pivot Quicksort and Beyond Analysis of Multiway Partitioning and Its Practical Potential Sebastian Wild Vortrag zur wissenschaftlichen Aussprache im Rahmen des Promotionsverfahrens 8. Juli 2016 Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 1 / 17

4. What is sorting? What do you think when you hear sorting? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 2 / 17

9. What is sorting? Two meanings of sorting 1 Separate diﬀerent sorts of things 2 Put items into order (alphabetical, numerical, ...) Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 2 / 17

13. Model of Computation How do computers sort? 1 2 3 4 5 6 7 8 9 50 2 75 17 10 78 33 72 98 Input given as array of n items index-based access (5th item A[5] is 10) we can read and write cells Only info about data: pairwise comparisons Is A[5] < A[8]? Yes, 10 < 72 only relative ranking of elements important may assume, e.g., numbers 1, . . . , n Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 3 / 17

14. Model of Computation How do computers sort? 1 2 3 4 5 6 7 8 9 50 2 75 17 10 78 33 72 98 Input given as array of n items index-based access (5th item A[5] is 10) we can read and write cells Only info about data: pairwise comparisons Is A[5] < A[8]? Yes, 10 < 72 only relative ranking of elements important may assume, e.g., numbers 1, . . . , n Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 3 / 17

15. Model of Computation How do computers sort? 1 2 3 4 5 6 7 8 9 50 2 75 17 10 78 33 72 98 Input given as array of n items e.g. Excel rows index-based access (5th item A[5] is 10) we can read and write cells Only info about data: pairwise comparisons Is A[5] < A[8]? Yes, 10 < 72 only relative ranking of elements important may assume, e.g., numbers 1, . . . , n Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 3 / 17

17. Model of Computation How do computers sort? 1 2 3 4 5 6 7 8 9 50 2 75 17 10 78 33 72 98 A[5] 10 Input given as array of n items e.g. Excel rows index-based access (5th item A[5] is 10) we can read and write cells Only info about data: pairwise comparisons Is A[5] < A[8]? Yes, 10 < 72 only relative ranking of elements important may assume, e.g., numbers 1, . . . , n Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 3 / 17

18. Model of Computation How do computers sort? 1 2 3 4 5 6 7 8 9 50 2 75 17 10 78 33 72 98 A[5] 10 Input given as array of n items e.g. Excel rows index-based access (5th item A[5] is 10) we can read and write cells Only info about data: pairwise comparisons Is A[5] < A[8]? Yes, 10 < 72 only relative ranking of elements important may assume, e.g., numbers 1, . . . , n Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 3 / 17

19. Model of Computation How do computers sort? 1 2 3 4 5 6 7 8 9 50 2 75 17 10 78 33 72 98 A[5] 10 A[8] 72 Input given as array of n items e.g. Excel rows index-based access (5th item A[5] is 10) we can read and write cells Only info about data: pairwise comparisons Is A[5] < A[8]? Yes, 10 < 72 only relative ranking of elements important may assume, e.g., numbers 1, . . . , n Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 3 / 17

20. Model of Computation How do computers sort? 1 2 3 4 5 6 7 8 9 50 2 75 17 10 78 33 72 98 A[5] 10 A[8] 72 Input given as array of n items e.g. Excel rows index-based access (5th item A[5] is 10) we can read and write cells Only info about data: pairwise comparisons Is A[5] < A[8]? Yes, 10 < 72 only relative ranking of elements important may assume, e.g., numbers 1, . . . , n Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 3 / 17

22. Model of Computation How do computers sort? 1 2 3 4 5 6 7 8 9 50 2 75 17 10 78 33 72 98 1 2 3 4 5 6 7 8 9 5 1 7 3 2 8 4 6 9 Input given as array of n items e.g. Excel rows index-based access (5th item A[5] is 10) we can read and write cells Only info about data: pairwise comparisons Is A[5] < A[8]? Yes, 10 < 72 only relative ranking of elements important may assume, e.g., numbers 1, . . . , n Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 3 / 17

23. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. 5 1 7 3 2 8 4 6 9 Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

24. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. 5 1 7 3 2 8 4 6 915 min ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

25. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. 5 1 7 3 2 8 4 6 9 ? 1 min 5 Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

26. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. 5 1 7 3 2 8 4 6 9 ? 71 min ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

27. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. 5 1 7 3 2 8 4 6 9 ? ? 31 min ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

28. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. 5 1 7 3 2 8 4 6 9 ? ? ? 21 min ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

29. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. 5 1 7 3 2 8 4 6 9 ? ? ? ? 81 min ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

30. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. 5 1 7 3 2 8 4 6 9 ? ? ? ? ? 41 min ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

31. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. 5 1 7 3 2 8 4 6 9 ? ? ? ? ? ? 61 min ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

32. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. 5 1 7 3 2 8 4 6 9 ? ? ? ? ? ? ? 91 min ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

33. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. 5 1 7 3 2 8 4 6 9 ? ? ? ? ? ? ? ? 1 min 5 Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

34. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. 5 1 7 3 2 8 4 6 9 ? ? ? ? ? ? ? ? 5 7 3 2 8 4 6 95 min 1 Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

35. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 5 7 3 2 8 4 6 91 Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

36. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 5 7 3 2 8 4 6 91 to do (by same procedure) Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

37. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 5 7 3 2 8 4 6 91 75 min ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

38. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 5 7 3 2 8 4 6 91 ? 35 min ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

39. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 5 7 3 2 8 4 6 91 ? ? 5 3 min Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

40. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 5 7 3 2 8 4 6 91 ? ? 23 min ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

41. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 5 7 3 2 8 4 6 91 ? ? ? 2 min 3 Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

42. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 5 7 3 2 8 4 6 91 ? ? ? 82 min ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

43. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 5 7 3 2 8 4 6 91 ? ? ? ? 42 min ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

44. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 5 7 3 2 8 4 6 91 ? ? ? ? ? 62 min ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

45. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 5 7 3 2 8 4 6 91 ? ? ? ? ? ? 92 min ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

46. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 5 7 3 2 8 4 6 91 ? ? ? ? ? ? ? 2 min 5 Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

47. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 7 3 5 8 4 6 952 min Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

48. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 7 3 5 8 4 6 92 Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

49. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 3 7 5 8 4 6 9 ? ? ? ? ? ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

50. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? 3 4 5 8 7 6 9 ? ? ? ? ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

51. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? 3 4 5 8 7 6 9 ? ? ? ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

52. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

53. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

54. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Each round puts one item in place. Remaining list one smaller. Can we do better? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

55. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Each round puts one item in place. Remaining list one smaller. Can we do better? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

56. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Each round puts one item in place. Remaining list one smaller. Can we do better? Yes! Finding the minimum costs the same as computing the rank number of elements smaller than given element of any element. Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

57. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

58. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal place of some item (pivot), put small items left, others right (partition). Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

59. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

60. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). 5 1 7 3 2 8 4 6 9 Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

61. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). 5 1 7 3 2 8 4 6 9 Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

62. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). 5 1 7 3 2 8 4 6 9 k g Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

63. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). < 5 1 7 3 2 8 4 6 9 k g ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

64. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). < 5 1 7 3 2 8 4 6 9 ? k g Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

65. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). < 5 1 7 3 2 8 4 6 9 ? k g ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

66. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). < > 5 1 7 3 2 8 4 6 9 ? k g ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

67. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). < > > 5 1 7 3 2 8 4 6 9 ? k g ? ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

68. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). < > >> 5 1 7 3 2 8 4 6 9 ? ? ? k g ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

69. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). < > >>< 5 1 7 3 2 8 4 6 9 ? ? ?? k g ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

70. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). < >>< > ? ? ?? k g ? 5 1 3 2 8 6 94 7 Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

71. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). < >>< >< ? ? ?? g ? 5 1 4 3 2 8 7 6 9 k g ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

72. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). < >>< >< < ? ? ?? g ? 5 1 4 3 2 8 7 6 9 ? k g ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

73. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). < >>< >< < > ? ? ??? 5 1 4 3 2 8 7 6 9 ? ? gk ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

74. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). < >>< >< < > ? ? ??? 5 1 4 3 2 8 7 6 9 ? ? ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

75. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). < >>< >< >< ? ? ???? ? ? 1 4 3 8 7 6 92 5 Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

76. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). < >>< >< >< ? ? ???? ? ? 2 1 4 3 8 7 6 95 reached ﬁnal place! Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

77. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). < >>< >< >< ? ? ???? ? ? 2 1 4 3 8 7 6 95 to do to do Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

78. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). ? ? ???? ? ? 2 1 4 3 8 7 6 95 to do Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

79. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). ? ? ???? ? ? 2 1 4 3 8 7 6 95 to do Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

80. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). < > > ? ? ???? ? ? 2 1 4 3 8 7 6 95 to do ? ? ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

81. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). < > > ? ? ???? ? ? to do ? ? ? 1 2 4 3 5 8 7 6 9 to do Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

82. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). ? ? ???? ? ? to do ? ? ? 1 2 4 3 5 8 7 6 9 to do Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

83. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). ? ? ???? ? ? to do ? ? ? 1 2 3 4 5 8 7 6 9 ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

84. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). ? ? ???? ? ? to do ? ? ? 1 2 3 4 5 8 7 6 9 ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

85. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). < < > ? ? ???? ? ? ? ? ? 1 2 3 4 5 8 7 6 9 ? ? ? ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

86. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). < < > ? ? ???? ? ? ? ? ? ? ? ? ? 1 2 3 4 5 6 7 8 9 Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

87. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). ? ? ???? ? ? ? ? ? ? ? ? ? 1 2 3 4 5 6 7 8 9 to do Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

88. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). ? ? ???? ? ? ? ? ? ? ? ? ? 1 2 3 4 5 6 7 8 9 Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

89. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). ? ? ???? ? ? ? ? ? ? ? ? ? 1 2 3 4 5 6 7 8 9 ? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

90. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). ? ? ???? ? ? ? ? ? ? ? ? ? 1 2 3 4 5 6 7 8 9 ? 16 Comparisons Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

91. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). ? ? ???? ? ? ? ? ? ? ? ? ? 1 2 3 4 5 6 7 8 9 ? 16 Comparisons Much less comparisons in Quicksort ...unless pivots are max/min! But average/expected behavior is good. Quicksort is method of choice in practice. Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

92. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). ? ? ???? ? ? ? ? ? ? ? ? ? 1 2 3 4 5 6 7 8 9 ? 16 Comparisons Much less comparisons in Quicksort ...unless pivots are max/min! But average/expected behavior is good. Quicksort is method of choice in practice. Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

93. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). ? ? ???? ? ? ? ? ? ? ? ? ? 1 2 3 4 5 6 7 8 9 ? 16 Comparisons Much less comparisons in Quicksort ...unless pivots are max/min! But average/expected (actually: almost always good) behavior is good. Quicksort is method of choice in practice. Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

94. What is Quicksort? How to make a computer sort? Selectionsort Find smallest item and put it in place. ? ? ? ? ? ? ? ? 1 ? ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 3 4 5 6 7 8 9 ? ? ? ? ? ? 36 Comparisons Quicksort Find ﬁnal kth smallest A[k] place of some item (pivot), put small items left, others right (partition). ? ? ???? ? ? ? ? ? ? ? ? ? 1 2 3 4 5 6 7 8 9 ? 16 Comparisons Much less comparisons in Quicksort ...unless pivots are max/min! But average/expected (actually: almost always good) behavior is good. Quicksort is method of choice in practice. Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 4 / 17

95. Variants of Quicksort It is tempting to try to develop ways to improve quicksort: a faster sorting algorithm is computer science’s “better mousetrap,” and quicksort is a venerable method that seems to invite tinkering. R. Sedgewick and K. Wayne, Algorithms 4th ed. Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 5 / 17

96. Variants of Quicksort It is tempting to try to develop ways to improve quicksort: a faster sorting algorithm is computer science’s “better mousetrap,” and quicksort is a venerable method that seems to invite tinkering. R. Sedgewick and K. Wayne, Algorithms 4th ed. Many variants tried Two important tuning options: 1 Choose pivots wisely (pivot sampling) for example: middle element of three (median-of-three) pivot never min or max! 2 Split into s 2 subproblems at once (multiway partitioning) for example: 2 pivots 3 classes: small, medium and large subproblems are smaller, but partitioning is more expensive. Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 5 / 17

97. Variants of Quicksort It is tempting to try to develop ways to improve quicksort: a faster sorting algorithm is computer science’s “better mousetrap,” and quicksort is a venerable method that seems to invite tinkering. R. Sedgewick and K. Wayne, Algorithms 4th ed. Many variants tried Two important tuning options: 1 Choose pivots wisely (pivot sampling) for example: middle element of three (median-of-three) pivot never min or max! 2 Split into s 2 subproblems at once (multiway partitioning) for example: 2 pivots 3 classes: small, medium and large subproblems are smaller, but partitioning is more expensive. Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 5 / 17

98. Variants of Quicksort It is tempting to try to develop ways to improve quicksort: a faster sorting algorithm is computer science’s “better mousetrap,” and quicksort is a venerable method that seems to invite tinkering. R. Sedgewick and K. Wayne, Algorithms 4th ed. Many variants tried ... many not helpful Two important tuning options: 1 Choose pivots wisely (pivot sampling) for example: middle element of three (median-of-three) pivot never min or max! 2 Split into s 2 subproblems at once (multiway partitioning) for example: 2 pivots 3 classes: small, medium and large subproblems are smaller, but partitioning is more expensive. Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 5 / 17

108. Variants of Quicksort It is tempting to try to develop ways to improve quicksort: a faster sorting algorithm is computer science’s “better mousetrap,” and quicksort is a venerable method that seems to invite tinkering. R. Sedgewick and K. Wayne, Algorithms 4th ed. Many variants tried ... many not helpful Two important tuning options: 1 Choose pivots wisely (pivot sampling) for example: middle element of three (median-of-three) pivot never min or max! 2 Split into s 2 subproblems at once (multiway partitioning) for example: 2 pivots 3 classes: small, medium and large subproblems are smaller, but partitioning is more expensive. conventional wisdom Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 5 / 17

109. Variants of Quicksort It is tempting to try to develop ways to improve quicksort: a faster sorting algorithm is computer science’s “better mousetrap,” and quicksort is a venerable method that seems to invite tinkering. R. Sedgewick and K. Wayne, Algorithms 4th ed. Many variants tried ... many not helpful Two important tuning options: 1 Choose pivots wisely (pivot sampling) for example: middle element of three (median-of-three) pivot never min or max! 2 Split into s 2 subproblems at once (multiway partitioning) for example: 2 pivots 3 classes: small, medium and large subproblems are smaller, but partitioning is more expensive. conventional wisdom till 2009! Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 5 / 17

110. Java 7’s Dual-Pivot Quicksort What happened in 2009? Java used e.g. for Android apps 7 sorting method was released! till 2009, Java used classic Quicksort Java 7 uses new dual-pivot Quicksort started as hobby project by Vladimir Yaroslavskiy 20 % faster than highly-tuned Java 6! Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 6 / 17

113. Java 7’s Dual-Pivot Quicksort What happened in 2009? Java used e.g. for Android apps 7 sorting method was released! till 2009, Java used classic as in example before + median-of-three + Insertionsort Quicksort Java 7 uses new dual-pivot Quicksort started as hobby project by Vladimir Yaroslavskiy 20 % faster than highly-tuned Java 6! Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 6 / 17

114. Java 7’s Dual-Pivot Quicksort What happened in 2009? Java used e.g. for Android apps 7 sorting method was released! till 2009, Java used classic as in example before + median-of-three + Insertionsort Quicksort Java 7 uses new dual-pivot Quicksort started as hobby project by Vladimir Yaroslavskiy 20 % faster than highly-tuned Java 6! 0 0.5 1 1.5 2 ·106 7 8 9 n time 10−6·nlnn Java 6 sorting method (121ms to sort 106 items) Normalized Java runtimes (in ms). Average and standard deviation of 1000 random permutations per size. Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 6 / 17

117. Java 7’s Dual-Pivot Quicksort What happened in 2009? Java used e.g. for Android apps 7 sorting method was released! till 2009, Java used classic as in example before + median-of-three + Insertionsort Quicksort Java 7 uses new dual-pivot Quicksort started as hobby project by Vladimir Yaroslavskiy 20 % faster than highly-tuned Java 6! 0 0.5 1 1.5 2 ·106 7 8 9 n time 10−6·nlnn Java 6 sorting method Java 7 sorting method (121ms to sort 106 items) (93ms to sort 106 items) Normalized Java runtimes (in ms). Average and standard deviation of 1000 random permutations per size. Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 6 / 17

118. Java 7’s Dual-Pivot Quicksort What happened in 2009? Java used e.g. for Android apps 7 sorting method was released! till 2009, Java used classic as in example before + median-of-three + Insertionsort Quicksort Java 7 uses new dual-pivot Quicksort started as hobby project by Vladimir Yaroslavskiy 20 % faster than highly-tuned Java 6! 0 0.5 1 1.5 2 ·106 7 8 9 n time 10−6·nlnn Java 6 sorting method Java 7 sorting method (121ms to sort 106 items) (93ms to sort 106 items) Normalized Java runtimes (in ms). Average and standard deviation of 1000 random permutations per size. No theoretical explanation for running time known in 2009! Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 6 / 17

119. Java 7’s Dual-Pivot Quicksort What happened in 2009? Java used e.g. for Android apps 7 sorting method was released! till 2009, Java used classic as in example before + median-of-three + Insertionsort Quicksort Java 7 uses new dual-pivot Quicksort started as hobby project by Vladimir Yaroslavskiy 20 % faster than highly-tuned Java 6! 0 0.5 1 1.5 2 ·106 7 8 9 n time 10−6·nlnn Java 6 sorting method Java 7 sorting method (121ms to sort 106 items) (93ms to sort 106 items) Normalized Java runtimes (in ms). Average and standard deviation of 1000 random permutations per size. No theoretical explanation for running time known in 2009! Only lucky experiments? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 6 / 17

120. Java 7’s Dual-Pivot Quicksort What happened in 2009? Java used e.g. for Android apps 7 sorting method was released! till 2009, Java used classic as in example before + median-of-three + Insertionsort Quicksort Java 7 uses new dual-pivot Quicksort started as hobby project by Vladimir Yaroslavskiy 20 % faster than highly-tuned Java 6! 0 0.5 1 1.5 2 ·106 7 8 9 n time 10−6·nlnn Java 6 sorting method Java 7 sorting method (121ms to sort 106 items) (93ms to sort 106 items) Normalized Java runtimes (in ms). Average and standard deviation of 1000 random permutations per size. No theoretical explanation for running time known in 2009! Only lucky experiments? Why did no-one come up with this earlier? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 6 / 17

121. Java 7’s Dual-Pivot Quicksort What happened in 2009? Java used e.g. for Android apps 7 sorting method was released! till 2009, Java used classic as in example before + median-of-three + Insertionsort Quicksort Java 7 uses new dual-pivot Quicksort started as hobby project by Vladimir Yaroslavskiy 20 % faster than highly-tuned Java 6! 0 0.5 1 1.5 2 ·106 7 8 9 n time 10−6·nlnn Java 6 sorting method Java 7 sorting method (121ms to sort 106 items) (93ms to sort 106 items) Normalized Java runtimes (in ms). Average and standard deviation of 1000 random permutations per size. No theoretical explanation for running time known in 2009! Only lucky experiments? Why did no-one come up with this earlier? That was the starting point for my work! Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 6 / 17

122. Overview of my work My Thesis Multiway partitioning has genuine potential to speed up Quicksort. Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 7 / 17

123. Overview of my work My Thesis Multiway partitioning has genuine potential to speed up Quicksort. Mathematical Analysis prove eternal truths not experiments Mathematical Analysis prove eternal truths not experiments Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 7 / 17

124. Overview of my work My Thesis Multiway partitioning has genuine potential to speed up Quicksort. Mathematical Analysis prove eternal truths not experiments What is multiway partitioning? generalize existing Quicksort variants generic s-way one-pass partitioning with pivot sampling What is multiway partitioning? generalize existing Quicksort variants generic s-way one-pass partitioning with pivot sampling Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 7 / 17

125. Overview of my work My Thesis Multiway partitioning has genuine potential to speed up Quicksort. Mathematical Analysis prove eternal truths not experiments What is multiway partitioning? generalize existing Quicksort variants generic s-way one-pass partitioning with pivot sampling Uniﬁed Analysis of Quicksort costs on average for large inputs parametric (algorithm and cost measure) Uniﬁed Analysis of Quicksort costs on average for large inputs parametric (algorithm and cost measure) Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 7 / 17

126. Overview of my work My Thesis Multiway partitioning has genuine potential to speed up Quicksort. Mathematical Analysis prove eternal truths not experiments What is multiway partitioning? generalize existing Quicksort variants generic s-way one-pass partitioning with pivot sampling Uniﬁed Analysis of Quicksort costs on average for large inputs parametric (algorithm and cost measure) Cost Measures generalize models of cost comparisons write accesses / swaps cache misses / scanned elements branch mispredictions element-wise charging schemes Cost Measures generalize models of cost comparisons write accesses / swaps cache misses / scanned elements branch mispredictions element-wise charging schemes Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 7 / 17

127. Overview of my work My Thesis Multiway partitioning has genuine potential to speed up Quicksort. Mathematical Analysis prove eternal truths not experiments What is multiway partitioning? generalize existing Quicksort variants generic s-way one-pass partitioning with pivot sampling Unified Analysis of Quicksort costs on average for large inputs parametric (algorithm and cost measure) Cost Measures generalize models of cost comparisons write accesses / swaps cache misses / scanned elements branch mispredictions element-wise charging schemes Discussion of Results influence of number of pivots pivot sampling organization of indices Discussion of Results influence of number of pivots pivot sampling organization of indices Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 7 / 17

128. Overview of my work My Thesis Multiway partitioning has genuine potential to speed up Quicksort. Mathematical Analysis prove eternal truths not experiments What is multiway partitioning? generalize existing Quicksort variants generic s-way one-pass partitioning with pivot sampling Uniﬁed Analysis of Quicksort costs on average for large inputs parametric (algorithm and cost measure) Cost Measures generalize models of cost comparisons write accesses / swaps cache misses / scanned elements branch mispredictions element-wise charging schemes Discussion of Results inﬂuence of number of pivots pivot sampling organization of indices Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 7 / 17

129. Entropy What is entropy? 1 Thermodynamics: unusable part of thermal energy 2 Mathematics: information content of a random variable Deﬁnition (Entropy) Let X be a random variable with P[X = i] = pi for i ∈ {1, . . . , u}. Its entropy is deﬁned as H(X) = u i=1 pi log2(1/pi). probability that X = i ...all clear? Let’s see an application. 1 2 3 4 5 6 7 8 9 10 11 12 Urn with colored balls. Alice draws ball X uniformly at random from urn. Bob wants to guess its color C by Yes/No-Questions. Theorem (Shannon’s Source Coding Theorem) On average, Bob can’t do with less than H(C) questions. We say: We must acquire H(C) bits of information to know C. Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 8 / 17

132. Entropy What is entropy? 1 Thermodynamics: unusable part of thermal energy (Physicists’ party? Next door, please.) 2 Mathematics: information content of a random variable Deﬁnition (Entropy) Let X be a random variable with P[X = i] = pi for i ∈ {1, . . . , u}. Its entropy is deﬁned as H(X) = u i=1 pi log2(1/pi). probability that X = i ...all clear? Let’s see an application. 1 2 3 4 5 6 7 8 9 10 11 12 Urn with colored balls. Alice draws ball X uniformly at random from urn. Bob wants to guess its color C by Yes/No-Questions. Theorem (Shannon’s Source Coding Theorem) On average, Bob can’t do with less than H(C) questions. We say: We must acquire H(C) bits of information to know C. Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 8 / 17

140. Entropy What is entropy? 1 Thermodynamics: unusable part of thermal energy (Physicists’ party? Next door, please.) 2 Mathematics: information content of a random variable Deﬁnition (Entropy) Let X be a random variable with P[X = i] = pi for i ∈ {1, . . . , u}. Its entropy is deﬁned as H(X) = u i=1 pi log2(1/pi). probability that X = i ...all clear? Let’s see an application. 1 2 3 4 5 6 7 8 9 10 11 12 Urn with colored balls. Alice draws ball X uniformly at random from urn. Bob wants to guess its color C P[C = ] = 3/12 = 0.25 P[C = ] = 3/12 = 0.25 P[C = ] = 3/12 = 0.25 P[C = ] = 3/12 = 0.25 by Yes/No-Questions. Theorem (Shannon’s Source Coding Theorem) On average, Bob can’t do with less than H(C) questions. We say: We must acquire H(C) bits of information to know C. Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 8 / 17

141. Entropy What is entropy? 1 Thermodynamics: unusable part of thermal energy (Physicists’ party? Next door, please.) 2 Mathematics: information content of a random variable Deﬁnition (Entropy) Let X be a random variable with P[X = i] = pi for i ∈ {1, . . . , u}. Its entropy is deﬁned as H(X) = u i=1 pi log2(1/pi). probability that X = i ...all clear? Let’s see an application. 1 2 3 4 5 6 7 8 9 10 11 12 Urn with colored balls. Alice draws ball X uniformly at random from urn. Bob wants to guess its color C P[C = ] = 3/12 = 0.25 P[C = ] = 3/12 = 0.25 P[C = ] = 3/12 = 0.25 P[C = ] = 3/12 = 0.25 by Yes/No-Questions. Theorem (Shannon’s Source Coding Theorem) On average, Bob can’t do with less than H(C) questions. We say: We must acquire H(C) bits of information to know C. Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 8 / 17

142. Entropy What is entropy? 1 Thermodynamics: unusable part of thermal energy (Physicists’ party? Next door, please.) 2 Mathematics: information content of a random variable Deﬁnition (Entropy) Let X be a random variable with P[X = i] = pi for i ∈ {1, . . . , u}. Its entropy is deﬁned as H(X) = u i=1 pi log2(1/pi). probability that X = i ...all clear? Let’s see an application. 1 2 3 4 5 6 7 8 9 10 11 12 Urn with colored balls. Alice draws ball X uniformly at random from urn. Bob wants to guess its color C P[C = ] = 3/12 = 0.25 P[C = ] = 3/12 = 0.25 P[C = ] = 3/12 = 0.25 P[C = ] = 3/12 = 0.25 by Yes/No-Questions. Theorem (Shannon’s Source Coding Theorem) On average, Bob can’t do with less than H(C) questions. = 4 · 1 4 log2(4) = 2 We say: We must acquire H(C) bits of information to know C. Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 8 / 17

143. Entropy What is entropy? 1 Thermodynamics: unusable part of thermal energy (Physicists’ party? Next door, please.) 2 Mathematics: information content of a random variable Deﬁnition (Entropy) Let X be a random variable with P[X = i] = pi for i ∈ {1, . . . , u}. Its entropy is deﬁned as H(X) = u i=1 pi log2(1/pi). probability that X = i ...all clear? Let’s see an application. 1 2 3 4 5 6 7 8 9 10 11 12 Urn with colored balls. Alice draws ball X uniformly at random from urn. Bob wants to guess its color C P[C = ] = 3/12 = 0.25 P[C = ] = 3/12 = 0.25 P[C = ] = 3/12 = 0.25 P[C = ] = 3/12 = 0.25 by Yes/No-Questions. Theorem (Shannon’s Source Coding Theorem) On average, Bob can’t do with less than H(C) questions. = 4 · 1 4 log2(4) = 2 We say: We must acquire H(C) bits of information to know C. Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 8 / 17

144. Information Theory & Sorting Where is the connection to sorting? 1 2 3 4 5 6 7 8 9 10 11 12 Now, Alice draws one of the n! n(n − 1) · · · 2 · 1 possible orderings of n items. Bob guesses the order by asking comparisons. Theorem (Information-Theoretic Lower Bound for Sorting) We cannot sort with less than log2(n!) = n log2(n) ± O(n) comparisons. Assumptions: (1) All orderings equally likely. (2) Only comparisons allowed.asymptotic error term for n → ∞ H(C) To achieve this bound, every comparison must yield 1 bit of information! Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 9 / 17

145. Information Theory & Sorting Where is the connection to sorting? 1 2 3 4 5 6 7 8 9 10 11 12 Now, Alice draws one of the n! n(n − 1) · · · 2 · 1 possible orderings of n items. Bob guesses the order by asking comparisons. Theorem (Information-Theoretic Lower Bound for Sorting) We cannot sort with less than log2(n!) = n log2(n) ± O(n) comparisons. Assumptions: (1) All orderings equally likely. (2) Only comparisons allowed.asymptotic error term for n → ∞ H(C) To achieve this bound, every comparison must yield 1 bit of information! Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 9 / 17

146. Information Theory & Sorting Where is the connection to sorting? 1 2 3 4 5 6 7 8 9 10 11 12 Now, Alice draws one of the n! n(n − 1) · · · 2 · 1 possible orderings = colors of n items. Bob guesses the order by asking comparisons. Theorem (Information-Theoretic Lower Bound for Sorting) We cannot sort with less than log2(n!) = n log2(n) ± O(n) comparisons. Assumptions: (1) All orderings equally likely. (2) Only comparisons allowed.asymptotic error term for n → ∞ H(C) To achieve this bound, every comparison must yield 1 bit of information! Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 9 / 17

150. Information Theory & Sorting Where is the connection to sorting? 1 2 3 4 5 6 7 8 9 10 11 12 Now, Alice draws one of the n! n(n − 1) · · · 2 · 1 possible orderings = colors of n items. Bob guesses the order by asking comparisons. Theorem (Information-Theoretic Lower Bound for Sorting) We cannot sort with less than log2(n!) = n log2(n) ± O(n) comparisons. Assumptions: (1) All orderings equally likely. (2) Only comparisons allowed.asymptotic error term for n → ∞ H(C) To achieve this bound, every comparison must yield 1 bit of information! We can reverse this idea to analyze Quicksort. Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 9 / 17

151. Comparisons in Quicksort How many comparisons does Quicksort use on average? Entropy-based analysis: 1 Compute the average information yield: x bit comparison . (0 < x 1) What is x? ...stay tuned. 2 Need to learn log2(n!) bits Use log2(n!) x = 1 x n log2(n) ± O(n) comparisons. Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 10 / 17

157. Comparisons in Quicksort How many comparisons does Quicksort use on average? Entropy-based analysis: 1 Compute the average information yield: x bit comparison . (0 < x 1) What is x? ...stay tuned. 2 Need to learn log2(n!) bits not mathematically rigorous ... but correct result! rigorous proof → dissertation Use log2(n!) x = 1 x n log2(n) ± O(n) comparisons. Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 10 / 17

158. Partitioning Entropy What is x? Classic Quicksort: Always compare with the pivot P < < < < > > > > 2 1 4 3 5 8 7 6 9 Outcome C is either small or large . Information yield is H(C) = −P[ < ] log2(P[ < ]) − P[ > ] log2(P[ > ]) Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 11 / 17

162. Partitioning Entropy What is x? Classic Quicksort: Always compare with the pivot P < < < < > > > > 2 1 4 3 5 8 7 6 9 Outcome C is either small or large . Information yield is H(C) = −P[ < ] prob. of item to be ] log2(P[ > ]) Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 11 / 17

163. Partitioning Entropy What is x? Classic Quicksort: Always compare with the pivot P < < < < > > > > 2 1 4 3 5 8 7 6 9 Outcome C is either small or large . Information yield is H(C) = −P[ < ] prob. of item to be ] log2(P[ > ]) Note: pivot value P is random. What is its distribution? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 11 / 17

164. Partitioning Entropy What is x? Classic Quicksort: Always compare with the pivot P < < < < > > > > 2 1 4 3 5 8 7 6 9 Outcome C is either small or large . Information yield is H(C) = −P[ < ] prob. of item to be ] log2(P[ > ]) Note: pivot value P is random. What is its distribution? Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 11 / 17

165. Partitioning Entropy What is x? Classic Quicksort: Always compare with the pivot P < < < < > > > > 2 1 4 3 5 8 7 6 9 Outcome C is either small or large . Information yield is H(C) = −P[ < ] prob. of item to be ] log2(P[ > ]) Note: pivot value P is random. What is its distribution? Excursion: Input Models Typical: Random Permutations items: 1, . . . , n all orderings equally likely We use the equivalent Uniform Model n numbers drawn i.i.d. Uniform(0, 1) i.i.d. random relative ranking P[ < ] = P, P[ > ] = 1 − P Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 11 / 17

170. Partitioning Entropy What is x? Classic Quicksort: Always compare with the pivot P < < < < > > > > 2 1 4 3 5 8 7 6 9 Outcome C is either small or large . Information yield is H(C) = −P[ < ] prob. of item to be ] log2(P[ > ]) Note: pivot value P is random. What is its distribution? Excursion: Input Models Typical: Random Permutations items: 1, . . . , n all orderings equally likely We use the equivalent Uniform Model n numbers drawn i.i.d. Uniform(0, 1) i.i.d. random relative ranking P[ < ] = P, P[ > ] = 1 − P same average sorting costs Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 11 / 17

171. Partitioning Entropy What is x? Classic Quicksort: Always compare with the pivot P < < < < > > > > 2 1 4 3 5 8 7 6 9 Outcome C is either small or large . Information yield is H(C) = −P[ < ] prob. of item to be ] log2(P[ > ]) Note: pivot value P is random. What is its distribution? Excursion: Input Models Typical: Random Permutations items: 1, . . . , n all orderings equally likely We use the equivalent Uniform Model n numbers drawn i.i.d. Uniform(0, 1) i.i.d. random relative ranking P[ < ] = P, P[ > ] = 1 − P same average sorting costs Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 11 / 17

172. Partitioning Entropy What is x? Classic Quicksort: Always compare with the pivot P < < < < > > > > 2 1 4 3 5 8 7 6 9 Outcome C is either small or large . Information yield is H(C) = −P[ < ] prob. of item to be ] log2(P[ > ]) Note: pivot value P is random. What is its distribution? Excursion: Input Models Typical: Random Permutations items: 1, . . . , n all orderings equally likely We use the equivalent Uniform Model n numbers drawn i.i.d. Uniform(0, 1) i.i.d. random relative ranking P[ < ] = P, P[ > ] = 1 − P same average sorting costs independent & identically distributed Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 11 / 17

173. Partitioning Entropy What is x? Classic Quicksort: Always compare with the pivot P < < < < > > > > 2 1 4 3 5 8 7 6 9 Outcome C is either small or large . Information yield is H(C) = −P[ < ] prob. of item to be ] log2(P[ > ]) Note: pivot value P is random. What is its distribution? Excursion: Input Models Typical: Random Permutations items: 1, . . . , n all orderings equally likely We use the equivalent Uniform Model n numbers drawn i.i.d. Uniform(0, 1) i.i.d. random relative ranking P[ < ] = P, P[ > ] = 1 − P same average sorting costs independent & identically distributed Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 11 / 17

174. Partitioning Entropy What is x? Classic Quicksort: Always compare with the pivot P < < < < > > > > 2 1 4 3 5 8 7 6 9 Outcome C is either small or large . Information yield is H(C) = −P[ < ] prob. of item to be ] log2(P[ > ]) Note: pivot value P is random. What is its distribution? Excursion: Input Models Typical: Random Permutations items: 1, . . . , n all orderings equally likely We use the equivalent Uniform Model n numbers drawn i.i.d. Uniform(0, 1) i.i.d. random relative ranking P[ < ] = P, P[ > ] = 1 − P 0 1P0 1 P[ < ] P[ > ] same average sorting costs independent & identically distributed Answer: P D = Uniform(0, 1) Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 11 / 17

175. Partitioning Entropy What is x? Classic Quicksort: Always compare with the pivot P < < < < > > > > 2 1 4 3 5 8 7 6 9 Outcome C is either small or large . Information yield is H(C) = −P[ < ] log2(P[ < ]) − P[ > ] log2(P[ > ]) Excursion: Input Models Typical: Random Permutations items: 1, . . . , n all orderings equally likely We use the equivalent Uniform Model n numbers drawn i.i.d. Uniform(0, 1) i.i.d. random relative ranking P[ < ] = P, P[ > ] = 1 − P 0 1P0 1 P[ < ] P[ > ] same average sorting costs independent & identically distributed E Average over P [H(C)] = E −P log2(P) − (1 − P) log2(1 − P) = −2 1 0 z log2(z) dz = 1 2 log2(e) ≈ 0.721348 Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 11 / 17

179. Partitioning Entropy What is x? Classic Quicksort: Always compare with the pivot P < < < < > > > > 2 1 4 3 5 8 7 6 9 Outcome C is either small or large . Information yield is H(C) = −P[ < ] log2(P[ < ]) − P[ > ] log2(P[ > ]) E Average over P [H(C)] = E −P log2(P) − (1 − P) log2(1 − P) = −2 1 0 z log2(z) dz = 1 2 log2(e) ≈ 0.721348 Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 11 / 17

180. Partitioning Entropy What is x? Classic Quicksort: Always compare with the pivot P < < < < > > > > 2 1 4 3 5 8 7 6 9 Outcome C is either small or large . Information yield is H(C) = −P[ < ] log2(P[ < ]) − P[ > ] log2(P[ > ]) E Average over P [H(C)] = E −P log2(P) − (1 − P) log2(1 − P) = −2 1 0 z log2(z) dz = 1 2 log2(e) ≈ 0.721348 s-way Quicksort with pivot sampling: s − 1 pivots: P1, . . . , Ps−1 s classes c1, . . . , cs Pivot Sampling: parameter t = (t1, . . . , ts) ∈ Ns 0 Example: s = 3, t = (1, 2, 3) choose pivots from k = 8 sample items 0 10 1 D D = Dirichlet(t1 + 1, . . . , ts + 1) E[H(C)] = s r=1 tr + 1 k + 1 Hk+1 − H 1 1 + 1 2 + 1 3 + · · · + 1 tr+1 tr+1 · log2(e) Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 11 / 17

187. Partitioning Entropy What is x? Classic Quicksort: Always compare with the pivot P < < < < > > > > 2 1 4 3 5 8 7 6 9 Outcome C is either small or large . Information yield is H(C) = −P[ < ] log2(P[ < ]) − P[ > ] log2(P[ > ]) E Average over P [H(C)] = E −P log2(P) − (1 − P) log2(1 − P) = −2 1 0 z log2(z) dz = 1 2 log2(e) ≈ 0.721348 s-way Quicksort with pivot sampling: s − 1 pivots: P1, . . . , Ps−1 s classes c1, . . . , cs Pivot Sampling: parameter t = (t1, . . . , ts) ∈ Ns 0 Example: s = 3, t = (1, 2, 3) choose pivots from k = 8 sample items 0 1 U1 0 1 D D = Dirichlet(t1 + 1, . . . , ts + 1) E[H(C)] = s r=1 tr + 1 k + 1 Hk+1 − H 1 1 + 1 2 + 1 3 + · · · + 1 tr+1 tr+1 · log2(e) Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 11 / 17

188. Partitioning Entropy What is x? Classic Quicksort: Always compare with the pivot P < < < < > > > > 2 1 4 3 5 8 7 6 9 Outcome C is either small or large . Information yield is H(C) = −P[ < ] log2(P[ < ]) − P[ > ] log2(P[ > ]) E Average over P [H(C)] = E −P log2(P) − (1 − P) log2(1 − P) = −2 1 0 z log2(z) dz = 1 2 log2(e) ≈ 0.721348 s-way Quicksort with pivot sampling: s − 1 pivots: P1, . . . , Ps−1 s classes c1, . . . , cs Pivot Sampling: parameter t = (t1, . . . , ts) ∈ Ns 0 Example: s = 3, t = (1, 2, 3) choose pivots from k = 8 sample items 0 1 U1 U2 0 1 D D = Dirichlet(t1 + 1, . . . , ts + 1) E[H(C)] = s r=1 tr + 1 k + 1 Hk+1 − H 1 1 + 1 2 + 1 3 + · · · + 1 tr+1 tr+1 · log2(e) Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 11 / 17

189. Partitioning Entropy What is x? Classic Quicksort: Always compare with the pivot P < < < < > > > > 2 1 4 3 5 8 7 6 9 Outcome C is either small or large . Information yield is H(C) = −P[ < ] log2(P[ < ]) − P[ > ] log2(P[ > ]) E Average over P [H(C)] = E −P log2(P) − (1 − P) log2(1 − P) = −2 1 0 z log2(z) dz = 1 2 log2(e) ≈ 0.721348 s-way Quicksort with pivot sampling: s − 1 pivots: P1, . . . , Ps−1 s classes c1, . . . , cs Pivot Sampling: parameter t = (t1, . . . , ts) ∈ Ns 0 Example: s = 3, t = (1, 2, 3) choose pivots from k = 8 sample items 0 1 U1 U2U3 0 1 D D = Dirichlet(t1 + 1, . . . , ts + 1) E[H(C)] = s r=1 tr + 1 k + 1 Hk+1 − H 1 1 + 1 2 + 1 3 + · · · + 1 tr+1 tr+1 · log2(e) Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 11 / 17

190. Partitioning Entropy What is x? Classic Quicksort: Always compare with the pivot P < < < < > > > > 2 1 4 3 5 8 7 6 9 Outcome C is either small or large . Information yield is H(C) = −P[ < ] log2(P[ < ]) − P[ > ] log2(P[ > ]) E Average over P [H(C)] = E −P log2(P) − (1 − P) log2(1 − P) = −2 1 0 z log2(z) dz = 1 2 log2(e) ≈ 0.721348 s-way Quicksort with pivot sampling: s − 1 pivots: P1, . . . , Ps−1 s classes c1, . . . , cs Pivot Sampling: parameter t = (t1, . . . , ts) ∈ Ns 0 Example: s = 3, t = (1, 2, 3) choose pivots from k = 8 sample items 0 1 U1 U2U3 U4 0 1 D D = Dirichlet(t1 + 1, . . . , ts + 1) E[H(C)] = s r=1 tr + 1 k + 1 Hk+1 − H 1 1 + 1 2 + 1 3 + · · · + 1 tr+1 tr+1 · log2(e) Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 11 / 17

191. Partitioning Entropy What is x? Classic Quicksort: Always compare with the pivot P < < < < > > > > 2 1 4 3 5 8 7 6 9 Outcome C is either small or large . Information yield is H(C) = −P[ < ] log2(P[ < ]) − P[ > ] log2(P[ > ]) E Average over P [H(C)] = E −P log2(P) − (1 − P) log2(1 − P) = −2 1 0 z log2(z) dz = 1 2 log2(e) ≈ 0.721348 s-way Quicksort with pivot sampling: s − 1 pivots: P1, . . . , Ps−1 s classes c1, . . . , cs Pivot Sampling: parameter t = (t1, . . . , ts) ∈ Ns 0 Example: s = 3, t = (1, 2, 3) choose pivots from k = 8 sample items 0 1 U1 U2U3 U4U5 0 1 D D = Dirichlet(t1 + 1, . . . , ts + 1) E[H(C)] = s r=1 tr + 1 k + 1 Hk+1 − H 1 1 + 1 2 + 1 3 + · · · + 1 tr+1 tr+1 · log2(e) Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 11 / 17

192. Partitioning Entropy What is x? Classic Quicksort: Always compare with the pivot P < < < < > > > > 2 1 4 3 5 8 7 6 9 Outcome C is either small or large . Information yield is H(C) = −P[ < ] log2(P[ < ]) − P[ > ] log2(P[ > ]) E Average over P [H(C)] = E −P log2(P) − (1 − P) log2(1 − P) = −2 1 0 z log2(z) dz = 1 2 log2(e) ≈ 0.721348 s-way Quicksort with pivot sampling: s − 1 pivots: P1, . . . , Ps−1 s classes c1, . . . , cs Pivot Sampling: parameter t = (t1, . . . , ts) ∈ Ns 0 Example: s = 3, t = (1, 2, 3) choose pivots from k = 8 sample items 0 1 U1 U2U3 U4U5 U6 0 1 D D = Dirichlet(t1 + 1, . . . , ts + 1) E[H(C)] = s r=1 tr + 1 k + 1 Hk+1 − H 1 1 + 1 2 + 1 3 + · · · + 1 tr+1 tr+1 · log2(e) Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 11 / 17

193. Partitioning Entropy What is x? Classic Quicksort: Always compare with the pivot P < < < < > > > > 2 1 4 3 5 8 7 6 9 Outcome C is either small or large . Information yield is H(C) = −P[ < ] log2(P[ < ]) − P[ > ] log2(P[ > ]) E Average over P [H(C)] = E −P log2(P) − (1 − P) log2(1 − P) = −2 1 0 z log2(z) dz = 1 2 log2(e) ≈ 0.721348 s-way Quicksort with pivot sampling: s − 1 pivots: P1, . . . , Ps−1 s classes c1, . . . , cs Pivot Sampling: parameter t = (t1, . . . , ts) ∈ Ns 0 Example: s = 3, t = (1, 2, 3) choose pivots from k = 8 sample items 0 1 U1 U2U3 U4U5 U6 U7 0 1 D D = Dirichlet(t1 + 1, . . . , ts + 1) E[H(C)] = s r=1 tr + 1 k + 1 Hk+1 − H 1 1 + 1 2 + 1 3 + · · · + 1 tr+1 tr+1 · log2(e) Sebastian Wild Dual-Pivot Quicksort and Beyond 2016-07-08 11 / 17

Dual-Pivot Quicksort and Beyond: Analysis of Multiway Partitioning and Its Practical Potential

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