Average cost of QuickXsort with pivot sampling

1. Average Cost of QuickXsort with Pivot Sampling Sebastian Wild wild@uwaterloo.ca AofA 2018 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 0 / 12

2. Outline 1 What is QuickXsort? 1 What is QuickXsort? 2 Recurrence 2 Recurrence 3 Toll function3 Toll function 4 Result4 Result Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 0 / 12

3. What is QuickXsort? QuickXsort = Quicksort + X Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 1 / 12

4. What is QuickXsort? QuickXsort = Quicksort + X QUICKXSORT(A[l..r]) 1 Choose pivot P. 2 i = PARTITION(A[l..r], P) // subproblems: A[l..i − 1] and A[i + 1..r] 3 Sort one subproblem with X. 4 Sort the other subproblem with QUICKXSORT. Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 1 / 12

5. What is QuickXsort? QuickXsort = Quicksort + X QUICKXSORT(A[l..r]) 1 Choose pivot P. 2 i = PARTITION(A[l..r], P) // subproblems: A[l..i − 1] and A[i + 1..r] 3 Sort one subproblem with X. 4 Sort the other subproblem with QUICKXSORT. Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 1 / 12

6. What is QuickXsort? QuickXsort = Quicksort + X QUICKXSORT(A[l..r]) 1 Choose pivot P median of k = 2t + 1 . 2 i = PARTITION(A[l..r], P) // subproblems: A[l..i − 1] and A[i + 1..r] 3 Sort one subproblem with X. 4 Sort the other subproblem with QUICKXSORT. Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 1 / 12

10. What is QuickXsort? QuickXsort = Quicksort + X QUICKXSORT(A[l..r]) 1 Choose pivot P median of k = 2t + 1 . 2 i = PARTITION(A[l..r], P) // subproblems: A[l..i − 1] and A[i + 1..r] 3 Sort one subproblem with X. 4 Sort the other subproblem with QUICKXSORT. sort by X Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 1 / 12

11. What is QuickXsort? QuickXsort = Quicksort + X QUICKXSORT(A[l..r]) 1 Choose pivot P median of k = 2t + 1 . 2 i = PARTITION(A[l..r], P) // subproblems: A[l..i − 1] and A[i + 1..r] 3 Sort one Which one? subproblem with X. 4 Sort the other subproblem with QUICKXSORT. sort by X sort recursively Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 1 / 12

12. What is QuickXsort? QuickXsort = Quicksort + X QUICKXSORT(A[l..r]) 1 Choose pivot P median of k = 2t + 1 . 2 i = PARTITION(A[l..r], P) // subproblems: A[l..i − 1] and A[i + 1..r] 3 Sort one Which one? subproblem with X And why should that help?! . 4 Sort the other subproblem with QUICKXSORT. sort by X sort recursively Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 1 / 12

13. What is QuickXsort? QuickXsort = Quicksort + X QUICKXSORT(A[l..r]) 1 Choose pivot P median of k = 2t + 1 . 2 i = PARTITION(A[l..r], P) // subproblems: A[l..i − 1] and A[i + 1..r] 3 Sort one Which one? subproblem with X And why should that help?! , using the other subproblem as scratch space. 4 Sort the other subproblem with QUICKXSORT. sort by X sort recursively Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 1 / 12

14. What is QuickXsort? QuickXsort = Quicksort + X QUICKXSORT(A[l..r]) 1 Choose pivot P median of k = 2t + 1 . 2 i = PARTITION(A[l..r], P) // subproblems: A[l..i − 1] and A[i + 1..r] 3 Sort one Which one? subproblem with X And why should that help?! , using the other subproblem as scratch space. 4 Sort the other subproblem with QUICKXSORT. sort by X sort recursively QuickXsort can turn X into an in-place sorting method! Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 1 / 12

15. What is QuickXsort? QuickXsort = Quicksort + X QUICKXSORT(A[l..r]) 1 Choose pivot P median of k = 2t + 1 . 2 i = PARTITION(A[l..r], P) // subproblems: A[l..i − 1] and A[i + 1..r] 3 Sort one subproblem with X, using the other subproblem as scratch space. 4 Sort the other subproblem with QUICKXSORT. sort by X sort recursively QuickXsort can turn X into an in-place sorting method! Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 1 / 12

16. QuickMergesort Good candidate for X: Mergesort #comparisons almost optimal but: merging uses extra space as buﬀer merging possible without extra space but not really practical! “Worst merge lanes in the D.C. area” Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 2 / 12

20. QuickMergesort Good candidate for X: Mergesort #comparisons almost optimal but: merging uses extra space as buﬀer merging possible without extra space but not really practical! For this talk: X = Mergesort Goal: inherit good comparison count, but in-place! “Worst merge lanes in the D.C. area” Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 2 / 12

21. QuickMergesort Good candidate for X: Mergesort #comparisons almost optimal but: merging uses extra space as buﬀer merging possible without extra space but not really practical! For this talk: X = Mergesort Goal: inherit good comparison count, but in-place! (and without collisions ...) “Worst merge lanes in the D.C. area” Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 2 / 12

22. Merging with swaps Important: Buffer elements must stay intact. run1 run2 buffer 1 Swap run1 and buffer 2 Merge back (swap min) 3 runs merged, buffer restored (but permuted) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 3 / 12

41. Which side with Mergesort? QUICKMERGESORT(A[l..r]) 1 Choose pivot P. 2 i = PARTITION(A[l..r], P) 5 if ... 6 MERGESORT(A[i + 1..r]) with buﬀer A[l..i − 1] Mergesort QuickMergesort 7 QUICKMERGESORT(A[l..i − 1]). 8 else 9 MERGESORT(A[l..i − 1]) with buﬀer A[i + 1..r] Mergesort QuickMergesort 10 QUICKMERGESORT(A[i..r]). 11 end if Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 4 / 12

42. Which side with Mergesort? QUICKMERGESORT(A[l..r]) 1 Choose pivot P. 2 i = PARTITION(A[l..r], P) 3 J1 := i − l; J2 := r − i 5 if ... 6 MERGESORT(A[i + 1..r]) with buﬀer A[l..i − 1] Mergesort QuickMergesort 7 QUICKMERGESORT(A[l..i − 1]). 8 else 9 MERGESORT(A[l..i − 1]) with buﬀer A[i + 1..r] Mergesort QuickMergesort 10 QUICKMERGESORT(A[i..r]). 11 end if Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 4 / 12

43. Which side with Mergesort? QUICKMERGESORT(A[l..r]) 1 Choose pivot P. 2 i = PARTITION(A[l..r], P) 3 J1 := i − l; J2 := r − i 4 largerWithX := min{J1, J2} 1 2 max{J1, J2} 5 if ... 6 MERGESORT(A[i + 1..r]) with buﬀer A[l..i − 1] Mergesort QuickMergesort 7 QUICKMERGESORT(A[l..i − 1]). 8 else 9 MERGESORT(A[l..i − 1]) with buﬀer A[i + 1..r] Mergesort QuickMergesort 10 QUICKMERGESORT(A[i..r]). 11 end if Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 4 / 12

44. Which side with Mergesort? QUICKMERGESORT(A[l..r]) 1 Choose pivot P. 2 i = PARTITION(A[l..r], P) 3 J1 := i − l; J2 := r − i 4 largerWithX := min{J1, J2} 1 2 max{J1, J2} 5 if largerWithX ∧ J2 J1 6 MERGESORT(A[i + 1..r]) with buﬀer A[l..i − 1] Mergesort QuickMergesort 7 QUICKMERGESORT(A[l..i − 1]). 8 else 9 MERGESORT(A[l..i − 1]) with buﬀer A[i + 1..r] Mergesort QuickMergesort 10 QUICKMERGESORT(A[i..r]). 11 end if Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 4 / 12

45. Which side with Mergesort? QUICKMERGESORT(A[l..r]) 1 Choose pivot P. 2 i = PARTITION(A[l..r], P) 3 J1 := i − l; J2 := r − i 4 largerWithX := min{J1, J2} 1 2 max{J1, J2} 5 if largerWithX ∧ J2 J1 ∨ ¬largerWithX ∧ J2 < J1 6 MERGESORT(A[i + 1..r]) with buﬀer A[l..i − 1] Mergesort QuickMergesort 7 QUICKMERGESORT(A[l..i − 1]). 8 else 9 MERGESORT(A[l..i − 1]) with buﬀer A[i + 1..r] Mergesort QuickMergesort 10 QUICKMERGESORT(A[i..r]). 11 end if Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 4 / 12

46. Which side with Mergesort? QUICKMERGESORT(A[l..r]) 1 Choose pivot P. 2 i = PARTITION(A[l..r], P) 3 J1 := i − l; J2 := r − i 4 largerWithX := min{J1, J2} 1 2 max{J1, J2} 5 if largerWithX ∧ J2 J1 ∨ ¬largerWithX ∧ J2 < J1 6 MERGESORT(A[i + 1..r]) with buffer A[l..i − 1] Mergesort QuickMergesort 7 QUICKMERGESORT(A[l..i − 1]). 8 else 9 MERGESORT(A[l..i − 1]) with buffer A[i + 1..r] Mergesort QuickMergesort 10 QUICKMERGESORT(A[i..r]). 11 end if Sort recursively iff relative size fulfills J1 n − 1 ∈ 1 3 , 1 2 ∪ 2 3 , 1 Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 4 / 12

47. Which side with Mergesort? QUICKMERGESORT(A[l..r]) 1 Choose pivot P. 2 i = PARTITION(A[l..r], P) 3 J1 := i − l; J2 := r − i 4 largerWithX := min{J1, J2} 1 2 max{J1, J2} 5 if largerWithX ∧ J2 J1 ∨ ¬largerWithX ∧ J2 < J1 6 MERGESORT(A[i + 1..r]) with buffer A[l..i − 1] Mergesort QuickMergesort 7 QUICKMERGESORT(A[l..i − 1]). 8 else 9 MERGESORT(A[l..i − 1]) with buffer A[i + 1..r] Mergesort QuickMergesort 10 QUICKMERGESORT(A[i..r]). 11 end if Sort recursively iff relative size fulfills J1 n − 1 ∈ 1 3 , 1 2 ∪ 2 3 , 1 J1 n − 1 ∈ 1 3 1 2 2 3 10 Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 4 / 12

48. Which side with Mergesort? QUICKMERGESORT(A[l..r]) 1 Choose pivot P. 2 i = PARTITION(A[l..r], P) 3 J1 := i − l; J2 := r − i 4 largerWithX := min{J1, J2} 1 2 max{J1, J2} 5 if largerWithX ∧ J2 J1 ∨ ¬largerWithX ∧ J2 < J1 6 MERGESORT(A[i + 1..r]) with buffer A[l..i − 1] Mergesort QuickMergesort 7 QUICKMERGESORT(A[l..i − 1]). 8 else 9 MERGESORT(A[l..i − 1]) with buffer A[i + 1..r] Mergesort QuickMergesort 10 QUICKMERGESORT(A[i..r]). 11 end if Sort recursively iff relative size fulfills J1 n − 1 ∈ 1 3 , 1 2 ∪ 2 3 , 1 J1 n − 1 ∈ 1 3 1 2 2 3 10 Similarly: mergesort iff J1 n − 1 ∈ 0 1 3 1 2 2 3 1 Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 4 / 12

50. Costs of QuickXsort Assumptions: average costs #comparisons for X: x(n) = an lg n + bn ± O(n1−ε) αn buﬀer space X = Mergesort: x(n) = n lg n + [−1.26, −1.24 ]n ± O(1) 1 2 n buﬀer space Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 5 / 12

51. Costs of QuickXsort Assumptions: average costs #comparisons for X: x(n) = an lg n + bn ± O(n1−ε) αn buﬀer space X = Mergesort: x(n) = n lg n + [−1.26, −1.24 ]n ± O(1) 1 2 n buﬀer space Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 5 / 12

52. Costs of QuickXsort Assumptions: average costs #comparisons for X: x(n) = an lg n + bn ± O(n1−ε) αn buﬀer space X = Mergesort: x(n) = n lg n + [−1.26, −1.24 ]n ± O(1) 1 2 n buﬀer space 211 212 213 214 215 216 −1.26 −1.25 −1.24 x(n) − n lg n n (log-plot) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 5 / 12

53. Costs of QuickXsort Assumptions: average costs #comparisons for X: x(n) = an lg n + bn ± O(n1−ε) αn buﬀer space X = Mergesort: x(n) = n lg n + [−1.26, −1.24]n ± O(1) 1 2 n buﬀer space Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 5 / 12

54. Costs of QuickXsort Assumptions: average costs #comparisons for X: x(n) = an lg n + bn ± O(n1−ε) αn buﬀer space X = Mergesort: x(n) = n lg n + [−1.26, −1.24]n ± O(1) 1 2 n buﬀer space c(n) = (n − k) + b(k) + E J1 n−1 ∈ 1 3 , 1 2 ∪ 2 3 , 1 · x(J2) + c(J1) + E J2 n−1 ∈ 1 3 , 1 2 ∪ 2 3 , 1 · x(J1) + c(J2) (n > k) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 5 / 12

55. Costs of QuickXsort Assumptions: average costs #comparisons for X: x(n) = an lg n + bn ± O(n1−ε) αn buﬀer space X = Mergesort: x(n) = n lg n + [−1.26, −1.24]n ± O(1) 1 2 n buﬀer space c(n) = (n − k) partition + b(k) + E J1 n−1 ∈ 1 3 , 1 2 ∪ 2 3 , 1 · x(J2) + c(J1) + E J2 n−1 ∈ 1 3 , 1 2 ∪ 2 3 , 1 · x(J1) + c(J2) (n > k) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 5 / 12

56. Costs of QuickXsort Assumptions: average costs #comparisons for X: x(n) = an lg n + bn ± O(n1−ε) αn buﬀer space X = Mergesort: x(n) = n lg n + [−1.26, −1.24]n ± O(1) 1 2 n buﬀer space c(n) = (n − k) + b(k) pivot sampling + E J1 n−1 ∈ 1 3 , 1 2 ∪ 2 3 , 1 · x(J2) + c(J1) + E J2 n−1 ∈ 1 3 , 1 2 ∪ 2 3 , 1 · x(J1) + c(J2) (n > k) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 5 / 12

57. Costs of QuickXsort Assumptions: average costs #comparisons for X: x(n) = an lg n + bn ± O(n1−ε) αn buﬀer space X = Mergesort: x(n) = n lg n + [−1.26, −1.24]n ± O(1) 1 2 n buﬀer space c(n) = (n − k) + b(k) + E J1 n−1 ∈ 1 3 , 1 2 ∪ 2 3 , 1 · x(J2) Mergesort right + c(J1) recurse left + E J2 n−1 ∈ 1 3 , 1 2 ∪ 2 3 , 1 · x(J1) + c(J2) (n > k) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 5 / 12

58. Costs of QuickXsort Assumptions: average costs #comparisons for X: x(n) = an lg n + bn ± O(n1−ε) αn buﬀer space X = Mergesort: x(n) = n lg n + [−1.26, −1.24]n ± O(1) 1 2 n buﬀer space c(n) = (n − k) + b(k) + E J1 n−1 ∈ 1 3 , 1 2 ∪ 2 3 , 1 · x(J2) + c(J1) + E J2 n−1 ∈ 1 3 , 1 2 ∪ 2 3 , 1 · x(J1) Mergesort left + c(J2) recurse right (n > k) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 5 / 12

59. Costs of QuickXsort Assumptions: average costs #comparisons for X: x(n) = an lg n + bn ± O(n1−ε) αn buﬀer space X = Mergesort: x(n) = n lg n + [−1.26, −1.24]n ± O(1) 1 2 n buﬀer space c(n) = (n − k) + b(k) + E J1 n−1 ∈ · x(J2) + c(J1) + E J2 n−1 ∈ · x(J1) + c(J2) (n > k) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 5 / 12

60. Costs of QuickXsort Assumptions: average costs #comparisons for X: x(n) = an lg n + bn ± O(n1−ε) αn buﬀer space X = Mergesort: x(n) = n lg n + [−1.26, −1.24]n ± O(1) 1 2 n buﬀer space c(n) = (n − k) + b(k) + E J1 n−1 ∈ · x(J2) + c(J1) + E J2 n−1 ∈ · x(J1) + c(J2) (n > k) + base cases Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 5 / 12

61. Costs of QuickXsort Assumptions: average costs #comparisons for X: x(n) = an lg n + bn ± O(n1−ε) αn buﬀer space X = Mergesort: x(n) = n lg n + [−1.26, −1.24]n ± O(1) 1 2 n buﬀer space c(n) = (n − k) + b(k) + E J1 n−1 ∈ · x(J2) + c(J1) + E J2 n−1 ∈ · x(J1) + c(J2) (n > k) = t(n) + n−1 j=0 wn,j · c(j) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 5 / 12

62. Costs of QuickXsort Assumptions: average costs #comparisons for X: x(n) = an lg n + bn ± O(n1−ε) αn buﬀer space X = Mergesort: x(n) = n lg n + [−1.26, −1.24]n ± O(1) 1 2 n buﬀer space c(n) = (n − k) + b(k) + E J1 n−1 ∈ · x(J2) + c(J1) + E J2 n−1 ∈ · x(J1) + c(J2) (n > k) = t(n) + n−1 j=0 wn,j · c(j) shape for Roura’s continuous master theorem: • c(n) not dominated by “leaves” of recursion • t(n) “standard scale”: t(n) ∼ Knα logβ n • wn,j “smooth”: wn,zn ≈ w(z) c(n) ∼ solution of c(n) = t(n) + w(z)c(zn)dz (which is known explicitly) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 5 / 12

68. Costs of QuickXsort Assumptions: average costs #comparisons for X: x(n) = an lg n + bn ± O(n1−ε) αn buﬀer space X = Mergesort: x(n) = n lg n + [−1.26, −1.24]n ± O(1) 1 2 n buﬀer space c(n) = (n − k) + b(k) + E J1 n−1 ∈ · x(J2) + c(J1) + E J2 n−1 ∈ · x(J1) + c(J2) (n > k) = t(n) + n−1 j=0 wn,j · c(j) shape for Roura’s continuous master theorem: • c(n) not dominated by “leaves” of recursion • t(n) “standard scale”: t(n) ∼ Knα logβ n ( ) • wn,j “smooth”: wn,zn ≈ w(z) c(n) ∼ solution of c(n) = t(n) + w(z)c(zn)dz (which is known explicitly) t(n) = (n − k) + b(k) + E J1 n−1 ∈ x(J1) + E J2 n−1 ∈ x(J2) wn,j =    1 · P[J1 = j] if j n−1 = 1 2 2 · P[J1 = j] if j n−1 ∈ 0 otherwise Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 5 / 12

69. Costs of QuickXsort Assumptions: average costs #comparisons for X: x(n) = an lg n + bn ± O(n1−ε) αn buﬀer space X = Mergesort: x(n) = n lg n + [−1.26, −1.24]n ± O(1) 1 2 n buﬀer space c(n) = (n − k) + b(k) + E J1 n−1 ∈ · x(J2) + c(J1) + E J2 n−1 ∈ · x(J1) + c(J2) (n > k) = t(n) + n−1 j=0 wn,j · c(j) shape for Roura’s continuous master theorem: • c(n) not dominated by “leaves” of recursion • t(n) “standard scale”: t(n) ∼ Knα logβ n ( ) • wn,j “smooth”: wn,zn ≈ w(z) ( ) c(n) ∼ solution of c(n) = t(n) + w(z)c(zn)dz (which is known explicitly) t(n) = (n − k) + b(k) + E J1 n−1 ∈ x(J1) + E J2 n−1 ∈ x(J2) wn,j =    1 · P[J1 = j] if j n−1 = 1 2 2 · P[J1 = j] if j n−1 ∈ 0 otherwise Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 5 / 12

70. Costs of QuickXsort Assumptions: average costs #comparisons for X: x(n) = an lg n + bn ± O(n1−ε) αn buﬀer space X = Mergesort: x(n) = n lg n + [−1.26, −1.24]n ± O(1) 1 2 n buﬀer space c(n) = (n − k) + b(k) + E J1 n−1 ∈ · x(J2) + c(J1) + E J2 n−1 ∈ · x(J1) + c(J2) (n > k) = t(n) + n−1 j=0 wn,j · c(j) shape for Roura’s continuous master theorem: • c(n) not dominated by “leaves” of recursion • t(n) “standard scale”: t(n) ∼ Knα logβ n ( ) • wn,j “smooth”: wn,zn ≈ w(z) ( ) c(n) ∼ solution of c(n) = t(n) + w(z)c(zn)dz (which is known explicitly) t(n) = (n − k) + b(k) + E J1 n−1 ∈ x(J1) + E J2 n−1 ∈ x(J2) wn,j =    1 · P[J1 = j] if j n−1 = 1 2 2 · P[J1 = j] if j n−1 ∈ 0 otherwise 0 10 20 30 40 50 w51,j for median-of-3 QuickMergesort Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 5 / 12

71. Shape Function wn,j “smooth” precise criterion: ∃w the “shape function” : [0, 1] → R 0 : n−1 j=0 wn,j − (j+1)/n j/n w(z) dz = O(n−1) (n→∞). subproblem size distribution: J1 D = J2 D = Binomial beta-binomial distribution (n, Beta(t + 1, t + 1)) ± O(1) binomial distribution highly concen Chernoﬀ etc. trated J1 n D ≈ P D = Beta median of 2t+1 i.i.d. Uniform(0, 1) r.v. (t + 1, t + 1) formally: nP J1 = zn = fBeta(z) = zt (1−z)t /B(t+1, t+1) ± O(n−1) w(z) = 2 z ∈ · fBeta(z) valid shape function Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 6 / 12

72. Shape Function wn,j “smooth” wn,j =    1 · P[J1 = j] if j n−1 = 1 2 2 · P[J1 = j] if j n−1 ∈ 0 otherwise precise criterion: ∃w the “shape function” : [0, 1] → R 0 : n−1 j=0 wn,j − (j+1)/n j/n w(z) dz = O(n−1) (n→∞). subproblem size distribution: J1 D = J2 D = Binomial beta-binomial distribution (n, Beta(t + 1, t + 1)) ± O(1) binomial distribution highly concen Chernoﬀ etc. trated J1 n D ≈ P D = Beta median of 2t+1 i.i.d. Uniform(0, 1) r.v. (t + 1, t + 1) formally: nP J1 = zn = fBeta(z) = zt (1−z)t /B(t+1, t+1) ± O(n−1) w(z) = 2 z ∈ · fBeta(z) valid shape function Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 6 / 12

76. Shape Function wn,j “smooth” wn,j =    1 · P[J1 = j] if j n−1 = 1 2 2 · P[J1 = j] if j n−1 ∈ 0 otherwise precise criterion: ∃w the “shape function” : [0, 1] → R 0 : n−1 j=0 wn,j − (j+1)/n j/n w(z) ≈ nwn,zn w(z) dz = O(n−1) (n→∞). subproblem size distribution: J1 D = J2 D = Binomial beta-binomial distribution (n, Beta(t + 1, t + 1)) ± O(1) binomial distribution highly concen Chernoﬀ etc. trated J1 n D ≈ P D = Beta median of 2t+1 i.i.d. Uniform(0, 1) r.v. (t + 1, t + 1) formally: nP J1 = zn = fBeta(z) = zt (1−z)t /B(t+1, t+1) ± O(n−1) w(z) = 2 z ∈ · fBeta(z) valid shape function Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 6 / 12

77. Shape Function wn,j “smooth” wn,j =    1 · P[J1 = j] if j n−1 = 1 2 2 · P[J1 = j] if j n−1 ∈ 0 otherwise precise criterion: ∃w the “shape function” : [0, 1] → R 0 : n−1 j=0 wn,j − (j+1)/n j/n w(z) ≈ nwn,zn w(z) dz = O(n−1) ...with guaranteed convergence rate (n→∞). subproblem size distribution: J1 D = J2 D = Binomial beta-binomial distribution (n, Beta(t + 1, t + 1)) ± O(1) binomial distribution highly concen Chernoﬀ etc. trated J1 n D ≈ P D = Beta median of 2t+1 i.i.d. Uniform(0, 1) r.v. (t + 1, t + 1) formally: nP J1 = zn = fBeta(z) = zt (1−z)t /B(t+1, t+1) ± O(n−1) w(z) = 2 z ∈ · fBeta(z) valid shape function Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 6 / 12

83. Shape Function wn,j “smooth” wn,j =    1 · P[J1 = j] if j n−1 = 1 2 2 · P[J1 = j] if j n−1 ∈ 0 otherwise precise criterion: ∃w the “shape function” : [0, 1] → R 0 : n−1 j=0 wn,j − (j+1)/n j/n w(z) ≈ nwn,zn w(z) dz = O(n−1) ...with guaranteed convergence rate (n→∞). subproblem size distribution: J1 D = J2 D = Binomial beta-binomial distribution (n, Beta(t + 1, t + 1)) ± O(1) binomial distribution highly concen Chernoﬀ etc. trated J1 n D ≈ P D = Beta median of 2t+1 i.i.d. Uniform(0, 1) r.v. (t + 1, t + 1) formally: nP J1 = zn = fBeta(z) = zt (1−z)t /B(t+1, t+1) ± O(n−1) w(z) = 2 z ∈ · fBeta(z) valid shape function 0 0.2 0.4 0.6 0.8 1 0 1 2 3 n · wn,zn vs. w(z) (n = 51, k = 3) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 6 / 12

86. Expected Subproblem Size Intuition behind shape function? w(z) = 2 z ∈ · fBeta(t+1,t+1)(z) w(z) is limiting density of relative subproblem size of recursive call ( 1 0 w(z)dz = 1) expected fraction of elements in recursive call: r := 1 0 z · w(z)dz = 2 z∈ incomplete beta integral z · zt(1 − z)t B(t + 1, t + 1) dz = 2E P ∈ · P Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 7 / 12

89. Expected Subproblem Size Intuition behind shape function? w(z) = 2 z ∈ · fBeta(t+1,t+1)(z) w(z) is limiting density of relative subproblem size of recursive call ( 1 0 w(z)dz = 1) expected fraction of elements in recursive call: r := 1 0 z · w(z)dz = 2 z∈ incomplete beta integral z · zt(1 − z)t B(t + 1, t + 1) dz = 2E P ∈ · P sample size fraction of elements in mergesort / recursive call fBeta(t+1,t+1) w(z) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 7 / 12

90. Expected Subproblem Size Intuition behind shape function? w(z) = 2 z ∈ · fBeta(t+1,t+1)(z) w(z) is limiting density of relative subproblem size of recursive call ( 1 0 w(z)dz = 1) expected fraction of elements in recursive call: r := 1 0 z · w(z)dz = 2 z∈ incomplete beta integral z · zt(1 − z)t B(t + 1, t + 1) dz = 2E P ∈ · P sample size fraction of elements in mergesort / recursive call fBeta(t+1,t+1) w(z) none 30.6% mergesort 69.4% recursive Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 7 / 12

91. Expected Subproblem Size Intuition behind shape function? w(z) = 2 z ∈ · fBeta(t+1,t+1)(z) w(z) is limiting density of relative subproblem size of recursive call ( 1 0 w(z)dz = 1) expected fraction of elements in recursive call: r := 1 0 z · w(z)dz = 2 z∈ incomplete beta integral z · zt(1 − z)t B(t + 1, t + 1) dz = 2E P ∈ · P sample size fraction of elements in mergesort / recursive call fBeta(t+1,t+1) w(z) none 30.6% mergesort 69.4% recursive median of 3 39.1% mergesort 60.9% recursive Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 7 / 12

92. Expected Subproblem Size Intuition behind shape function? w(z) = 2 z ∈ · fBeta(t+1,t+1)(z) w(z) is limiting density of relative subproblem size of recursive call ( 1 0 w(z)dz = 1) expected fraction of elements in recursive call: r := 1 0 z · w(z)dz = 2 z∈ incomplete beta integral z · zt(1 − z)t B(t + 1, t + 1) dz = 2E P ∈ · P sample size fraction of elements in mergesort / recursive call fBeta(t+1,t+1) w(z) none 30.6% mergesort 69.4% recursive median of 3 39.1% mergesort 60.9% recursive median of 5 43.7% mergesort 56.3% recursive Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 7 / 12

93. Expected Subproblem Size Intuition behind shape function? w(z) = 2 z ∈ · fBeta(t+1,t+1)(z) w(z) is limiting density of relative subproblem size of recursive call ( 1 0 w(z)dz = 1) expected fraction of elements in recursive call: r := 1 0 z · w(z)dz = 2 z∈ incomplete beta integral z · zt(1 − z)t B(t + 1, t + 1) dz = 2E P ∈ · P sample size fraction of elements in mergesort / recursive call fBeta(t+1,t+1) w(z) none 30.6% mergesort 69.4% recursive median of 3 39.1% mergesort 60.9% recursive median of 5 43.7% mergesort 56.3% recursive median of 7 46.6% mergesort 53.4% recursive Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 7 / 12

94. Expected Subproblem Size Intuition behind shape function? w(z) = 2 z ∈ · fBeta(t+1,t+1)(z) w(z) is limiting density of relative subproblem size of recursive call ( 1 0 w(z)dz = 1) expected fraction of elements in recursive call: r := 1 0 z · w(z)dz = 2 z∈ incomplete beta integral z · zt(1 − z)t B(t + 1, t + 1) dz = 2E P ∈ · P sample size fraction of elements in mergesort / recursive call fBeta(t+1,t+1) w(z) none 30.6% mergesort 69.4% recursive median of 3 39.1% mergesort 60.9% recursive median of 5 43.7% mergesort 56.3% recursive median of 7 46.6% mergesort 53.4% recursive median of 41 55.1% mergesort 44.9% recursive Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 7 / 12

95. Expected Subproblem Size Intuition behind shape function? w(z) = 2 z ∈ · fBeta(t+1,t+1)(z) w(z) is limiting density of relative subproblem size of recursive call ( 1 0 w(z)dz = 1) expected fraction of elements in recursive call: r := 1 0 z · w(z)dz = 2 z∈ incomplete beta integral z · zt(1 − z)t B(t + 1, t + 1) dz = 2E P ∈ · P sample size fraction of elements in mergesort / recursive call fBeta(t+1,t+1) w(z) none 30.6% mergesort 69.4% recursive median of 3 39.1% mergesort 60.9% recursive median of 5 43.7% mergesort 56.3% recursive median of 7 46.6% mergesort 53.4% recursive median of 41 55.1% mergesort 44.9% recursive exact median 50% mergesort 50% recursive Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 7 / 12

97. The leading term t(n) ∼ Knα logβ n t(n) = E J1 n−1 ∈ x(J1) + E J2 n−1 ∈ x(J2) +(n−k)+b(k) E J1 n−1 ∈ x(J1) Recall: x(n) = n lg n − 1.24n ± O(1) J1 D ≈ Pn where P D = Beta(t + 1, t + 1) ∼ E P ∈ Pn lg(Pn) ∼ E P ∈ Pn lg( n ) = E P ∈ · P · n lg n = E P ∈ · (1 − P) · n lg n Recall: r = 2E P ∈ · P = 1 2 (1 − r) · n lg n c(n) ∼ solution of c(n) = t(n) + w(z)c(zn)dz Ansatz: c(n) ∼ γ · n lg n γ · n lg n = (1 − r) · n lg n + γr · n lg n Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 8 / 12

109. The leading term t(n) ∼ Knα logβ n t(n) = E J1 n−1 ∈ x(J1) + E J2 n−1 ∈ x(J2) +(n−k)+b(k) E J1 n−1 ∈ x(J1) Recall: x(n) = n lg n − 1.24n ± O(1) J1 D ≈ Pn where P D = Beta(t + 1, t + 1) ∼ E P ∈ Pn lg(Pn) ∼ E P ∈ Pn lg( n ) = E P ∈ · P · n lg n = E P ∈ · (1 − P) · n lg n Recall: r = 2E P ∈ · P = 1 2 (1 − r) · n lg n c(n) ∼ solution of c(n) = t(n) + w(z)c(zn)dz Ansatz: c(n) ∼ γ · n lg n γ · n lg n = (1 − r) · n lg n + γr · n lg n γ = 1 c(n) ∼ x(n)! Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 8 / 12

110. The leading term t(n) ∼ Knα logβ n t(n) = E J1 n−1 ∈ x(J1) + E J2 n−1 ∈ x(J2) +(n−k)+b(k) E J1 n−1 ∈ x(J1) Recall: x(n) = n lg n − 1.24n ± O(1) J1 D ≈ Pn where P D = Beta(t + 1, t + 1) ∼ E P ∈ Pn lg(Pn) ∼ E P ∈ Pn lg( n ) = E P ∈ · P · n lg n = E P ∈ · (1 − P) · n lg n Recall: r = 2E P ∈ · P = 1 2 (1 − r) · n lg n c(n) ∼ solution of c(n) = t(n) + w(z)c(zn)dz Ansatz: c(n) ∼ γ · n lg n γ · n lg n = (1 − r) · n lg n + γr · n lg n γ = 1 c(n) ∼ x(n)! QuickMergesort is optimal w.r.t. leading term! Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 8 / 12

111. The leading term t(n) ∼ Knα logβ n t(n) = E J1 n−1 ∈ x(J1) + E J2 n−1 ∈ x(J2) +(n−k)+b(k) E J1 n−1 ∈ x(J1) Recall: x(n) = n lg n − 1.24n ± O(1) J1 D ≈ Pn where P D = Beta(t + 1, t + 1) ∼ E P ∈ Pn lg(Pn) ∼ E P ∈ Pn lg( n ) = E P ∈ · P · n lg n = E P ∈ · (1 − P) · n lg n Recall: r = 2E P ∈ · P = 1 2 (1 − r) · n lg n c(n) ∼ solution of c(n) = t(n) + w(z)c(zn)dz Ansatz: c(n) ∼ γ · n lg n γ · n lg n = (1 − r) · n lg n + γr · n lg n γ = 1 c(n) ∼ x(n)! QuickMergesort is optimal w.r.t. leading term! Is that surprising? Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 8 / 12

112. The leading term t(n) ∼ Knα logβ n t(n) = E J1 n−1 ∈ x(J1) + E J2 n−1 ∈ x(J2) +(n−k)+b(k) E J1 n−1 ∈ x(J1) Recall: x(n) = n lg n − 1.24n ± O(1) J1 D ≈ Pn where P D = Beta(t + 1, t + 1) ∼ E P ∈ Pn lg(Pn) ∼ E P ∈ Pn lg( n ) = E P ∈ · P · n lg n = E P ∈ · (1 − P) · n lg n Recall: r = 2E P ∈ · P = 1 2 (1 − r) · n lg n c(n) ∼ solution of c(n) = t(n) + w(z)c(zn)dz Ansatz: c(n) ∼ γ · n lg n γ · n lg n = (1 − r) · n lg n + γr · n lg n γ = 1 c(n) ∼ x(n)! QuickMergesort is optimal w.r.t. leading term! Is that surprising? recurse recurse recurse recursesize of mergesort parts sum to n x(αn) ∼ α · x(n) c(n) ∼ x(n) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 8 / 12

113. The leading term t(n) ∼ Knα logβ n t(n) = E J1 n−1 ∈ x(J1) + E J2 n−1 ∈ x(J2) +(n−k)+b(k) E J1 n−1 ∈ x(J1) Recall: x(n) = n lg n − 1.24n ± O(1) J1 D ≈ Pn where P D = Beta(t + 1, t + 1) ∼ E P ∈ Pn lg(Pn) ∼ E P ∈ Pn lg( n ) = E P ∈ · P · n lg n = E P ∈ · (1 − P) · n lg n Recall: r = 2E P ∈ · P = 1 2 (1 − r) · n lg n c(n) ∼ solution of c(n) = t(n) + w(z)c(zn)dz Ansatz: c(n) ∼ γ · n lg n γ · n lg n = (1 − r) · n lg n + γr · n lg n γ = 1 c(n) ∼ x(n)! QuickMergesort is optimal w.r.t. leading term! Is that surprising? recurse recurserecurserecurse size of mergesort parts sum to n x(αn) ∼ α · x(n) c(n) ∼ x(n) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 8 / 12

114. The leading term t(n) ∼ Knα logβ n t(n) = E J1 n−1 ∈ x(J1) + E J2 n−1 ∈ x(J2) +(n−k)+b(k) E J1 n−1 ∈ x(J1) Recall: x(n) = n lg n − 1.24n ± O(1) J1 D ≈ Pn where P D = Beta(t + 1, t + 1) ∼ E P ∈ Pn lg(Pn) ∼ E P ∈ Pn lg( n ) = E P ∈ · P · n lg n = E P ∈ · (1 − P) · n lg n Recall: r = 2E P ∈ · P = 1 2 (1 − r) · n lg n c(n) ∼ solution of c(n) = t(n) + w(z)c(zn)dz Ansatz: c(n) ∼ γ · n lg n γ · n lg n = (1 − r) · n lg n + γr · n lg n γ = 1 c(n) ∼ x(n)! QuickMergesort is optimal w.r.t. leading term! Is that surprising? recurse recurserecurserecurse size of mergesort parts sum to n x(αn) ∼ α · x(n) (leading term behaves linear) c(n) ∼ x(n) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 8 / 12

115. The leading term t(n) ∼ Knα logβ n t(n) = E J1 n−1 ∈ x(J1) + E J2 n−1 ∈ x(J2) +(n−k)+b(k) E J1 n−1 ∈ x(J1) Recall: x(n) = n lg n − 1.24n ± O(1) J1 D ≈ Pn where P D = Beta(t + 1, t + 1) ∼ E P ∈ Pn lg(Pn) ∼ E P ∈ Pn lg( n ) = E P ∈ · P · n lg n = E P ∈ · (1 − P) · n lg n Recall: r = 2E P ∈ · P = 1 2 (1 − r) · n lg n c(n) ∼ solution of c(n) = t(n) + w(z)c(zn)dz Ansatz: c(n) ∼ γ · n lg n γ · n lg n = (1 − r) · n lg n + γr · n lg n γ = 1 c(n) ∼ x(n)! QuickMergesort is optimal w.r.t. leading term! Is that surprising? recurse recurserecurserecurse size of mergesort parts sum to n x(αn) ∼ α · x(n) (leading term behaves linear) c(n) ∼ x(n) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 8 / 12

116. The leading term t(n) ∼ Knα logβ n t(n) = E J1 n−1 ∈ x(J1) + E J2 n−1 ∈ x(J2) +(n−k)+b(k) E J1 n−1 ∈ x(J1) Recall: x(n) = n lg n − 1.24n ± O(1) J1 D ≈ Pn where P D = Beta(t + 1, t + 1) ∼ E P ∈ Pn lg(Pn) ∼ E P ∈ Pn lg( n ) = E P ∈ · P · n lg n = E P ∈ · (1 − P) · n lg n Recall: r = 2E P ∈ · P = 1 2 (1 − r) · n lg n c(n) ∼ solution of c(n) = t(n) + w(z)c(zn)dz Ansatz: c(n) ∼ γ · n lg n γ · n lg n = (1 − r) · n lg n + γr · n lg n γ = 1 c(n) ∼ x(n)! QuickMergesort is optimal w.r.t. leading term! Is that surprising? recurse recurserecurserecurse size of mergesort parts sum to n x(αn) ∼ α · x(n) (leading term behaves linear) c(n) ∼ that was known previously x(n) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 8 / 12

117. The leading term t(n) ∼ Knα logβ n t(n) = E J1 n−1 ∈ x(J1) + E J2 n−1 ∈ x(J2) +(n−k)+b(k) E J1 n−1 ∈ x(J1) Recall: x(n) = n lg n − 1.24n ± O(1) J1 D ≈ Pn where P D = Beta(t + 1, t + 1) ∼ E P ∈ Pn lg(Pn) ∼ E P ∈ Pn lg( n ) = E P ∈ · P · n lg n = E P ∈ · (1 − P) · n lg n Recall: r = 2E P ∈ · P = 1 2 (1 − r) · n lg n c(n) ∼ solution of c(n) = t(n) + w(z)c(zn)dz Ansatz: c(n) ∼ γ · n lg n γ · n lg n = (1 − r) · n lg n + γr · n lg n γ = 1 c(n) ∼ x(n)! QuickMergesort is optimal w.r.t. leading term! Is that surprising? recurse recurserecurserecurse size of mergesort parts sum to n x(αn) ∼ α · x(n) (leading term behaves linear) c(n) ∼ that was known previously x(n) ... algorithm details do not matter! Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 8 / 12

118. The linear term Linear term depends on pivot sampling partitioning method linear term of mergesort but not on how sma unlike for Quicksort! ll subproblems solved! Consider c (n) = c(n) − n lg n Can show: Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 9 / 12

124. The linear term Linear term depends on pivot sampling partitioning method linear term of mergesort but not on how sma unlike for Quicksort! ll subproblems solved! Consider c (n) = c(n) − n lg n Can show: c (n) = c(n) − an lg n = 2 r=1 E Ar(Jr)c(Jr) − an lg n + t(n) = 2 r=1 E Ar(Jr) c(Jr) − aJr lg Jr + a E Ar(Jr)Jr lg Jr − an lg n + (n − 1) + E A2(J2) · x(J1) + E A1(J1) · x(J2) = 2 r=1 E Ar(Jr)c (Jr) + (n − 1) − an lg n + a E A1(J1) + A2(J2) J1 lg J1 + b E[A2(J2)J1] + a E A2(J2) + A1(J1) J2 lg J2 + b E[A1(J1)J2] ± O(n1−ε ) Since J1 D = J2 we can simplify E A1(J1) + A2(J2) J1 lg J1 + E A2(J2) + A1(J1) J2 lg J2 = E A1(J1) + A2(J2) J1 lg J1 + E A2(J1) + A1(J2) J1 lg J1 = E J1 lg J1 · A1(J1) + A1(J2) + A2(J1) + A2(J2) = 2E[J lg J] = 2 E[ J n ] · n lg n + 2 · 1 ln 2 E[ J n ln J n ] · n = n lg n − 1 ln 2 Hk+1 − Ht+1 n ± O(n1−ε ). Plugging this back into our equation for c (n), we ﬁnd c (n) = 2 r=1 E Ar(Jr)c (Jr) + (n − 1) − an lg n + a n lg n − 1 ln 2 Hk+1 − Ht+1 n + b I0, 1 3 (t + 2, t + 1) + I1 2 , 2 3 (t + 2, t + 1) · n ± O(n1−ε ) = 2 r=1 E Ar(Jr)c (Jr) + t (n) where t (n) = b n ± O(n1−ε ) b = 1 − a ln 2 Hk+1 − Ht+1 + b · H Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 9 / 12

125. The linear term Linear term depends on pivot sampling partitioning method linear term of mergesort but not on how sma unlike for Quicksort! ll subproblems solved! Consider c (n) = c(n) − n lg n Can show: c (n) = c(n) − an lg n = 2 r=1 E Ar(Jr)c(Jr) − an lg n + t(n) = 2 r=1 E Ar(Jr) c(Jr) − aJr lg Jr + a E Ar(Jr)Jr lg Jr − an lg n + (n − 1) + E A2(J2) · x(J1) + E A1(J1) · x(J2) = 2 r=1 E Ar(Jr)c (Jr) + (n − 1) − an lg n + a E A1(J1) + A2(J2) J1 lg J1 + b E[A2(J2)J1] + a E A2(J2) + A1(J1) J2 lg J2 + b E[A1(J1)J2] ± O(n1−ε ) Since J1 D = J2 we can simplify E A1(J1) + A2(J2) J1 lg J1 + E A2(J2) + A1(J1) J2 lg J2 = E A1(J1) + A2(J2) J1 lg J1 + E A2(J1) + A1(J2) J1 lg J1 = E J1 lg J1 · A1(J1) + A1(J2) + A2(J1) + A2(J2) = 2E[J lg J] = 2 E[ J n ] · n lg n + 2 · 1 ln 2 E[ J n ln J n ] · n = n lg n − 1 ln 2 Hk+1 − Ht+1 n ± O(n1−ε ). Plugging this back into our equation for c (n), we ﬁnd c (n) = 2 r=1 E Ar(Jr)c (Jr) + (n − 1) − an lg n + a n lg n − 1 ln 2 Hk+1 − Ht+1 n + b I0, 1 3 (t + 2, t + 1) + I1 2 , 2 3 (t + 2, t + 1) · n ± O(n1−ε ) = 2 r=1 E Ar(Jr)c (Jr) + t (n) where t (n) = b n ± O(n1−ε ) b = 1 − a ln 2 Hk+1 − Ht+1 + b · H c (n) = t (n) + n−1 j=0 wn,j · c (j) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 9 / 12

126. The linear term Linear term depends on pivot sampling partitioning method linear term of mergesort but not on how sma unlike for Quicksort! ll subproblems solved! Consider c (n) = c(n) − n lg n Can show: c (n) = c(n) − an lg n = 2 r=1 E Ar(Jr)c(Jr) − an lg n + t(n) = 2 r=1 E Ar(Jr) c(Jr) − aJr lg Jr + a E Ar(Jr)Jr lg Jr − an lg n + (n − 1) + E A2(J2) · x(J1) + E A1(J1) · x(J2) = 2 r=1 E Ar(Jr)c (Jr) + (n − 1) − an lg n + a E A1(J1) + A2(J2) J1 lg J1 + b E[A2(J2)J1] + a E A2(J2) + A1(J1) J2 lg J2 + b E[A1(J1)J2] ± O(n1−ε ) Since J1 D = J2 we can simplify E A1(J1) + A2(J2) J1 lg J1 + E A2(J2) + A1(J1) J2 lg J2 = E A1(J1) + A2(J2) J1 lg J1 + E A2(J1) + A1(J2) J1 lg J1 = E J1 lg J1 · A1(J1) + A1(J2) + A2(J1) + A2(J2) = 2E[J lg J] = 2 E[ J n ] · n lg n + 2 · 1 ln 2 E[ J n ln J n ] · n = n lg n − 1 ln 2 Hk+1 − Ht+1 n ± O(n1−ε ). Plugging this back into our equation for c (n), we ﬁnd c (n) = 2 r=1 E Ar(Jr)c (Jr) + (n − 1) − an lg n + a n lg n − 1 ln 2 Hk+1 − Ht+1 n + b I0, 1 3 (t + 2, t + 1) + I1 2 , 2 3 (t + 2, t + 1) · n ± O(n1−ε ) = 2 r=1 E Ar(Jr)c (Jr) + t (n) where t (n) = b n ± O(n1−ε ) b = 1 − a ln 2 Hk+1 − Ht+1 + b · H c (n) = t (n) + n−1 j=0 w same shape as c(n) n,j · c (j) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 9 / 12

127. The linear term Linear term depends on pivot sampling partitioning method linear term of mergesort but not on how sma unlike for Quicksort! ll subproblems solved! Consider c (n) = c(n) − n lg n Can show: c (n) = c(n) − an lg n = 2 r=1 E Ar(Jr)c(Jr) − an lg n + t(n) = 2 r=1 E Ar(Jr) c(Jr) − aJr lg Jr + a E Ar(Jr)Jr lg Jr − an lg n + (n − 1) + E A2(J2) · x(J1) + E A1(J1) · x(J2) = 2 r=1 E Ar(Jr)c (Jr) + (n − 1) − an lg n + a E A1(J1) + A2(J2) J1 lg J1 + b E[A2(J2)J1] + a E A2(J2) + A1(J1) J2 lg J2 + b E[A1(J1)J2] ± O(n1−ε ) Since J1 D = J2 we can simplify E A1(J1) + A2(J2) J1 lg J1 + E A2(J2) + A1(J1) J2 lg J2 = E A1(J1) + A2(J2) J1 lg J1 + E A2(J1) + A1(J2) J1 lg J1 = E J1 lg J1 · A1(J1) + A1(J2) + A2(J1) + A2(J2) = 2E[J lg J] = 2 E[ J n ] · n lg n + 2 · 1 ln 2 E[ J n ln J n ] · n = n lg n − 1 ln 2 Hk+1 − Ht+1 n ± O(n1−ε ). Plugging this back into our equation for c (n), we ﬁnd c (n) = 2 r=1 E Ar(Jr)c (Jr) + (n − 1) − an lg n + a n lg n − 1 ln 2 Hk+1 − Ht+1 n + b I0, 1 3 (t + 2, t + 1) + I1 2 , 2 3 (t + 2, t + 1) · n ± O(n1−ε ) = 2 r=1 E Ar(Jr)c (Jr) + t (n) where t (n) = b n ± O(n1−ε ) b = 1 − a ln 2 Hk+1 − Ht+1 + b · H c (n) = t diﬀerent toll function (n) + n−1 j=0 w same shape as c(n) n,j · c (j) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 9 / 12

128. The linear term Linear term depends on pivot sampling partitioning method linear term of mergesort but not on how sma unlike for Quicksort! ll subproblems solved! Consider c (n) = c(n) − n lg n Can show: c (n) = c(n) − an lg n = 2 r=1 E Ar(Jr)c(Jr) − an lg n + t(n) = 2 r=1 E Ar(Jr) c(Jr) − aJr lg Jr + a E Ar(Jr)Jr lg Jr − an lg n + (n − 1) + E A2(J2) · x(J1) + E A1(J1) · x(J2) = 2 r=1 E Ar(Jr)c (Jr) + (n − 1) − an lg n + a E A1(J1) + A2(J2) J1 lg J1 + b E[A2(J2)J1] + a E A2(J2) + A1(J1) J2 lg J2 + b E[A1(J1)J2] ± O(n1−ε ) Since J1 D = J2 we can simplify E A1(J1) + A2(J2) J1 lg J1 + E A2(J2) + A1(J1) J2 lg J2 = E A1(J1) + A2(J2) J1 lg J1 + E A2(J1) + A1(J2) J1 lg J1 = E J1 lg J1 · A1(J1) + A1(J2) + A2(J1) + A2(J2) = 2E[J lg J] = 2 E[ J n ] · n lg n + 2 · 1 ln 2 E[ J n ln J n ] · n = n lg n − 1 ln 2 Hk+1 − Ht+1 n ± O(n1−ε ). Plugging this back into our equation for c (n), we find c (n) = 2 r=1 E Ar(Jr)c (Jr) + (n − 1) − an lg n + a n lg n − 1 ln 2 Hk+1 − Ht+1 n + b I0, 1 3 (t + 2, t + 1) + I1 2 , 2 3 (t + 2, t + 1) · n ± O(n1−ε ) = 2 r=1 E Ar(Jr)c (Jr) + t (n) where t (n) = b n ± O(n1−ε ) b = 1 − a ln 2 Hk+1 − Ht+1 + b · H c (n) = t different toll function (n) + n−1 j=0 w same shape as c(n) n,j · c (j) t (n) ∼ 1 − 1 ln 2 H =1+ 1 2 + 1 3 +···+ 1 k+1 k+1 − Ht+1 + b linear coeff. of mergesort ≈ −1.24 (1 − r) · n Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 9 / 12

129. The linear term Linear term depends on pivot sampling partitioning method linear term of mergesort but not on how sma unlike for Quicksort! ll subproblems solved! Consider c (n) = c(n) − n lg n Can show: c (n) = t diﬀerent toll function (n) + n−1 j=0 w same shape as c(n) n,j · c (j) t (n) ∼ 1 − 1 ln 2 H =1+ 1 2 + 1 3 +···+ 1 k+1 k+1 − Ht+1 + b linear coeﬀ. of mergesort ≈ −1.24 (1 − r) · n Use continuous master theorem for c (n): wn,j smooth t(n) ∼ Knα logβ n c (n) not dominated by leaves of recursion Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 9 / 12

136. The linear term Linear term depends on pivot sampling partitioning method linear term of mergesort but not on how sma unlike for Quicksort! ll subproblems solved! Consider c (n) = c(n) − n lg n Can show: c (n) = t diﬀerent toll function (n) + n−1 j=0 w same shape as c(n) n,j · c (j) t (n) ∼ 1 − 1 ln 2 H =1+ 1 2 + 1 3 +···+ 1 k+1 k+1 − Ht+1 + b linear coeﬀ. of mergesort ≈ −1.24 (1 − r) · n Use continuous master theorem for c (n): wn,j smooth t(n) ∼ Knα logβ n c (n) not dominated by leaves of recursion c (n) ∼ t (n) 1 − r Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 9 / 12

138. Result Comparisons in QuickMergesort: c(n) = x(n) + qt · n ± O(log n) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 10 / 12

139. Result Comparisons in QuickMergesort: c(n) = x(n) + qt · n ± O(log n) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 10 / 12

140. Result Comparisons in QuickMergesort: c(n) = x(n) same as X ... + qt · n ± O(log n) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 10 / 12

141. Result Comparisons in QuickMergesort: c(n) = x(n) same as X ... + qt ... + “QuickXsort penalty” · n ± O(log n) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 10 / 12

142. Result Comparisons in QuickMergesort: c(n) = x(n) same as X ... + qt ... + “QuickXsort penalty” · n ± O(log n) t = 0 0 0.2 0.4 0.6 0.8 1 1.2 α = 1 2 (our QuickMergesort) 0.91 Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 10 / 12

143. Result Comparisons in QuickMergesort: c(n) = x(n) same as X ... + qt ... + “QuickXsort penalty” · n ± O(log n) t = 0 t = 1 0 0.2 0.4 0.6 0.8 1 1.2 α = 1 2 (our QuickMergesort) 0.91 0.41 Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 10 / 12

144. Result Comparisons in QuickMergesort: c(n) = x(n) same as X ... + qt ... + “QuickXsort penalty” · n ± O(log n) t = 0 t = 1 t = 2 0 0.2 0.4 0.6 0.8 1 1.2 α = 1 2 (our QuickMergesort) 0.91 0.41 0.25 Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 10 / 12

145. Result Comparisons in QuickMergesort: c(n) = x(n) same as X ... + qt ... + “QuickXsort penalty” · n ± O(log n) t = 0 t = 1 t = 2 t = 3 0 0.2 0.4 0.6 0.8 1 1.2 α = 1 2 (our QuickMergesort) 0.91 0.41 0.25 0.18 Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 10 / 12

146. Result Comparisons in QuickMergesort: c(n) = x(n) same as X ... + qt ... + “QuickXsort penalty” · n ± O(log n) t = 0 t = 1 t = 2 t = 3 t = 10 0 0.2 0.4 0.6 0.8 1 1.2 α = 1 2 (our QuickMergesort) 0.91 0.41 0.25 0.18 0.06 Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 10 / 12

147. Result Comparisons in QuickMergesort: c(n) = x(n) same as X ... + qt ... + “QuickXsort penalty” · n ± O(log n) t = 0 t = 1 t = 2 t = 3 t = 10 t → ∞ 0 0.2 0.4 0.6 0.8 1 1.2 α = 1 2 (our QuickMergesort) 0.91 0.41 0.25 0.18 0.06 0 Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 10 / 12

148. Result Comparisons in QuickMergesort: c(n) = x(n) same as X ... + qt ... + “QuickXsort penalty” · n ± O(log n) t = 0 t = 1 t = 2 t = 3 t = 10 t → ∞ 0 0.2 0.4 0.6 0.8 1 1.2 α = 1 2 (our QuickMergesort) α = 1 (always recurse on larger half) 0.91 0.41 0.25 0.18 0.06 0 1.11 0.51 0.32 0.23 0.08 0 Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 10 / 12

149. Conclusion Findings QuickMergesort sorts in-place with close-to-optimal #comparisons (but not a stable sort) QuickXsort penalty rapidly decreases with sample size Skewed pivots not helpful (worse despite sorting a larger part by X!) Extensions variance inﬂuence of ﬂuctuations in Mergesort? limit distributions? Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 11 / 12

156. Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 12 / 12

157. Icons made by Freepik and Gregor Cresnar from www.flaticon.com. Title background by Phil’s 1stPix (CC BY-BC-SA 2.0) Sebastian Wild Average Cost of QuickXsort with Pivot Sampling 2018-06-28 13 / 12

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