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Administrative

Administrative. Sep. 20 (today) – HW1 due Sep. 21 8am – problem session Sep. 25 – HW3 (=QUIZ #1) due Sep. 27 – HW4 due Sep. 28 8am – problem session Oct. 2 Oct. 4 – QUIZ #2. (pages 45-79 of DPV). Merge 2 sorted lists. M ERGE INSTANCE: 2 lists x i , y i such that

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Administrative

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  1. Administrative Sep. 20 (today) – HW1 due Sep. 21 8am – problem session Sep. 25 – HW3 (=QUIZ #1) due Sep. 27 – HW4 due Sep. 28 8am – problem session Oct. 2 Oct. 4 – QUIZ #2 (pages 45-79 of DPV)

  2. Merge 2 sorted lists MERGE INSTANCE: 2 lists xi, yi such that x1  x2  …  xn y1  y2  …  ym SOLUTION: ordered merge

  3. Merge 2 sorted lists 1 i  1, j  1 2 while i  n and j  n do 3 if xi  yj then 4 output xi, i  i + 15 else 6 output yj, j  j + 1 7 output remaining elements

  4. Mergesort MERGE-SORT(a,l,r) if l < r then m  (l+r)/2  MERGE-SORT(a,I,m) MERGE-SORT(a,m+1,r) MERGE(a,l,m,r)

  5. Mergesort Running time?

  6. Mergesort Running time? [ ... n … ] [ … n/2 … ] [ … n/2 … ] [ … n/4 … ] [ … n/4 … ] [ … n/4 … ] [ … n/4 … ] Depth ?

  7. Mergesort Running time? [ ... n … ] [ … n/2 … ] [ … n/2 … ] [ … n/4 … ] [ … n/4 … ] [ … n/4 … ] [ … n/4 … ] Depth = log n

  8. Mergesort Time spent on merge? [ ... n … ] [ … n/2 … ] [ … n/2 … ] [ … n/4 … ] [ … n/4 … ] [ … n/4 … ] [ … n/4 … ] Depth = log n

  9. Mergesort [ ... n … ] [ … n/2 … ] [ … n/2 … ] [ … n/4 … ] [ … n/4 … ] [ … n/4 … ] [ … n/4 … ] Time spent on merge? O(n) O(n) O(n) Depth = log n O(n.logn)

  10. Mergesort recurrence T(n)= T(n/2) + (n) if n>1 T(1)= (1)

  11. Recurrences T(n)  T( n/2 ) + T( n/2 ) + c.n We “showed” : T(n)=O(n log n) for T(n)  2 T( n/2 ) + c.n

  12. Recurrences T(n)  2 T( n/2 ) + c.n T(1) = O(1) Proposition: T(n)  d.n.lg n Proof: T(n) 2 T( n/2 ) + c.n  2 d (n/2).lg (n/2) + c.n = d.n.( lg n – 1 ) + cn = d.n.lg n + (c-d).n  d.n.lg n

  13. Recurrences T(n)  T( n/2 ) + T( n/2 ) + c.n T(n)  2 T( n/2 ) + c.n G(n) = T(n+2) G(n) = T(n+2)  T(n/2+1) + T(n/2+1) + c.n = G(n/2-1) + G(n/2-1) + c.n

  14. Recurrences • useful • guess – substitute (prove) • or use “Master theorem” • T(n) = a T(n/b) + f(n) • If f(n) = O(nc-) then T(n) =(nc) • If f(n) = (nc) then T(n) =(nc.log n) • If f(n) = (nc+) then T(n)=(f(n)) • if a.f(n/b)  d.f(n) for some d<1 and n>n0 c=logb a

  15. Karatsuba-Offman a=2n/2 a1 + a0 b=2n/2 b1 + b0 ab=(2n/2a1+a0)(2n/2b1+b0) = 2n a1 b1 + 2n/2 (a1 b0 + a0 b1) + a0 b0

  16. Karatsuba-Offman a=2n/2 a1 + a0 b=2n/2 b1 + b0 Multiply(a,b,n) if n=1 return a*b else R1 Multiply(a1,b1,n/2) R2 Multiply(a0,b1,n/2) R3 Multiply(a1,b0,n/2) R4 Multiply(a0,b0,n/2) return 2n R1+ 2n/2 (R2+R3) + R4

  17. Karatsuba-Offman Multiply(a,b,n) if n=1 return a*b else R1 Multiply(a1,b1,n/2) R2 Multiply(a0,b1,n/2) R3 Multiply(a1,b0,n/2) R4 Multiply(a0,b0,n/2) return 2n R1+ 2n/2 (R2+R3) + R4 Recurrence?

  18. Karatsuba-Offman Multiply(a,b,n) if n=1 return a*b else R1 Multiply(a1,b1,n/2) R2 Multiply(a0,b1,n/2) R3 Multiply(a1,b0,n/2) R4 Multiply(a0,b0,n/2) return 2n R1+ 2n/2 (R2+R3) + R4 Recurrence? T(n) = 4T(n/2) + O(n)

  19. Karatsuba-Offman T(n) = 4T(n/2) + O(n) T(n)=O(n2)

  20. Karatsuba-Offman ab=(2n/2a1+a0)(2n/2b1+b0) = 2na1 b1 + 2n/2 (a1 b0 + a0 b1) + a0 b0 Can compute in less than 4 multiplications?

  21. Karatsuba-Offman ab=(2n/2a1+a0)(2n/2b1+b0) = 2na1 b1 + 2n/2 (a1 b0 + a0 b1) + a0 b0 Can compute using 3 multiplications: (a0+a1)(b0+b1) = a0b0 + (a1 b0 + a0 b1) + a1 b1

  22. Karatsuba-Offman Multiply(a,b,n) if n=1 return a*b else R1 Multiply(a1,b1,n/2) R2 Multiply(a0,b0,n/2) R3 Multiply(a1+a0,b1+b0,n/2+1) R4 R3 – R2 – R1 return 2n R1+ 2n/2 R3 + R2 Recurrence?

  23. Karatsuba-Offman Multiply(a,b,n) if n=1 return a*b else R1 Multiply(a1,b1,n/2) R2 Multiply(a0,b0,n/2) R3 Multiply(a1+a0,b1+b0,n/2+1) R4 R3 – R2 – R1 return 2n R1+ 2n/2 R3 + R2 Recurrence? T(n) = 3T(n/2) + O(n)

  24. Recurrences • T(n) = a T(n/b) + f(n) • If f(n) = O(nc-) then T(n) =(nc) • If f(n) = (nc) then T(n) =(nc.log n) • If f(n) = (nc+) then T(n)=(f(n)) • if a.f(n/b)  d.f(n) for some d<1 and n>n0 c=logb a T(n) = 3 T(n/2) + (n) T(n) = 2T(n/2) + (n.log n)

  25. Karatsuba-Offman T(n) = 3T(n/2) + O(n) T(n)=O(nC) C=log2 3  1.58

  26. Finding the minimum min  A[1] for i from 2 to n do if A[i]<min then min  A[i] How many comparisons?

  27. Finding the minimum How many comparisons? comparison based algorithm: The only allowed operation is comparing the elements

  28. Finding the minimum

  29. Finding the k-th smallest element k = n/2 = MEDIAN

  30. 8 2 6 5 8 8 9 2 3 6 3 1 7 1 1 6 2 8 3 6 9 1 1 1 7 5 2 8 8 3 Finding the k-th smallest element

  31. Finding the k-th smallest element 8 2 6 5 8 8 9 2 3 6 3 1 7 1 1 6 2 8 3 6 9 1 1 1 7 5 2 8 8 3 1) sort each 5-tuple

  32. Finding the k-th smallest element 6 2 8 3 6 9 1 1 1 7 5 2 8 8 3 1) sort each 5-tuple

  33. Finding the k-th smallest element 1 2 8 3 6 9 6 1 1 7 5 2 8 8 3 1) sort each 5-tuple

  34. Finding the k-th smallest element 1 2 8 3 6 9 6 1 1 7 5 2 8 8 3 1) sort each 5-tuple

  35. Finding the k-th smallest element 1 1 8 3 2 9 6 5 1 7 6 2 8 8 3 1) sort each 5-tuple

  36. Finding the k-th smallest element TIME = ? 1 1 1 3 2 2 6 5 3 7 6 7 8 8 9 1) sort each 5-tuple

  37. 1 1 1 3 2 2 6 5 3 7 6 7 8 8 9 Finding the k-th smallest element TIME = (n) 1) sort each 5-tuple

  38. Finding the k-th smallest element 2) find median of the middle n/5 elements TIME = ? 1 1 1 3 2 2 6 5 3 7 6 7 8 8 9

  39. Finding the k-th smallest element 2) find median of the middle n/5 elements TIME = T(n/5) 1 1 1 3 2 2 6 5 3 7 6 7 8 8 9

  40. Finding the k-th smallest element At least ? Many elements in the array are  X 1 1 1 3 2 2 6 5 3 7 6 7 8 8 9

  41. Finding the k-th smallest element At least ? Many elements in the array are  X 1 1 1 2 2 3 3 5 6 7 6 7 9 8 8

  42. Finding the k-th smallest element At least 3n/10 elements in the array are  X 1 1 1 2 2 3 3 5 6 7 6 7 9 8 8

  43. Finding the k-th smallest element 8 2 6 5 8 8 9 2 3 6 3 1 7 1 1 1 1 1 2 2 3 3 5 6 7 6 7 9 8 8 At least 3n/10 elements in the array are  X

  44. Finding the k-th smallest element 8 2 6 5 8 8 9 2 3 6 3 1 7 1 1 >X X 3 2 1 5 8 8 9 7 8 3 1 2 1 6 6 At least 3n/10 elements in the array are  X

  45. Finding the k-th smallest element 8 2 6 5 8 8 9 2 3 6 3 1 7 1 1 >X X 3 2 1 5 8 8 9 7 8 3 1 2 1 6 6 Recurse, time ? At least 3n/10 elements in the array are  X

  46. Finding the k-th smallest element 8 2 6 5 8 8 9 2 3 6 3 1 7 1 1 >X X 3 2 1 5 8 8 9 7 8 3 1 2 1 6 6 Recurse, time  T(7n/10) At least 3n/10 elements in the array are  X

  47. 8 2 6 5 8 8 9 2 3 6 3 1 7 1 1 1 1 6 1 1 2 8 1 1 3 3 3 6 2 2 2 2 9 6 1 6 1 5 5 3 3 1 7 7 7 5 6 6 7 7 2 8 8 8 8 8 8 9 9 3 Finding the k-th smallest element 1 2 5 8 2 6 3 7 8 6 1 8 9 1 3 X >X 1 2 1 8 7 3 2 3 1 5 8 9 6 8 6 recurse

  48. 8 2 6 5 8 8 9 2 3 6 3 1 7 1 1 1 6 1 1 1 2 8 1 1 3 3 3 2 2 6 2 2 9 1 6 6 5 5 1 3 1 3 7 7 7 6 5 6 7 2 7 8 8 8 8 8 8 9 3 9 Finding the k-th smallest element 1 2 5 8 2 6 3 7 8 6 1 8 9 1 3 (n) X >X (n) 1 2 1 8 7 3 2 3 1 5 8 9 6 8 6 recurse T(7n/10) T(n/5)

  49. Finding the k-th smallest element T(n)  T(n/5) + T(7n/10) + O(n)

  50. Finding the k-th smallest element T(n)  T(n/5) + T(7n/10) + O(n) T(n)  d.n Induction step: T(n)  T(n/5) + T(7n/10) + O(n)  d.(n/5) + d.(7n/10) + O(n)  d.n + (O(n) – dn/10)  d.n

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