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Lecture 6

Lecture 6. Merge Sort: Algorithm . Merge-Sort (A, left, right) if left ≥ right return else middle ← ( left+right )/2  Merge-Sort (A, left, middle) Merge-Sort (A, middle+1, right) Merge (A, left, middle, right). Recursive Call. Merge-Sort: Merge. Sorted. A:. merge.

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Lecture 6

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  1. Lecture 6

  2. Merge Sort: Algorithm • Merge-Sort(A, left, right) • if left ≥ right return • else • middle ←(left+right)/2 • Merge-Sort(A, left, middle) • Merge-Sort(A, middle+1, right) • Merge(A, left, middle, right) Recursive Call

  3. Merge-Sort: Merge Sorted A: merge Sorted SecondPart SortedFirstPart A: A[right] A[left] A[middle]

  4. 5 15 28 30 6 10 14 5 2 3 7 8 1 4 5 6 R: L: 3 5 15 28 6 10 14 22 Temporary Arrays Merge-Sort: Merge Example A:

  5. Merge-Sort: Merge Example A: 3 1 5 15 28 30 6 10 14 k=0 R: L: 3 2 15 3 7 28 30 8 6 1 10 4 5 14 22 6 i=0 j=0

  6. Merge-Sort: Merge Example A: 2 1 5 15 28 30 6 10 14 k=1 R: L: 2 3 5 3 15 7 8 28 6 1 10 4 14 5 6 22 i=0 j=1

  7. Merge-Sort: Merge Example A: 3 1 2 15 28 30 6 10 14 k=2 R: L: 2 3 7 8 6 1 10 4 5 14 22 6 i=1 j=1

  8. Merge-Sort: Merge Example A: 1 2 3 6 10 14 4 k=3 R: L: 2 3 7 8 1 6 10 4 14 5 6 22 j=1 i=2

  9. Merge-Sort: Merge Example A: 1 2 3 4 6 10 14 5 k=4 R: L: 2 3 7 8 1 6 10 4 14 5 6 22 i=2 j=2

  10. Merge-Sort: Merge Example A: 6 1 2 3 4 5 6 10 14 k=5 R: L: 2 3 7 8 6 1 4 10 5 14 6 22 i=2 j=3

  11. Merge-Sort: Merge Example A: 7 1 2 3 4 5 6 14 k=6 R: L: 2 3 7 8 1 6 10 4 14 5 22 6 i=2 j=4

  12. Merge-Sort: Merge Example A: 8 1 2 3 4 5 6 7 14 k=7 R: L: 2 3 5 3 7 15 28 8 6 1 10 4 5 14 22 6 i=3 j=4

  13. Merge-Sort: Merge Example A: 1 2 3 4 5 6 7 8 k=8 R: L: 3 2 3 5 15 7 8 28 1 6 10 4 14 5 22 6 j=4 i=4

  14. Merge(A, left, middle, right) • n1 ← middle – left + 1 • n2 ← right – middle • create array L[n1], R[n2] • for i ← 0 to n1-1 do L[i] ← A[left +i] • for j ← 0 to n2-1 do R[j] ← A[middle+j] • k ← i ← j ← 0 • while i < n1 & j < n2 • if L[i] < R[j] • A[k++] ← L[i++] • else • A[k++] ← R[j++] • while i < n1 • A[k++] ← L[i++] • while j < n2 • A[k++] ← R[j++] n = n1+n2 Space: n Time : cn for some constant c

  15. Merge-Sort(A, 0, 7) Divide A: 3 7 5 1 6 2 8 4 6 2 8 4 3 7 5 1

  16. Merge-Sort(A, 0, 7) , divide Merge-Sort(A, 0, 3) A: 3 7 5 1 8 4 6 2 6 2 8 4

  17. Merge-Sort(A, 0, 7) , divide Merge-Sort(A, 0, 1) A: 3 7 5 1 8 4 2 6 6 2

  18. Merge-Sort(A, 0, 7) , base case Merge-Sort(A, 0, 0) A: 3 7 5 1 8 4 2 6

  19. Merge-Sort(A, 0, 7) Merge-Sort(A, 0, 0), return A: 3 7 5 1 8 4 2 6

  20. Merge-Sort(A, 0, 7) , base case Merge-Sort(A, 1, 1) A: 3 7 5 1 8 4 6 2

  21. Merge-Sort(A, 0, 7) Merge-Sort(A, 1, 1), return A: 3 7 5 1 8 4 2 6

  22. Merge-Sort(A, 0, 7) Merge(A, 0, 0, 1) A: 3 7 5 1 8 4 2 6

  23. Merge-Sort(A, 0, 7) Merge-Sort(A, 0, 1), return A: 3 7 5 1 8 4 2 6

  24. Merge-Sort(A, 0, 7) , divide Merge-Sort(A, 2, 3) A: 3 7 5 1 2 6 4 8 8 4

  25. Merge-Sort(A, 0, 7) Merge-Sort(A, 2, 2), base case A: 3 7 5 1 2 6 4 8

  26. Merge-Sort(A, 0, 7) Merge-Sort(A, 2, 2),return A: 3 7 5 1 2 6 4 8

  27. Merge-Sort(A, 0, 7) Merge-Sort(A, 3, 3), base case A: 2 6 8 4

  28. Merge-Sort(A, 0, 7) Merge-Sort(A, 3, 3), return A: 3 7 5 1 2 6 4 8

  29. Merge-Sort(A, 0, 7) Merge(A, 2, 2, 3) A: 3 7 5 1 2 6 4 8

  30. Merge-Sort(A, 0, 7) Merge-Sort(A, 2, 3), return A: 3 7 5 1 4 8 2 6

  31. Merge-Sort(A, 0, 7) Merge(A, 0, 1, 3) A: 3 7 5 1 2 4 6 8

  32. Merge-Sort(A, 0, 7) Merge-Sort(A, 0, 3), return A: 2 4 6 8 3 7 5 1

  33. Merge-Sort(A, 0, 7) Merge-Sort(A, 4, 7) A: 2 4 6 8 3 7 5 1

  34. Merge-Sort(A, 0, 7) Merge (A, 4, 5, 7) A: 2 4 6 8 1 3 5 7

  35. Merge-Sort(A, 0, 7) Merge-Sort(A, 4, 7), return A: 2 4 6 8 1 3 5 7

  36. Merge-Sort(A, 0, 7) Merge-Sort(A, 0, 7), done! Merge(A, 0, 3, 7) A: 1 2 3 4 5 6 7 8

  37. Merge-Sort Analysis cn n 2 × cn/2 = cn n/2 n/2 log n levels 4 × cn/4 = cn n/4 n/4 n/4 n/4 n/2 × 2c = cn 2 2 2 Total: cn log n • Total running time:(nlogn) • Total Space: (n)

  38. Merge-Sort Summary Approach: divide and conquer Time • Most of the work is in the merging • Total time: (n log n) Space: • (n), more space than other sorts.

  39. Merge Sort Analysis • Suppose n is a power of two, so that we always split into even halves. • For n = 1 the time to merge sort is1 otherwise • The time to merge sort n numbers is equal to the time to do two recursive merge sorts of size n/2, plus the time to merge, which is linear. T(1) = 1 T(n) = 2T(n/2) + n • Solve this recurrence to find out running time.

  40. Solution by Telescoping Sum T(n) = 2T(n/2) + n Divide the recurrence relation through by n. T(n) / n = T(n/2) / n/2 + 1 This equation is valid for any n that is a power of 2. So T(n/2) / n/2 = T(n/4) / n/4 + 1 T(n/4) / n/4 = T(n/8) / n/8 + 1 ….. T(2) / 2 = T(1) / 1 + 1 Add both sides of all the above equations T(n) / n + T(n/2) / n/2 + T(n/4) / n/4 + ….+ T(2) / 2 = T(n/2) / 2 + 1 + T(n/4) / n/4 + 1 + T(n/8) / n/8 + 1 +…..+ T(1) / 1 + 1

  41. Solution by Telescoping Sum T(n) / n + T(n/2) / n/2 + T(n/4) / n/4 + ….+ T(2) / 2 = T(n/2) / 2 + 1 + T(n/4) / n/4 + 1 + T(n/8) / n/8 + 1 +…..+ T(1) / 1 + 1 • All the terms other than T(n) / n on right hand side cancel the terms on left hand side. • Since n is a power of 2, these equations add up to lgn (Tree has lgn levels). T(n) / n = T(1) / 1 + lgn T(n) = n + nlgn T(n) = O(nlgn)

  42. Solution by Substitution Method T(n) = 2 T(n/2) + n • Substitute n/2 into the main equation 2T(n/2) = 2(2(T(n/4)) + n/2) = 4T(n/4) + n And T(n) = 4T (n/4) + 2n • Again by substituting n/4, we can see 4T(n/4) = 4(2T(n/8)) + (n/4) = 8T(n/8) + n And T(n) = 8T(n/8) + 3n • Continuing in this manner, we obtain T(n) = 2k T(n/2k) + k.n Using k = lgn T(n) = nT(1) + nlgn = nlgn + n T(n) = O(nlgn)

  43. Problem with Merge Sort • Merging two sorted lists requires linear extra memory. • Additional work spent copying to the temporary array and back through out the algorithm. • This problem slows down the sort considerably.

  44. Overview • Divide and Conquer • Merge Sort • Quick Sort

  45. Quick Sort • Divide: • Pick any element p as the pivot, e.g, the first element • Partition the remaining elements into FirstPart,which contains all elements< p SecondPart, which contains all elements ≥ p • Recursively sort the FirstPart and SecondPart • Combine: no work is necessary since sorting is done in place

  46. Partition Recursive call p p p p ≤ x p ≤ x Sorted FirstPart Sorted SecondPart x < p Quick Sort A: pivot FirstPart SecondPart x < p Sorted

  47. Quick Sort Quick-Sort(A, left, right) ifleft ≥ right return else middle ← Partition(A, left, right) Quick-Sort(A, left, middle–1 ) Quick-Sort(A, middle+1, right) end if

  48. p x < p p p ≤ x x < p p ≤ x p Partition A: A: A: p

  49. Partition Example A: 4 8 6 3 5 1 7 2

  50. Partition Example i=0 A: 4 8 6 3 5 1 7 2 j=1

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