Lecture 5

1. Lecture 5

2. Quiz 2 Input: A nonnegative integer n S:=0 for i:=0 to n do S:=S + i*i return S • What Doest this Algorithm Compute • What is its basic operation • How many times is the basic operation executed? • What is the efficiency class of this algorithm • Suggest an improvement or a better algorithm and its efficiency class

3. Divide and Conquer • Merge Sort • Quick Sort

4. Divide and Conquer • Base Case, solve the problem directly if it is small enough • Divide the problem into two or more similar and smaller subproblems • Recursively solve the subproblems • Combine solutions to the subproblems

5. Divide and Conquer - Sort Problem: • Input: A[left..right] – unsorted array of integers • Output:A[left..right] – sorted in non-decreasing order

6. Divide and Conquer - Sort • Base case • at most one element (left ≥ right), return • DivideA into twosubarrays: FirstPart, SecondPart • Two Subproblems: • sort the FirstPart • sort the SecondPart • Recursively • sort FirstPart • sort SecondPart • Combinesorted FirstPart and sorted SecondPart

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

8. Merge Merge Sort: Idea Divide into two halves A FirstPart SecondPart Recursively sort SecondPart FirstPart A is sorted!

9. 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

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

11. 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:

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

43. 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

44. 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)

45. 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.

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

47. 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

48. 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

49. 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

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