Presentation is loading. Please wait.

Presentation is loading. Please wait.

1 Overview Divide and Conquer Merge Sort Quick Sort.

Published byFranklin Atkins Modified over 8 years ago

Similar presentations

Presentation on theme: "1 Overview Divide and Conquer Merge Sort Quick Sort."— Presentation transcript:

1 1 Overview Divide and Conquer Merge Sort Quick Sort

2 2 Divide and Conquer 1.Base Case, solve the problem directly if it is small enough 2.Divide the problem into two or more similar and smaller subproblems 3.Recursively solve the subproblems 4.Combine solutions to the subproblems

3 3 Divide and Conquer - Sort Problem: Input: A[left..right] – unsorted array of integers Output: A[left..right] – sorted in non-decreasing order

4 4 Divide and Conquer - Sort 1. Base case at most one element (left ≥ right), return 2. Divide A into two subarrays: FirstPart, SecondPart Two Subproblems: sort the FirstPart sort the SecondPart 3. Recursively sort FirstPart sort SecondPart 4. Combine sorted FirstPart and sorted SecondPart

5 5 Overview Divide and Conquer Merge Sort Quick Sort

6 6 Merge Sort: Idea Merge Recursively sort Divide into two halves FirstPart SecondPart FirstPart SecondPart A A is sorted!

7 7 Merge Sort: Algorithm Merge-Sort (A, left, right) if left ≥ right return else middle ← b (left+right)/2  Merge-Sort(A, left, middle) Merge-Sort(A, middle+1, right) Merge(A, left, middle, right) Recursive Call

8 8 A[middle] A[left] Sorted FirstPart Sorted SecondPart Merge-Sort: Merge A[right] merge A: A: Sorted

9 9 6101422 351528 L:R: Temporary Arrays 515283061014 5 Merge-Sort: Merge Example 23781456 A:

10 10 Merge-Sort: Merge Example 3515283061014 L: A: 3152830 6101422 R: i=0 j=0 k=0 2378 1456 1

11 11 Merge-Sort: Merge Example 1515283061014 L: A: 351528 6101422 R: k=1 2378 1456 2 i=0 j=1

12 12 Merge-Sort: Merge Example 1215283061014 L: A: 6101422 R: i=1 k=2 2378 1456 3 j=1

13 13 Merge-Sort: Merge Example 12361014 L: A: 6101422 R: i=2 j=1 k=3 2378 1456 4

14 14 Merge-Sort: Merge Example 123461014 L: A: 6101422 R: j=2 k=4 2378 1456 i=2 5

15 15 Merge-Sort: Merge Example 1234561014 L: A: 6101422 R: i=2 j=3 k=5 2378 1456 6

16 16 Merge-Sort: Merge Example 12345614 L: A: 6101422 R: k=6 2378 1456 7 i=2 j=4

17 17 Merge-Sort: Merge Example 123456714 L: A: 351528 6101422 R: 2378 1456 8 i=3 j=4 k=7

18 18 Merge-Sort: Merge Example 12345678 L: A: 351528 6101422 R: 2378 1456 i=4 j=4 k=8

19 19 Merge(A, left, middle, right) 1. n 1 ← middle – left + 1 2. n 2 ← right – middle 3. create array L[n 1 ], R[n 2 ] 4. for i ← 0 to n 1 -1 do L[i] ← A[left +i] 5. for j ← 0 to n 2 -1 do R[j] ← A[middle+j] 6. k ← i ← j ← 0 7. while i < n 1 & j < n 2 8. if L[i] < R[j] 9. A[k++] ← L[i++] 10. else 11. A[k++] ← R[j++] 12. while i < n 1 13. A[k++] ← L[i++] 14. while j < n 2 15. A[k++] ← R[j++] n = n 1 +n 2 Space: n Time : cn for some constant c

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

42 42 Merge-Sort Analysis cn 2 × cn/2 = cn 4 × cn/4 = cn n/2 × 2c = cn log n levels Total running time:  (nlogn) Total Space:  (n) Total: cn log n n n/2 n/4 2 2 2

43 43 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.

44 44 Overview Divide and Conquer Merge Sort Quick Sort

45 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 46 Quick Sort x < p p p ≤ x Partition FirstPart SecondPart p pivot A: Recursive call x < p p p ≤ x Sorted FirstPart Sorted SecondPart Sorted

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

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

49 49 Partition ExampleA: 48635172

50 50 Partition ExampleA: 48635172 i=0 j=1

51 51 Partition ExampleA: j=1 48635172 i=0 8

52 52 Partition ExampleA: 486351726 i=0 j=2

53 53 Partition ExampleA: 4 8 635172 i=0 3 83 j=3i=1

54 54 Partition ExampleA: 43685172i=1 5 j=4

55 55 Partition ExampleA: 43685172i=1 1 j=5

56 56 Partition ExampleA: 43685172i=2 16 j=5

57 57 Partition ExampleA: 438572i=2 16 7 j=6

58 58 Partition ExampleA: 438572i=2 16 2 2 8i=3j=7

59 59 Partition ExampleA: 432678i=3 15 j=8

60 60 Partition ExampleA: 41678i=3 25 4 2 3

61 61 A: 3 678 1 5 42 x < 4 4 ≤ x pivot in correct position Partition Example

62 62 Partition(A, left, right) 1. x ← A[left] 2. i ← left 3. for j ← left+1 to right 4. if A[j] < x then 5. i ← i + 1 6. swap(A[i], A[j]) 7. end if 8. end for j 9. swap(A[i], A[left]) 10. return i n = right – left +1 Time: cn for some constant c Space: constant

63 63 4 8 6 3 5 1 7 2 2 3 1 5 6 7 8 4 Quick-Sort(A, 0, 7) Partition A:

64 64 2 3 1 5 6 7 8 4 2 13 Quick-Sort(A, 0, 7) Quick-Sort(A, 0, 2) A:, partition

65 65 2 5 6 7 8 4 1 1 3 Quick-Sort(A, 0, 7) Quick-Sort(A, 0, 0), base case, return

66 66 2 5 6 7 8 4 1 3 3 Quick-Sort(A, 0, 7) Quick-Sort(A, 1, 1), base case

67 67 5 6 7 8 4 2 1 3 2 1 3 Quick-Sort(A, 0, 7) Quick-Sort(A, 2, 2), return Quick-Sort(A, 0, 2), return

68 68 4 2 1 3 5 6 7 86 7 8 5 Quick-Sort(A, 0, 7) Quick-Sort(A, 2, 2), return Quick-Sort(A, 4, 7), partition

69 69 4 5 6 7 8 7 8 6 6 2 1 3 Quick-Sort(A, 0, 7) Quick-Sort(A, 5, 7), partition

70 70 4 5 6 7 8 8 7 2 1 3 Quick-Sort(A, 0, 7) Quick-Sort(A, 6, 7), partition

71 71 4 5 6 7 2 1 3 Quick-Sort(A, 0, 7) Quick-Sort(A, 7, 7) 8, return, base case 8

72 72 4 5 6 8 7 2 1 3 Quick-Sort(A, 0, 7) Quick-Sort(A, 6, 7), return

73 73 4 5 2 1 3 Quick-Sort(A, 0, 7) Quick-Sort(A, 5, 7), return 6 8 7

74 74 4 2 1 3 Quick-Sort(A, 0, 7) Quick-Sort(A, 4, 7), return 5 6 8 7

75 75 4 2 1 3 Quick-Sort(A, 0, 7), done! 5 6 8 7

Download ppt "1 Overview Divide and Conquer Merge Sort Quick Sort."

Similar presentations

Algorithms Analysis Lecture 6 Quicksort. Quick Sort 88 14 98 25 62 52 79 30 23 31 Divide and Conquer.

Algorithms Analysis Lecture 6 Quicksort. Quick Sort Divide and Conquer.
Comp 122, Spring 2004 Divide and Conquer (Merge Sort)

Comp 122, Spring 2004 Divide and Conquer (Merge Sort)
1 Divide & Conquer Algorithms. 2 Recursion Review A function that calls itself either directly or indirectly through another function Recursive solutions.

1 Divide & Conquer Algorithms. 2 Recursion Review A function that calls itself either directly or indirectly through another function Recursive solutions.
Stephen P. Carl - CS 2421 Recursive Sorting Algorithms Reading: Chapter 5.

Stephen P. Carl - CS 2421 Recursive Sorting Algorithms Reading: Chapter 5.
Divide and Conquer Technique General Method:  The Divide and Conquer Technique splits n inputs into k subsets, 1< k ≤ n, yielding k subproblems.  These.

Divide and Conquer Technique General Method:  The Divide and Conquer Technique splits n inputs into k subsets, 1< k ≤ n, yielding k subproblems.  These.
Chapter 4: Divide and Conquer Master Theorem, Mergesort, Quicksort, Binary Search, Binary Trees The Design and Analysis of Algorithms.

Chapter 4: Divide and Conquer Master Theorem, Mergesort, Quicksort, Binary Search, Binary Trees The Design and Analysis of Algorithms.
21/3/00SEM107- Kamin & ReddyClass 15 - Recursive Sorting - 1 Class 15 - Recursive sorting methods r Processing arrays by recursion r Divide-and-conquer.

21/3/00SEM107- Kamin & ReddyClass 15 - Recursive Sorting - 1 Class 15 - Recursive sorting methods r Processing arrays by recursion r Divide-and-conquer.
Lecture 2: Divide and Conquer algorithms Phan Thị Hà Dương

Lecture 2: Divide and Conquer algorithms Phan Thị Hà Dương
DIVIDE AND CONQUER APPROACH. General Method Works on the approach of dividing a given problem into smaller sub problems (ideally of same size).  Divide.

DIVIDE AND CONQUER APPROACH. General Method Works on the approach of dividing a given problem into smaller sub problems (ideally of same size).  Divide.
Quicksort CSE 331 Section 2 James Daly. Review: Merge Sort Basic idea: split the list into two parts, sort both parts, then merge the two lists 11324262152728.

Quicksort CSE 331 Section 2 James Daly. Review: Merge Sort Basic idea: split the list into two parts, sort both parts, then merge the two lists
Divide and Conquer Chapter 6. Divide and Conquer Paradigm Divide the problem into sub-problems of smaller sizes Conquer by recursively doing the same.

Divide and Conquer Chapter 6. Divide and Conquer Paradigm Divide the problem into sub-problems of smaller sizes Conquer by recursively doing the same.
Analysis of Algorithms CS 477/677 Instructor: Monica Nicolescu Lecture 5.

Analysis of Algorithms CS 477/677 Instructor: Monica Nicolescu Lecture 5.
Analysis of Algorithms CS 477/677 Sorting – Part B Instructor: George Bebis (Chapter 7)

Analysis of Algorithms CS 477/677 Sorting – Part B Instructor: George Bebis (Chapter 7)
Spring 2015 Lecture 5: QuickSort & Selection

Spring 2015 Lecture 5: QuickSort & Selection
CSC2100B Quick Sort and Merge Sort Xin 1. Quick Sort Efficient sorting algorithm Example of Divide and Conquer algorithm Two phases ◦ Partition phase.

CSC2100B Quick Sort and Merge Sort Xin 1. Quick Sort Efficient sorting algorithm Example of Divide and Conquer algorithm Two phases ◦ Partition phase.
25 May 20151 Quick Sort (11.2) CSE 2011 Winter 2011.

25 May Quick Sort (11.2) CSE 2011 Winter 2011.
Introduction to Algorithms Chapter 7: Quick Sort.

Introduction to Algorithms Chapter 7: Quick Sort.
1 Today’s Material Divide & Conquer (Recursive) Sorting Algorithms –QuickSort External Sorting.

1 Today’s Material Divide & Conquer (Recursive) Sorting Algorithms –QuickSort External Sorting.
Sorting Algorithms and Average Case Time Complexity

Sorting Algorithms and Average Case Time Complexity
CS 162 Intro to Programming II Quick Sort 1. Quicksort Maybe the most commonly used algorithm Quicksort is also a divide and conquer algorithm Advantage.

CS 162 Intro to Programming II Quick Sort 1. Quicksort Maybe the most commonly used algorithm Quicksort is also a divide and conquer algorithm Advantage.

Similar presentations

Ads by Google