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Lecture 5. 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 - PowerPoint PPT Presentation
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Lecture 5
2
Quiz 2Input: A nonnegative integer nS:=0for i:=0 to n do
S:=S + i*ireturn 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 Conquer1. 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
5
Divide and Conquer - SortProblem: • Input: A[left..right] – unsorted array of integers • Output: A[left..right] – sorted in non-decreasing
order
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Divide and Conquer - Sort1. 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
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Overview
• Divide and Conquer
• Merge Sort
• Quick Sort
8
Merge Sort: Idea
Merge
Recursively sort
Divide intotwo halves
FirstPart SecondPart
FirstPart SecondPart
A
A is sorted!
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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
10A[middle]A[left]
Sorted FirstPart
Sorted SecondPart
Merge-Sort: Merge
A[right]
merge
A:
A:
Sorted
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6 10 14 223 5 15 28L: R:
Temporary Arrays
5 15 28 30 6 10 145
Merge-Sort: Merge Example
2 3 7 8 1 4 5 6A:
12
Merge-Sort: Merge Example
3 5 15 28 30 6 10 14
L:
A:
3 15 28 30 6 10 14 22
R:
i=0 j=0
k=0
2 3 7 8 1 4 5 6
1
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Merge-Sort: Merge Example
1 5 15 28 30 6 10 14
L:
A:
3 5 15 28 6 10 14 22
R:
k=1
2 3 7 8 1 4 5 6
2
i=0 j=1
14
Merge-Sort: Merge Example
1 2 15 28 30 6 10 14
L:
A:
6 10 14 22
R:
i=1
k=2
2 3 7 8 1 4 5 6
3
j=1
15
Merge-Sort: Merge Example
1 2 3 6 10 14
L:
A:
6 10 14 22
R:
i=2 j=1
k=3
2 3 7 8 1 4 5 6
4
16
Merge-Sort: Merge Example
1 2 3 4 6 10 14
L:
A:
6 10 14 22
R:
j=2
k=4
2 3 7 8 1 4 5 6
i=2
5
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Merge-Sort: Merge Example
1 2 3 4 5 6 10 14
L:
A:
6 10 14 22
R:
i=2 j=3
k=5
2 3 7 8 1 4 5 6
6
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Merge-Sort: Merge Example
1 2 3 4 5 6 14
L:
A:
6 10 14 22
R:
k=6
2 3 7 8 1 4 5 6
7
i=2 j=4
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Merge-Sort: Merge Example
1 2 3 4 5 6 7 14
L:
A:
3 5 15 28 6 10 14 22
R:2 3 7 8 1 4 5 6
8
i=3 j=4
k=7
20
Merge-Sort: Merge Example
1 2 3 4 5 6 7 8
L:
A:
3 5 15 28 6 10 14 22
R:2 3 7 8 1 4 5 6
i=4 j=4
k=8
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Merge(A, left, middle, right)1. n1 ← middle – left + 12. n2 ← right – middle3. create array L[n1], R[n2]4. for i ← 0 to n1-1 do L[i] ← A[left +i]5. for j ← 0 to n2-1 do R[j] ← A[middle+j]6. k ← i ← j ← 07. while i < n1 & j < n2 8. if L[i] < R[j] 9. A[k++] ← L[i++]10. else11. A[k++] ← R[j++]12. while i < n1
13. A[k++] ← L[i++]14. while j < n2
15. A[k++] ← R[j++] n = n1+n2
Space: nTime : cn for some constant c
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6 2 8 4 3 7 5 16 2 8 4 3 7 5 1
Merge-Sort(A, 0, 7)
DivideA:
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6 2 8 4
3 7 5 1
6 2 8 4
Merge-Sort(A, 0, 3) , divideA:
Merge-Sort(A, 0, 7)
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3 7 5 1
8 4
6 26 2
Merge-Sort(A, 0, 1) , divideA:
Merge-Sort(A, 0, 7)
25
3 7 5 1
8 4
6
2
Merge-Sort(A, 0, 0) , base caseA:
Merge-Sort(A, 0, 7)
26
3 7 5 1
8 4
6 2
Merge-Sort(A, 0, 0), returnA:
Merge-Sort(A, 0, 7)
27
3 7 5 1
8 4
6
2
Merge-Sort(A, 1, 1) , base caseA:
Merge-Sort(A, 0, 7)
28
3 7 5 1
8 4
6 2
Merge-Sort(A, 1, 1), returnA:
Merge-Sort(A, 0, 7)
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3 7 5 1
8 4
2 6
Merge(A, 0, 0, 1)A:
Merge-Sort(A, 0, 7)
30
3 7 5 1
8 42 6
Merge-Sort(A, 0, 1), returnA:
Merge-Sort(A, 0, 7)
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3 7 5 1
8 4
2 6
Merge-Sort(A, 2, 3)
48
, divideA:
Merge-Sort(A, 0, 7)
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3 7 5 1
4
2 6
8
Merge-Sort(A, 2, 2), base caseA:
Merge-Sort(A, 0, 7)
33
3 7 5 1
4
2 6
8
Merge-Sort(A, 2, 2), returnA:
Merge-Sort(A, 0, 7)
34
4
2 6
8
Merge-Sort(A, 3, 3), base caseA:
Merge-Sort(A, 0, 7)
35
3 7 5 1
4
2 6
8
Merge-Sort(A, 3, 3), returnA:
Merge-Sort(A, 0, 7)
36
3 7 5 1
2 6
4 8
Merge(A, 2, 2, 3)A:
Merge-Sort(A, 0, 7)
37
3 7 5 1
2 6 4 8
Merge-Sort(A, 2, 3), returnA:
Merge-Sort(A, 0, 7)
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3 7 5 1
2 4 6 8
Merge(A, 0, 1, 3)A:
Merge-Sort(A, 0, 7)
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3 7 5 12 4 6 8Merge-Sort(A, 0, 3), return
A:
Merge-Sort(A, 0, 7)
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3 7 5 1
2 4 6 8Merge-Sort(A, 4, 7)
A:
Merge-Sort(A, 0, 7)
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1 3 5 7
2 4 6 8A:
Merge (A, 4, 5, 7)
Merge-Sort(A, 0, 7)
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1 3 5 72 4 6 8Merge-Sort(A, 4, 7), return
A:
Merge-Sort(A, 0, 7)
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1 2 3 4 5 6 7 8Merge(A, 0, 3, 7)
A:
Merge-Sort(A, 0, 7)Merge-Sort(A, 0, 7), done!
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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/2
n/4 n/4 n/4 n/4
2 2 2
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Merge-Sort SummaryApproach: divide and conquerTime
– Most of the work is in the merging– Total time: (n log n)
Space: – (n), more space than other sorts.
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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 < pSecondPart, which contains all elements ≥ p
• Recursively sort the FirstPart and SecondPart
• Combine: no work is necessary since sorting
is done in place
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Quick Sort
x < p p p ≤ x
PartitionFirstPart SecondPart
ppivot
A:
Recursive call
x < p p p ≤ x
SortedFirstPart
SortedSecondPart
Sorted
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Quick SortQuick-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
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Partitionp
p x < p p ≤ x
p p ≤ xx < p
A:
A:
A:p
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Partition Example
A: 4 8 6 3 5 1 7 2
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Partition Example
A: 4 8 6 3 5 1 7 2
i=0
j=1
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Partition Example
A:
j=1
4 8 6 3 5 1 7 2
i=0
8
54
Partition Example
A: 4 8 6 3 5 1 7 26
i=0
j=2
55
Partition Example
A: 4 8 6 3 5 1 7 2
i=0
383
j=3
i=1
56
Partition Example
A: 4 3 6 8 5 1 7 2
i=1
5
j=4
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Partition Example
A: 4 3 6 8 5 1 7 2
i=1
1
j=5
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Partition Example
A: 4 3 6 8 5 1 7 2
i=2
1 6
j=5
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Partition Example
A: 4 3 8 5 7 2
i=2
1 6 7
j=6
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Partition Example
A: 4 3 8 5 7 2
i=2
1 6 22 8
i=3
j=7
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Partition Example
A: 4 3 2 6 7 8
i=3
1 5
j=8
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Partition Example
A: 4 1 6 7 8
i=3
2 542 3
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A: 3 6 7 81 542
x < 4 4 ≤ x
pivot incorrect position
Partition Example
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Partition(A, left, right)1. x ← A[left]2. i ← left3. for j ← left+1 to right4. if A[j] < x then 5. i ← i + 16. swap(A[i], A[j])7. end if8. end for j9. swap(A[i], A[left])10. return i
n = right – left +1 Time: cn for some constant c Space: constant
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4 8 6 3 5 1 7 22 3 1 5 6 7 84
Quick-Sort(A, 0, 7)Partition
A:
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2 3 1
5 6 7 84
2 1 3
Quick-Sort(A, 0, 7)Quick-Sort(A, 0, 2)
A:
, partition
67
2
5 6 7 84
1
1 3
Quick-Sort(A, 0, 7)Quick-Sort(A, 0, 0) , base case, return
68
2
5 6 7 84
1
33
Quick-Sort(A, 0, 7)Quick-Sort(A, 1, 1) , base case
69
5 6 7 842 1 3
2 1 3
Quick-Sort(A, 0, 7)Quick-Sort(A, 2, 2), returnQuick-Sort(A, 0, 2), return
70
42 1 3
5 6 7 86 7 85
Quick-Sort(A, 0, 7)Quick-Sort(A, 2, 2), returnQuick-Sort(A, 4, 7) , partition
71
4
5
6 7 87 866
2 1 3
Quick-Sort(A, 0, 7)Quick-Sort(A, 5, 7) , partition
72
4
5
6
7 887
2 1 3
Quick-Sort(A, 0, 7)Quick-Sort(A, 6, 7) , partition
73
4
5
6
7
2 1 3
Quick-Sort(A, 0, 7)Quick-Sort(A, 7, 7)
8
, return, base case
8
74
4
5
6 87
2 1 3
Quick-Sort(A, 0, 7)Quick-Sort(A, 6, 7) , return
75
4
5
2 1 3
Quick-Sort(A, 0, 7)Quick-Sort(A, 5, 7) , return
6 87
76
42 1 3
Quick-Sort(A, 0, 7)Quick-Sort(A, 4, 7) , return
5 6 87
77
42 1 3
Quick-Sort(A, 0, 7)Quick-Sort(A, 0, 7) , done!
5 6 87
78
Quick-Sort: Best Case Even Partition
Total time: (nlogn)
cn
2 × cn/2 = cn
4 × c/4 = cn
n/3 × 3c = cn
log n levels
n
n/2 n/2
n/4
3 3 3
n/4n/4n/4
79
cn
c(n-1)
3c
2c
n
n-1
n-2
3
2
c(n-2)
Happens only if input is sortd input is reversely sorted
Quick-Sort: Worst Case Unbalanced Partition
Total time: (n2)
80
Quick-Sort: an Average Case• Suppose the split is 1/10 : 9/10
Quick-Sort: an Average Case
cn
cn
cn
≤cn
n
0.1n 0.9n
0.01n 0.09n 0.09n
Total time: (nlogn)
0.81n
2
2
log10n
log10/9n
≤cn
81
Quick-Sort Summary• Time
– Most of the work done in partitioning.– Average case takes (n log(n)) time.– Worst case takes (n2) time
Space– Sorts in-place, i.e., does not require additional
space
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Summary• Divide and Conquer
• Merge-Sort – Most of the work done in Merging– (n log(n)) time– (n) space
• Quick-Sort– Most of the work done in partitioning– Average case takes (n log(n)) time– Worst case takes (n2) time– (1) space
