2_Divide and Conquer
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Transcript of 2_Divide and Conquer
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Divide and Conquer
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Recommended Reading
Chapter 2 from the textbookIntroduction to Algorithms
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Divide and Conquer
Divide
Break the problem into smaller sub problems
Conquer
Solve the sub problems
Generally, this involves waiting for the problem to be
small enough that it is trivial to solve
Combine
Given the results of the solved sub problems, combinethem to generate a solution for the complete problem
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Divide and Conquer: some thoughts
Often, the sub-problem is the same as the original problem
Dividing the problem in half frequently does the job
May have to get creative about how the data is split
Splitting tends to generate run times with log n in them
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Divide and Conquer: Sorting
How should we split the data?
What are the sub problems we need to solve?
How do we combine the results from these sub problems?
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MergeSort
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MergeSort: Merge
Assuming L and R are sorted already, merge
the two to create a single sorted array
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Merge
R: 2 4 6 7L: 1 3 5 8
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Merge
R: 2 4 6 7L: 1 3 5 8
B:
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Merge
R: 2 4 6 7L: 1 3 5 8
B:
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Merge
R: 2 4 6 7L: 1 3 5 8
B:
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Merge
R: 2 4 6 7L: 1 3 5 8
B: 1
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Merge
R: 2 4 6 7L: 1 3 5 8
B: 1
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Merge
R: 2 4 6 7L: 1 3 5 8
B: 1 2
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Merge
R: 2 4 6 7L: 1 3 5 8
B: 1 2
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Merge
R: 2 4 6 7L: 1 3 5 8
B: 1 2 3
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Merge
R: 2 4 6 7L: 1 3 5 8
B: 1 2 3
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Merge
R: 2 4 6 7L: 1 3 5 8
B: 1 2 3 4
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Merge
R: 2 4 6 7L: 1 3 5 8
B: 12 3 4
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Merge
R: 2 4 6 7L: 1 3 5 8
B: 12 3 4 5
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Merge
R: 2 4 6 7L: 1 3 5 8
B: 1 2 3 4 5
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Merge
R: 2 4 6 7L: 1 3 5 8
B: 1 2 3 4 5 6
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Merge
R: 2 4 6 7L: 1 3 5 8
B: 1 2 34 5 6
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Merge
R: 2 4 6 7L: 1 3 5 8
B: 1 2 34 5 6 7
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Merge
R: 2 4 6 7L: 1 3 5 8
B: 1 2 34 5 6 7
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Merge
R: 2 4 6 7L: 1 3 5 8
B: 1 2 34 5 6 7 8
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Merge
Does the algorithm terminate?
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Merge
Is it correct?
Loop invariant: At the end of each iteration of the for
loop of lines 4-10 the subarray B[1..k] contains thesmallestkelements from L and R in sorted order.
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Merge
Is it correct?
Loop invariant: At the end of each iteration of thefor loop of lines 4-10 the subarray B[1..k]contains the smallestkelements from L and R insorted order.
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Merge
Running time?
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Merge
Running time? (n) - linear
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Merge-Sort: Divide
8 5 1 3 6 2 7 4
6 2 7 48 5 1 3
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Merge-Sort: Divide
8 5 1 3 6 2 7 4
6 2 7 48 5 1 3
1 38 5
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Merge-Sort: Divide
8 5 1 3 6 2 7 4
6 2 7 48 5 1 3
1 38 5
8 5
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Merge-Sort: Conquer
8 5 1 3 6 2 7 4
6 2 7 48 5 1 3
1 38 5
8 5
Sorting a list of
one element is
easy
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Merge-Sort: Combine
8 5 1 3 6 2 7 4
6 2 7 48 5 1 3
1 35 8
8 5
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Merge-Sort: Divide
8 5 1 3 6 2 7 4
6 2 7 48 5 1 3
1 35 8
1 3
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Merge-Sort: Divide
8 5 1 3 6 2 7 4
6 2 7 48 5 1 3
1 35 8
1 3
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Merge-Sort: Merge
8 5 1 3 6 2 7 4
6 2 7 48 5 1 3
1 35 8
1 3
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Merge-Sort: Merge
8 5 1 3 6 2 7 4
6 2 7 41 3 5 8
1 35 8
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Merge-Sort
8 5 1 3 6 2 7 4
6 2 7 41 3 5 8
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Merge-Sort
8 5 1 3 6 2 7 4
2 4 6 71 3 5 8
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Merge-Sort: Merge
1 2 3 4 5 6 7 8
2 4 6 71 3 5 8
Done!
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MergeSort
Running time?
otherwise)()()2/(2
smallisif)(
nCnDnT
ncnT
D(n): cost of splitting (dividing) the dataC(n): cost of merging/combining the data
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MergeSort
Running time?
otherwise)()()2/(2
smallisif)(
nCnDnT
ncnT
D(n): cost of splitting (dividing) the data - linear (n)C(n): cost of merging/combining the data linear (n)
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MergeSort
Running time?
otherwise)2/(2
smallisif)(
cnnT
ncnT
Which is?
M S t
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MergeSort
cn
T(n/2) T(n/2)
otherwise)2/(2
smallisif)(
cnnT
ncnT
M S t
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MergeSort
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
otherwise)2/(2
smallisif)(
cnnT
ncnT
M S t
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MergeSort
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
T(n/8) T(n/8) T(n/8) T(n/8) T(n/8) T(n/8) T(n/8) T(n/8)
otherwise)2/(2
smallisif)(
cnnT
ncnT
M S t
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MergeSort
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
cn/8 cn/8 cn/8 cn/8 cn/8 cn/8 cn/8 cn/8
c c c c c c c c c c c
otherwise)2/(2
smallisif)(
cnnT
ncnT
M S t
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MergeSort
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
cn/8 cn/8 cn/8 cn/8 cn/8 cn/8 cn/8 cn/8
c c c c c c c c c c c
cn
cn
cn
cn
cn
otherwise)2/(2
smallisif)(
cnnT
ncnT
M S t
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MergeSort
otherwise)2/(2
smallisif)(
cnnT
ncnT
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
cn/8 cn/8 cn/8 cn/8 cn/8 cn/8 cn/8 cn/8
c c c c c c c c c c c
cn
cn
cn
cn
cn
Depth? M S t
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MergeSort
We can calculate the depth, by determining when
the recursion is c At each level, we divide by 2
12
d
n
nd 2
nd
log2log
nd log2log
nd 2log
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MergeSort
Running time?
Each level costs cn
log n levels
cn log n = (n log n )
otherwise)2/(2
smallisif)(
cnnT
ncnT
