Lecture 5 Sorting Networks

1
Lecture 5Sorting Networks
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Comparator
comparator
3
A Sorting Network
2
2
9
5
6
5
9
5
2
5
6
2
6
9
6
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A sorting network is a comparison network which
output monotone nondecreasing sequence for every
input.
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Depth
2
2
9
5
6
5
9
5
2
5
6
2
6
9
6
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Depth is the maximum number of comparators on
a path from an input wire to an output wire.
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Depth parallel time
2
2
9
5
6
5
9
5
2
5
6
2
6
9
6
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Depth is the maximum number of comparators on
a path from an input wire to an output wire.
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Insertion Sort
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key
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Sorting network constructed from insertion sort.
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How to construct a sortingnetwork from merging
sort?
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Divide and Conquer

• Divide the problem into subproblems.
• Conquer the subproblems by solving them
  recursively.
• Combine the solutions to subproblems into the
  solution for original problem.

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Merge Sort
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Procedure
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Structure
Sorting network
Merging network
Sorting network
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Construction of Merging Network

• 0-1 principal.
• Bitonic sorter.
• Merging network.

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0-1 principal
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Lemma
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Proof of 0-1 Principal
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Bitonic Sequence
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Bitonic 0-1 Sequence
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Some Properties
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The half-cleaner
0
0
0
0
bitonic clean
0
1
1
0
bitonic
1
1
0
0
bitonic
1
0
0
1
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The half-cleaner
0
0
0
0
bitonic clean
1
1
1
0
bitonic
1
1
1
1
bitonic
1
1
0
1
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Lemma

• One of two halfs is bitonic clean.
• every number in the 1st half any element in
  the 2nd half.

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Proof (case 1)
0
0
1
0
1
1
0
0
0
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Proof (case 2)
0
0
1
0
1
0
0
1
0
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Proof (case 3)
0
0
1
0
1
1
1
0
0
0
1
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Proof (case 4)
0
0
1
1
1
0
0
1
1
0
0
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Proof (case 5)
1
1
0
0
1
1
1
1
0
1
1
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Proof (case 6)
1
1
0
1
1
1
0
1
1
0
1
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Proof (case 7)
1
0
1
1
0
0
1
1
0
0
1
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Proof (case 8)
1
0
1
0
1
0
1
0
0
1
1
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bitonic
sorted
Half cleaners
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0
0
0
0
sorted
0
0
1
0
sorted
0
0
0
1
sorted
1
1
1
1
Half cleaners
Merging Network
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Structure
Sorting network
Merging network
Sorting network
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Sorting Network
Merging Networks
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What we learnt in this lecture?

• What is sorting network?
• Depth parallel time.
• Sorting network from Merge sort.

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Permutation Network

• Switching network
• Rearrangeability
• Nework with 2x2 crossbars

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Crossbar Switch
A crossbar switch can realize any matching
between Inputs and outputs.
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3-stage Clos Network
1
n
n
n
n
m
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Rearrangeability
Theorem
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Network with 2x2 crossbars
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Puzzle