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Batcher Sorting Network, n = 4
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Batcher Sorting Network, n = 8
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Lemma 1 Any subsequence of a sorted sequence is a sorted sequence.
sorted sorted 1 1 1 1 1 1 1 1
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Lemma 2 For a sorted sequence, the number of 0’s in the even subsequence is either equal to, or one greater than, the number of 0’s in the odd subsequence. sorted 1 1 1 1 1 1 even odd
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Lemma 3 For two sorted sequences and :
denotes the the number of 0’s in denotes the even subsequence of denotes the odd subsequence of
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Lemma 3 1 1 1 x 1 E x 1 1 O x
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Lemma 3 For two sorted sequences and : (by Lemma 2) (by Lemma 2)
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Merge Network Merge[4] sorted sorted sorted
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Merge Network (pf.) sorted sorted sorted sorted Merge[4] (by Lemma 1)
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Merge Network (pf.) sorted sorted Merge[4] By Lemma 3 and
differ by at most 1 By Lemma 3 sorted
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Merge Network (pf.) Merge[4] sorted and differ by at most 1 By Lemma 3
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Merge Network (pf.) 1 1 1 1 1 1 Merge[4] By Lemma 3 and
Merge[4] 1 1 and differ by at most 1 By Lemma 3 1 1 1 1
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Batcher Sorting Network
Merge[8] sorted Sort[4]
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Batcher Sorting Network, n = 4
Merge[4]
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Batcher Sorting Network, n = 8
Merge[8] Sort[4]
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Sorting Networks AKS (Ajtai, Komlós, Szemerédi) Network:
based on expander graphs. AKS (Chvátal) Batcher AKS better for
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