Batcher

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Batchers merging network

• Efficient Parallel Algorithms
• COMP308

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Odd-even merging network

• We assume that n is exact power of 2, and we wish
  to merge the sorted sequence of elements on lines
  lta1,..,angt with those on lines ltan1,..,a2ngt
• We recursively construct two odd-even merging
  networks that merge sorted subsequences in
  parallel.

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Structure of the network

• The first merges the sequence on lines lta1,
  a3,..,an-1gt with sequences on lines ltan1,
  an2,..,a2n-1gt (the odd elements)
• The second merges the sequence on lines lta2,
  a4,..,angt with sequences on lines ltan2,
  an4,..,a2ngt (the even elements)
• To combine two sorted subsequences, we put a
  comparator between a2i-1 and a2i for all
  i1,2,,n

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8-input network
1 2 3 4 5 6
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Skeleton of the Proof

• Apply the zero-one principle to prove that
  Batchers sorting is correct, consider only
  merges of sequences of the form 0000011111 and
  00001111111
• In odd-even merging we send odd sequence to one
  merger and the even sequence to another merger.
  We can assume by induction that smaller mergers
  work correctly.
• The first merger can get one or two more zeros.
  than the second one. The outputs of these mergers
  are interleaved and can be 1 before 0 (dirty one).

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• After sorting separately on odd and even
  positions we will have everything sorted unless
  the situation like this happens
• 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1
• Then the last final touch (correction step)
  cleans the sequence.
• The situation is subtle, the last correction does
  not clean for example
• 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1
• But the dirty 1 (if there is any) should be on an
  even position, which is always the case