Some minimum merging networks

  • Authors:

  • Gembu Morohashi;Shigeki Iwata

  • Affiliations:

  • NTT Information Sharing Platform Labs., Yokosuka-shi, Kanagawa 239-0847 Japan and Department of Computer Science, The University of Electro-Communications, Chofu-shi, Tokyo 182-8585, Japan;Department of Computer Science, The University of Electro-Communications, Chofu-shi, Tokyo 182-8585, Japan

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2004

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Abstract

Let M(m, n) be the minimum number of comparators which constructs an (m, n)-merging network. Batcher's odd-even merge, which is a merging network constructed by his algorithm, provides the best upper bound for M(m, n) to date. Recently Iwata (Inform. and Comput. 168 (2001) 187) analyzed the property of leftmost comparators, and showed M(m1 + m2, n) ≥ [(M(m1, n) + M(m2, n) + m1 + m2 + n - 2)/2]. We extend Iwata's proofs and show that Batcher's (6, 8k + 7)-, (9, 16k + 9)-, (7, 8)-merging networks are optimal for all k ≥ 0.In Batcher's (m, n)-merging network, the ith smallest element out of m elements and another ith smallest element out of n elements are first compared for all i (1 ≤ i ≤ min{m, n}). Under an assumption of existence of such min{m, n} comparators in optimal (m, n)-merging networks, we show that M(n, n) = M(n - 1, n) + 1 = M(n - 2, n) + 3.