Superconcentrators, generalizers and generalized connectors with limited depth

  • Authors:

  • Danny Dolev;Cynthia Dwork;Nicholas Pippenger;Avi Wigderson

  • Affiliations:

  • -;-;-;-

  • Venue:
  • STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
  • Year:
  • 1983

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Abstract

We show that the minimum possible size of an n-superconcentrator with depth 2k≥4 is &thgr;(n&lgr;(k, n)), where &lgr;(k, .) is the inverse of a certain function at the k-th level of the primitive recursive hierarchy. It follows that the minimum possible depth of an n-superconcentrator with linear size is &thgr;(&bgr;(n)), where &bgr; is the inverse of a function growing more rapidly than any primitive recursive function. Similar results hold for generalizers. We give a simple explicit construction for a (d1...dk)-generalizer with depth k and size (d1+...+dk)d1...dk. This is applied to give a simple explicit construction for a generalized n-connector with depth 2k−3 and size (2d1+3d2+...+3dk−1+2dk) d1...dk. These are the best explicit constructions currently available. We also show that, for each fixed k≥2, the minimum possible size of a generalized n-connector with depth k is &Ohgr;(n1+1/k) and 0((n log n)1+1/k).