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Lower bounds for constant depth circuits for prefix problems

  • Ashok K. Chandra
  • Steven Fortune
  • Richard Lipton
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 154)

Abstract

A prefix-or circuit has n inputs and n outputs; the ith output is the OR of the first i inputs. A prefix-carry circuit has 2n inputs, interpreted as two n-bit numbers, and n outputs; the ith output is the carry in the ith position of the sum of the two numbers. We show a nonlinear lower bound for constant-depth, unboundedfanin implementations of prefix-or. However, with negation, linear size circuits are possible. For prefix-carry, we show nonlinear lower bounds for arbitrary circuits. In both cases the lower bounds exhibit a size/depth tradeoff: the circuit size must be at least Ω(nf d −1 d(n)) for depth a constant times d. Here the functions f d form an increasing hierarchy coextensive with the primitive recursive functions. The lower bounds match the known upper bounds for these problems, to within a constant factor for depth.

Keywords

Boolean Function Boolean Circuit Parallel Random Access Machine Output Vertex Primitive Recursive Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

  • Ashok K. Chandra
    • 1
  • Steven Fortune
    • 1
  • Richard Lipton
    • 2
  1. 1.Mathematical Sciences DepartmentIBM T.J. Watson Research CenterYorktown Heights
  2. 2.Department of EECSPrinceton UniversityPrinceton

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