Optimal Conclusive Sets for Comparator Networks

Abstract

A set of input vectors S is conclusive if correct functionality for all input vectors is implied by correct functionality over vectors in S. We consider four functionalities of comparator networks: sorting, merging of two equal length sorted vectors, sorting of bitonic vectors, and halving (i.e., separating values above and below the median). For each of these functionalities, we present tight lower and upper bounds on the size of conclusive sets. Bounds are given both for conclusive sets composed of binary vectors and of general vectors. The bounds for general vectors are smaller than the bounds for binary vectors implied by the 0-1 principle. Our results hold also for comparator networks with unbounded fanout.

Assume the network at hand has n inputs and outputs, where n is even. We present a conclusive set for sorting that contains $\binom{n}{n/2}$ nonbinary vectors. For merging, we present a conclusive set with $\frac n2 +1$ nonbinary vectors. For bitonic sorting, we present a conclusive set with n nonbinary vectors. For halving, we present $\binom{n}{n/2}$ binary vectors that constitute a conclusive set. We prove that all these conclusive sets are optimal.