Optimal conclusive sets for comparator networks

AbstractA set of input vectors S is conclusive for a certain functionality if, for every comparator network, correct functionality for all input vectors is implied by correct functionality for all vectors in S. We consider four functionalities of comparator networks: sorting, merging, sorting of bitonic vectors, and halving. For each of these functionalities, we present two conclusive sets of minimal cardinality. The members of the first set are restricted to be binary, while the members of the second set are unrestricted. For all the above functionalities, except halving, the unrestricted conclusive set is much smaller than the binary one

Optimal Conclusive Sets for Comparator Networks

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. Specifically, we present a conclusive set for sorting that contains � � n n/2 nonbinary vectors. For merging, we present a conclusive set with n 2 + 1 nonbinary vectors. For � bitonic sorting, we present a binary vectors that constitute conclusive set with n nonbinary vectors. For halving we present � n n/2 a conclusive set. We prove that all these conclusive sets are optimal