Comparative Analysis of the Efficiency of the Transformation Algorithm and the Cycles Decomposition Algorithm for the Synthesis of Ternary Reversible Circuits

Abstract

The design algorithm for binary reversible circuits based on the decomposition of permutations as cascade of disjoint cycles (CD algorithm) is extended to the ternary domain. Its performance is evaluated using as a benchmark all 362, 880 ternary reversible functions of two arguments, and is compared to that of the transformation based MMD algorithm. It is shown that in 96.71% of the cases the MMD algorithm leads to reversible circuits with a cost not higher than that obtained with the CD algorithm, and in 54.39% of the cases the MMD algorithm leads to reversible circuits with a cost lower than that obtained with the CD algorithm. Moreover the CD algorithm in 96% of the circuits reaches realizations with lower cost compared to the cost of decompositions into transpositions and in 69.5% of the circuits with respect to decompositions into 3-cycles. The results may be extended to the class of 2 x 2-based n x n ternary reversible functions.