Dynamic Programming Algorithms as Reversible Circuits: Symmetric Function Realization

The work starts with a general idea of how to realize a dynamic programming algorithm as a reversible circuit. This realization is not tied to a specific reversible design model and technology or a class of dynamic algorithms, it shows an approach for such synthesis. As an illustration of this approach, a class of all symmetric functions is realized in a dynamic programming algorithm manner as a reversible circuit of Toffoli elements. The garbage, quantum and reversible costs of the presented implementation are calculated and compared to the costs of previously described reversible synthesis methods. The summary of results of this comparison is as follows. The quantum cost of the proposed method is better than the quantum cost of any other known systematic approach. The garbage is usually lower (except for comparison with the synthesis methods, primary goal of which is synthesis with theoretically minimal garbage). For the large functions reversible cost is better or has the same asymptotic that of other methods. Although the reversible cost comparison may not look beneficial for the small functions, the possible technological implementation (quantum) shows that it is beneficial to use the presented approach even for the small functions.