A Note on Bennett's Time-Space Tradeoff for Reversible Computation

Given any irreversible program with running time T and space complexity S, and given any e > 0, Bennett shows how to construct an equivalent reversible program with running time O(T1+) and space complexity O(S In T). Although these loose upper bounds are formally correct, they are misleading due to a hidden constant factor in the space bound. It is shown that this constant factor is approximately e21/, which diverges exponentially as e approaches 0. Bennett’s analysis is simplified using recurrence equations and it is proven that the reversible program actually runs in time O(T+/S) and space O(S(1 + In (T/S))). Bennett claims that for any e > 0, the reversible program can be made to run in time O(T) and space O(ST ). This claim is corrected and tightened as follows: whenever T => 2S and for any e -> / (0.58 lg (T/S)), the reversible program can be made to run in time O(T) and space f(S(T/S)’/2)fqO(S(T/S)). For S <= T < 2S, Bennett’s 1973 simulation yields an equivalent reversible program that runs in time O(T) and space O(S). Key words, algorithms, reversible computation, time-space tradeoff AMS(MOS) subject classifications. 68Q05, 68Q15