On the Negation-Limited Circuit Complexity of Sorting and Inverting k-tonic Sequences

A binary sequence x1, ..., xn is called k-tonic if it contains at most k changes between 0 and 1, i.e., there are at most k indices such that xi ≠xi+1. A sequence ¬x1, ..., ¬xn is called an inversion of x1, ..., xn. In this paper, we investigate the size of a negation-limited circuit, which is a Boolean circuit with a limited number of NOT gates, that sorts or inverts k-tonic input sequences. We show that if k = O(1) and t = O(loglogn), a k-tonic sequence of length n can be sorted by a circuit with t NOT gates whose size is O((n logn)/ 2ct) where c > 0 is some constant. This generalizes a similar upper bound for merging by Amano, Maruoka and Tarui [4], which corresponds to the case k = 2. We also show that a k-tonic sequence of length n can be inverted by a circuit with O(k logn) NOT gates whose size is O(kn) and depth is O(k log2n). This reduces the size of the negation-limited inverter of size O(n logn) by Beals, Nishino and Tanaka [6] when k = o(logn). If k = O(1), our inverter has size O(n) and depth O(log2n) and contains O(logn) NOT gates. For this case, the size and the number of NOT gates are optimal up to a constant factor.