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

Abstract

AbstractA binary sequence x 1, ..., x n 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 x i ≠x i + 1. A sequence ¬x 1, ..., ¬x n is called an inversion of x 1, ..., x n . 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 log2 n). 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(log2 n) and contains O(logn) NOT gates. For this case, the size and the number of NOT gates are optimal up to a constant factor.KeywordsBoolean FunctionInput SequenceBinary SequenceBinary RepresentationCircuit ComplexityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.