On the Negation-Limited Circuit Complexity of Sorting and Inverting k-tonic Sequences
A 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)/ 2 ct ) where c > 0 is some constant. This generalizes a similar upper bound for merging by Amano, Maruoka and Tarui , 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  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 Function Input Sequence Binary Sequence Binary Representation Circuit Complexity
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