The monotone circuit complexity of Boolean functions
Combinatorica
The complexity of Boolean functions
The complexity of Boolean functions
More on the complexity of negation-limited circuits
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
On the Complexity of Negation-Limited Boolean Networks
SIAM Journal on Computing
On the Inversion Complexity of a System of Functions
Journal of the ACM (JACM)
Bounds to Complexities of Networks for Sorting and for Switching
Journal of the ACM (JACM)
Higher lower bounds on monotone size
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
On the negation-limited circuit complexity of merging
Discrete Applied Mathematics - Special issue: Special issue devoted to the fifth annual international computing and combinatories conference (COCOON'99) Tokyo, Japan 26-28 July 1999
Hauptvortrag: The complexity of negation-limited networks - A brief survey
Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
On the minimum number of negations leading to super-polynomial savings
Information Processing Letters
IEEE Transactions on Computers
Linear-size log-depth negation-limited inverter for k-tonic binary sequences
Theoretical Computer Science
Linear-size log-depth negation-limited inverter for k-tonic binary sequences
TAMC'07 Proceedings of the 4th international conference on Theory and applications of models of computation
Negation-Limited complexity of parity and inverters
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
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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.