A linear time algorithm for recognizing regular Boolean functions

A positive (or monotone) Boolean function is regular if its variables are naturally ordered, left to right, by decreasing strength, so that shifting the nonzero component of any true vector to the left always yields another true vector. This paper considers the problem of recognizing whether a positive function f is regular, where f is given by min T(f) (the set of all minimal true vectors of f). We propose a simple linear time (i.e., O(n|min T(f)|)- time) algorithm for it. This improves upon the previous algorithm by Provan and Ball which requires O(n2|min T(f)|) time. As a corollary, we also present an O(n(n + |min T(f)|))- time algorithm for the recognition problem of 2-monotonic functions.

[1]  Bruno Simeone,et al.  Polynomial-time algorithms for regular set-covering and threshold synthesis , 1985, Discret. Appl. Math..

[2]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, STOC '84.

[3]  Bruno Simeone,et al.  A O(nm)-Time Algorithm for Computing the Dual of a Regular Boolean Function , 1994, Discret. Appl. Math..

[4]  Lisa Hellerstein,et al.  Complexity theoretic hardness results for query learning , 1998, computational complexity.

[5]  Antonio Sassano,et al.  An O(mn) Algorithm for Regular Set-Covering Problems , 1987, Theor. Comput. Sci..

[6]  Kenneth H. Rosen,et al.  Applications of Discrete Mathematics , 1991 .

[7]  P. Hammer,et al.  Aggregation of inequalities in integer programming. , 1975 .

[8]  L. Khachiyan.  Polynomial algorithms in linear programming , 1980 .

[9]  K. Ramamurthy.  Coherent Structures and Simple Games , 1990 .

[10]  Toshihide Ibaraki,et al.  The Maximum Latency and Identification of Positive Boolean Functions , 1994, SIAM J. Comput..

[11]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[12]  Yves Crama,et al.  Dualization of regular Boolean functions , 1987, Discret. Appl. Math..

[13]  Toshihide Ibaraki,et al.  A Fast and Simple Algorithm for Identifying 2-Monotonic Positive Boolean Functions , 1995, J. Algorithms.

[14]  Saburo Muroga,et al.  Threshold logic and its applications , 1971 .

[15]  Robert O. Winder,et al.  Threshold logic , 1971, IEEE Spectrum.

[16]  L. G. H. Cijan.  A polynomial algorithm in linear programming , 1979 .

[17]  Kazuhisa Makino.  A Linear Time Algorithm for Recognizing Regular Boolean Functions , 1999, ISAAC.

[18]  Peter L. Hammer,et al.  An Algorithm to Dualize a Regular Switching Function , 1979, IEEE Transactions on Computers.

[19]  Toshihide Ibaraki,et al.  Polynomial-Time Recognition of 2-Monotonic Positive Boolean Functions Given by an Oracle , 1997, SIAM J. Comput..

[20]  J. Scott Provan,et al.  Disjoint Products and Efficient Computation of Reliability , 1988, Oper. Res..

[21]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .