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Reduction of Sizes of Decision Diagrams by Autocorrelation Functions
May 2003 (vol. 52 no. 5)
pp. 592-606

Abstract—This paper discusses optimization of decisions diagrams (DDs) by total autocorrelation functions. We present an efficient algorithm for construction of Linearly Transformed Binary Decision Diagrams (LT-BDDs) and Linearly transformed multiterminal BDDs (LT-MTBDDs) for systems of Boolean functions, based on linearization of these functions by the corresponding autocorrelation functions. Then, we present a method for reduction of sizes of DDs by a level-by-level reduction of the width of DDs using the total autocorrelation functions. The approach provides for a simple procedure for minimization of LT-BDDs and LT-MTBDDs and upper bounds on their sizes. Experimental results for benchmarks illustrate that the proposed method on average is very efficient.

[1] S. Agaian, J. Astola, and K. Egiazarian, Binary Polynomial Transforms and Nonlinear Digital Filters. Marcel Dekker, 1995.
[2] J. Bern, C. Meinel, and A. Slobodova, “Efficient OBDD-Based Manipulation in CAD Beyond Current Limits,” Proc. 32nd Design Automation Conf., pp. 408-413, 1995.
[3] B. Bollig and I. Wegener, “Improving the Variable Ordering of OBDDs is NP-Complete,” IEEE Trans. Computers, vol. 45, no. 9, pp. 993-1002, 1996.
[4] R.E. Bryant, “Graph-Based Algorithms for Boolean Functions Manipulation,” IEEE Trans. Computers, vol. 35, no. 8, pp. 667-691, 1986.
[5] E.M. Clarke, K.L. McMillan, X. Zhao, and M. Fujita, “Spectral Transforms for Extremely Large Boolean Functions,” Proc. IFIP WG 10.5 Workshop on Applications of the Reed-Muller Expression in Circuit Design, U. Kebschull, E. Schubert, and W. Rosenstiel, eds., pp. 86-90, Sept. 1993.
[6] R. Drechsler and B. Becker, Binary Decision Diagrams, Theory and Impementation, Kluwer Academic Publishers, 1998.
[7] M. Fujita, Y. Kukimoto, and R.K. Brayton, “BDD Minimization by Truth Table Permutation,” Proc. Int'l Symp. Circuits and Systems, vol. 4, pp. 596-599, May 1996.
[8] M. Fujita, Y. Matsunaga, and T. Kakuda, "On variable Ordering of Binary Decision Diagrams for the Application of Multi-Level Logic Synthesis," Proc. European Design Automation Conf., pp. 50-54, 1991.
[9] M. Fujita, J.C.-Y. Yang, M. Clarke, X. Zhao, and P. McGeer, “Fast Spectrum Computation for Logic Functions Using Binary Decision Diagrams,” Proc. Int'l Symp. Computer-Aided Surgery (ISCAS '94), pp. 275-278, 1994.
[10] W. Günther and R. Drechsler, “BDD Minimization by Linear Transforms,” Advanced Computer Systems, pp. 525-532, 1998.
[11] W. Günther and R. Drechsler, “Efficient Manipulation Algorithms for Linearly Transformed BDDs,” Proc. Fourth Int'l Workshop Applications of Reed-Muller Expansion in Circuit Design, pp. 225-232, May 1999.
[12] W. Günther and R. Drechsler, “Minimization of BDDs Using Linear Transformations Based on Evolutionary Techniques,” Proc. Int'l Symp. Circuit and Systems, 1999.
[13] M.G. Karpovsky, Finite Orthogonal Series in the Design of Digital Devices. John Wiley, 1976.
[14] M.G. Karpovsky and E.S. Moskalev, “Utilization of Autocorrelation Characteristics for the Realization of Systems of Logical Functions,” Avtomatika i Telmekhanika, no. 2, pp. 83-90, 1970.
[15] Spectral Techniques and Fault Detection. M.G. Karpovsky, ed., pp. 35-90, Academic Press, 1985.
[16] M.G. Karpovsky, R.S. Stanković, and J.T. Astola, “Spectral Techniques for Design and Testing of Computer Hardware,” Proc. Int'l Workshop Spectral Techniques in Logic Design, pp. 1-34, June 2000.
[17] R.J. Lechner and A. Moezzi, “Synthesis of Encoded PLAs,” Spectral Techniques and Fault Detection, 1985.
[18] C. Meinel, F. Somenzi, and T. Theobald, “Linear Sifting of Decision Diagrams,” Proc. Design Automation Conf., pp. 202-207, 1997.
[19] C. Meinel, F. Somenzi, and T. Theobald, Linear Sifting of Decision Diagrams and Its Application in Synthesis IEEE Trans. Computer Automated Design, vol. 19, no. 5, pp. 521-533, 2000.
[20] D. Milošević, R.S. Stanković, and C. Moraga, “Calculation of Dyadic Autocorrelation throguh Decision Diagrams,” Proc. Int'l Workshop Computational Intelligence and Information Technologies, pp. 129-134, June 2001.
[21] S. Minato, “Graph-Based Representations of Discrete Functions,” Representations of Discrete Functions, pp. 1-28, 1996.
[22] S. Panda and F. Somenzi, “Who Are the Variables in Your Neigborhood,” Proc. IEEE Int'l Conf. Computer-Aided Design, pp. 74-77, 1995.
[23] S. Panda, F. Somenzi, and B. Plessier, "Symmetry Detection and Dynamic Variable Ordering of Decision Diagrams," Proc. ICCAD94, pp. 628-631, 1994.
[24] J. Rice, M. Serra, and J.C. Muzio, “The use of Autocorrelation Coefficients for Variable Ordering for ROBDDs,” Proc. Fourth Int'l Workshop Applications of Reed-Muller Expansion in Circuit Design, pp. 185-196, Aug. 1999.
[25] R. Rudell, "Dynamic Variable Ordering for Ordered Binary Decision Diagrams," Proc. ICCAD-93, pp. 42-47, 1993.
[26] T. Sasao, Switching Theory for Logic Synthesis. Kluwer Academic Publishers, 1999.
[27] Representations of Discrete Functions, T. Sasao and M. Fujita, eds., Kluwer, 1996.
[28] M. Sauerhoff, I. Wegener, and R. Werchner, “Optimal Ordered Binary Decision Diagrams for Read-Once Formulas,” Discrete Applied Math., vol. 103, pp. 237-258, 2000.
[29] D. Sieling, “On the Existence of Polynomial Time Approximation Schemes for OBDD Minimization,” Proc. Symp. Theoretical Aspects of Computer Science, vol. 1373, pp. 205-215, 1998.
[30] F. Somenzi, CUDD—Colorado Univ. Decision Diagram Package, 1996.
[31] R.S. Stanković, Spectral Transform Decision Diagrams in Simple Questions and Simple Answers. Belgrade: Nauka, 1998.
[32] R.S. Stanković, “Some Remarks on Basic Characteristics of Decision Diagrams,” Proc. Fourth Int'l Workshop Applications of Reed-Muller Expansion in Circuit Design, pp. 139-146, Aug. 1999.
[33] R.S. Stanković, M. Bhattacharaya, and J.T. Astola, “Calulation of Dyadic Autocorrelation through Decision Diagrams,” Proc. European Conf. Circuit Theory and Design, pp. 337-340, Aug. 2001.
[34] R.S. Stanković and T. Sasao, “Decision Diagrams for Representation of Discrete Functions: Uniform Interpretation and Classification,” Proc. Asia and South Pacific Design Automation Conf., Feb. 1998.
[35] R.S. Stanković, T. Sasao, and C. Moraga, “Spectral Transform Decision Diagrams,” Representations of Discrete Functions, pp. 55-92, 1996.
[36] E.A. Trachtenberg, “SVD of Frobenius Matrices for Approximate and Multiobjective Signal Processing Tasks,” SVD and Signal Processing, E.F. Derettere, ed., pp. 331-345, Elsevier, 1988.
[37] I. Wegener, “Worst Case Examples for Operations over OBDDs,” Information Processing Letters, no. 74, pp. 91-94, 2000.

Index Terms:
Logic synthesis, spectral techniques, decision diagrams, linear transforms, autocorrelation functions.
Citation:
Mark G. Karpovsky, Radomir S. Stankovc, Jaakko T. Astola, "Reduction of Sizes of Decision Diagrams by Autocorrelation Functions," IEEE Transactions on Computers, vol. 52, no. 5, pp. 592-606, May 2003, doi:10.1109/TC.2003.1197126
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