On the Numbers of Products in Prefix SOPs for Interval Functions

SUMMARY First, this paper derives the prefix sum-of-products expression (PreSOP) and the number of products in a PreSOP for an interval function. Second, it derives Ψ(n ,τ p), the number of n-variable interval functions that can be represented with τp products. Finally, it shows that more than 99.9% of the n-variable interval functions can be represented with � 3n − 1� products, when n is sufficiently large. These results are use

[1]  K. Pagiamtzis,et al.  Content-addressable memory (CAM) circuits and architectures: a tutorial and survey , 2006, IEEE Journal of Solid-State Circuits.

[2]  Tsutomu Sasao.  On the Number of Products to Represent Interval Functions by SOPs with Four-Valued Variables , 2010, 2010 40th IEEE International Symposium on Multiple-Valued Logic.

[3]  Saeed Ghahramani.  Fundamentals of Probability , 1995 .

[4]  George Varghese,et al.  Scalable packet classification , 2001, SIGCOMM 2001.

[5]  Nick McKeown,et al.  Packet classification on multiple fields , 1999, SIGCOMM '99.

[6]  Jonathan S. Turner,et al.  Packet classification using extended TCAMs , 2003, 11th IEEE International Conference on Network Protocols, 2003. Proceedings..

[7]  Anand Rangarajan,et al.  Algorithms for advanced packet classification with ternary CAMs , 2005, SIGCOMM '05.

[8]  Shalabh.  Fundamentals of Probability: a First Course , 2011 .

[9]  Eric Torng,et al.  Topological transformation approaches to optimizing TCAM-based packet classification systems , 2009, SIGMETRICS '09.

[10]  Praveen Yalagandula,et al.  Minimizing Rulesets for TCAM Implementation , 2009, IEEE INFOCOM 2009.

[11]  R.N. Mahapatra,et al.  An Efficient Approach to On-Chip Logic Minimization , 2007, IEEE Transactions on Very Large Scale Integration (VLSI) Systems.

[12]  Timothy Sherwood,et al.  Modeling TCAM power for next generation network devices , 2006, 2006 IEEE International Symposium on Performance Analysis of Systems and Software.

[13]  Tsutomu Sasao.  On the Complexity of Classification Functions , 2008, 38th International Symposium on Multiple Valued Logic (ismvl 2008).

[14]  Tsutomu Sasao,et al.  Worst and Best Irredundant Sum-of-Products Expressions , 2001, IEEE Trans. Computers.

[15]  Jia Wang,et al.  Packet classifiers in ternary CAMs can be smaller , 2006, SIGMETRICS '06/Performance '06.

[16]  Isaac Keslassy,et al.  On the code length of TCAM coding schemes , 2010, 2010 IEEE International Symposium on Information Theory.

[17]  Tsutomu Sasao.  Memory-Based Logic Synthesis , 2011 .

[18]  Bernd Reusch,et al.  Generation of Prime Implicants from Subfunctions and a Unifying Approach to the Covering Problem , 1975, IEEE Transactions on Computers.

[19]  Danny Hendler,et al.  Space-Efficient TCAM-Based Classification Using Gray Coding , 2007, IEEE Transactions on Computers.

[20]  David E. Taylor.  Survey and taxonomy of packet classification techniques , 2005, CSUR.

[21]  Baruch Schieber,et al.  Computing the minimum DNF representation of Boolean functions defined by intervals , 2005, Discret. Appl. Math..

[22]  Frank Vahid,et al.  On-chip logic minimization , 2003, Proceedings 2003. Design Automation Conference (IEEE Cat. No.03CH37451).

[23]  László Lovász,et al.  Binomial Coefficients and Pascal’s Triangle , 2003 .

[24]  Tsutomu Sasao,et al.  Switching Theory for Logic Synthesis , 1999, Springer US.