Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights Farzad Parvaresh HP Labs, Palo Alto Joint work with Erik Ordentlich and Ron M. Roth Novermber 2011

2 Introducing the problem Consider all the 2x2 binary matrices:

3 Introducing the problem Consider all the 2x2 binary matrices: 7 How many binary matrices exist such that number of ones in each row or column is at most ?

4 Memristors Applications

5 Memristors Applications

6 Memristors Applications Drives too much current

7 Memristors Applications

8 Memristors Applications Drives too much current

9 Memristors Applications Do not want too many memristors in any row or column with low resistance state. Drives too much current Map binary data into matrices such that number of ones in each row or column is at most.  Each one in the matrix corresponds to a low resistance state. How many bits can be stored in an memory with this restriction?

10 First attempt Number of bounded row and column matrices E. Ordentlich, and R.M. Roth, “Low complexity two-dimensional weight-constrained codes”, ISIT, August, Efficient one-to-one mapping of bits to binary bounded row and column weight matrices. Are there more bounded row and column weight matrices?

11 Counting number of bounded row column matrices: First attempt Flipping Algorithm Add one row and one column of all zero vectors to the matrix.

12 Counting number of bounded row column matrices: First attempt Flipping Algorithm

13 Counting number of bounded row column matrices: First attempt Flipping Algorithm Flip

14 Counting number of bounded row column matrices: First attempt Flipping Algorithm Flip Notice that total number of ones in the matrix after each flipping will reduce at least by one.

15 Counting number of bounded row column matrices: First attempt Flipping Algorithm Flip Notice that total number of ones in the matrix after each flipping will reduce at least by one.

16 Counting number of bounded row column matrices: First attempt Flipping Algorithm Notice that total number of ones in the matrix after each flipping will reduce at least by one. Flipping will stop after at most steps. Flipping maps matrices to bounded row and column matrices.

Are there more bounded row and column matrices Count number of bounded row and column matrices for small. For even : For odd :

18 Main result Theorem: Let denote the standard normal cumulative distribution function, and then, Theorem: Let denote the standard normal cumulative distribution function, and then, Proof: In two parts. Show a lower bound and an upper bound for.

19 Previous work B.D. McKay, I.M. Wanless, and N.C. Wormald, “Asymptotic enumeration of graphs with a given bound on the maximum degree,” Comb. Probab. Comput., E.C. Posner and R.J. McEliece, “The number of stable points of and infinite-range spin glass memory,” Jet Propulsion Laboratory, Tech. Rep., Expected number of solutions to:

20 Canfield, Greenhill and McKay (CGM08) Lower bound Theorem[CGM08]: = Set of all binary matrices with row sum equal to column sum equal to.

21 Lower bound Enumerate bounded row and column matrices that satisfy assumptions of CGM theorem. Number of ones in each row or column is around the mean. Set of bounded row and column matrices:

22 Lower bound total number of ones in matrix

23 Lower bound Enlarge the set.

24 Lower bound

25 Lower bound where Approximate summation by integration

26 Lower bound where denotes the real n -dimensional cube,

27 Lower bound Looks like a multidimensional Gaussian distribution!

28 Lower bound Simulate Gaussians: Use saddle point

29 Upper bound Set of bounded row and column matrices: We have to enumerate rest of the matrices that do not satisfy assumptions of CGM theorem.

30 Majorization Lemma Upper bound Lemma: For any with and and, respectively, majorizing and,

31 Majorization Lemma Upper bound Lemma: For any with and and, respectively, majorizing and,

32 Majorization Lemma Upper bound Lemma: For any with and and, respectively, majorizing and, For any and find and that are majorized by and, and satisfy the assumptions of CGM theorem. Then use the Lemma to upper bound.

33 Upper bound After choosing the appropriate anchor point for majorization and simplification we can show: The Integral is equivalent to We can compute the expectation using the same techniques as lower bound: Same Gaussian as lower bound.

34 Main result Theorem: Let denote the standard normal cumulative distribution function, and then, Theorem: Let denote the standard normal cumulative distribution function, and then, Proof: Lower bound: Upper bound: Set

35 Future work Tighter enumeration of bounded row and column matrices. Efficient mapping of data to bounded row and column matrices that achieves optimal redundancy.