asymptotic enumeration of binary matrices with bounded row and column weights

Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Weights

Farzad ParvareshHP Labs, Palo Alto

Joint work with Erik Ordentlich and Ron M. RothNovermber 2011

2

Introducing the problem

Consider all the 2x2 binary matrices:

0 0

0 0

1 0

0 0

0 1

0 0

0 0

0 1

0 0

1 0

1 1

0 0

1 0

0 1

1 0

1 0

1 1

0 1

1 0

1 1

0 1

0 1

0 1

1 0

0 1

1 1

1 1

1 0

0 0

1 1

1 1

1 1

3

Introducing the problem

Consider all the 2x2 binary matrices:

0 0

0 0

1 0

0 0

0 1

0 0

0 0

0 1

0 0

1 0

1 0

0 1

0 1

1 0

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, 2011.

Efficient one-to-one mapping of bits to binary bounded row and column weight matrices.

Are there more bounded row and column weight matrices?

17

4 0.982601

5 1.409983

6 1.136897

7 1.424295

8 1.195001

9 1.424725

10 1.227649

11 1.424964

12 1.249322

13 1.425054

14 1.265102

15 1.425093

Are there more bounded row and column matricesCount 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,

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., 2002.

• E.C. Posner and R.J. McEliece, “The number of stable points of and infinite-range spin glass memory,” Jet Propulsion Laboratory, Tech. Rep., 1985.

Expected number of solutions to:

20

Canfield , Greenhill and McKay (CGM08)

Lower bound

Theorem[CGM08]:

1 0 0 1 1 1 0

1 1 0 0 0 0 1

0 0 0 1 1 0 0

0 0 1 0 1 0 0

1 1 1 1 0 1 0

0 1 1 0 1 1 1

1 0 0 0 0 0 0

1 1 1 0 0 1 0

= 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,

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.