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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. Introducing the problem. Consider all the 2x2 binary matrices:. Introducing the problem. - PowerPoint PPT Presentation
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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
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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 ,
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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.
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