Exact Repair problems with multiple sources: CISS 2014
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CISS 2014, Princeton NJ 1
Exact Repair Problems with Multiple Sources
Jayant Apte*, Congduan Li, John MacLaren Walsh, Steven Weber
ECE Dept. Drexel University
CISS 2014, Princeton NJ 2
Outline
● Problem Definition● Computer assisted proofs: General Structure● Polyhedral bounds on● Polyhedral computation interpretation of rate
region computation● A projection technique for computing
achievable rate region
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Outline
● Problem Definition● Computer assisted proofs: General structure● Polyhedral bounds on● Polyhedral computation interpretation of rate
region computation● A projection technique for computing
achievable rate region
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(n,k,d) Exact Repair with multiple sources
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(n,k,d) Exact Repair with multiple sources
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2-source (3,2,2) exact repair problem
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2-source (3,2,2) exact repair problem
2 sources
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2-source (3,2,2) exact repair problem
3 encoding functions
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2-source (3,2,2) exact repair problem
3 storage random variables
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2-source (3,2,2) exact repair problem
3 decoders with different demands
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2-source (3,2,2) exact repair problem
6 repair encodingfunctions
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2-source (3,2,2) exact repair problem
3 repair decodingfunctions
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2-source (3,2,2) exact repair problem
Total 11 random variables
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Implicit characterization of rate region(Yan et al.)
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Outline
● Problem Definition● Computer assisted proofs: General structure● Polyhedral bounds on● Polyhedral computation interpretation of rate
region computation● A projection technique for computing
achievable rate region
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Motivation
SourcesDecoderDemands
SoftwareNetwork
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Motivation
Software
SourcesDecoderDemands
NetworkRate Region
and optimal codes
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Software for computer assisted proofs
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Computer assisted converse
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Computer assisted converse
Inequalities obtained as an implication of linear Shannon-type,non-Shannon-type, non-linear non-Shannon type inequalities andnetwork constraints
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Computer assisted converse
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Computer assisted achievability
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Software for computer assisted proofs
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Outline
● Problem Definition● Computer assisted proofs: General structure● Polyhedral bounds on● Polyhedral computation interpretation of rate
region computation● A projection technique for computing
achievable rate region
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● Closure of set of all 'entropic' vectors arising from N-variable probability distributions
3-D rendition of
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● Closure of set of all 'entropic' vectors arising from N-variable probability distributions
● Each entropic vector is formed by stacking entropies of subsets of N random variables
3-D rendition of
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● Closure of set of all 'entropic' vectors arising from N-variable probability distributions
● Each entropic vector is formed by stacking entropies of subsets of N random variables
● Cone:
3-D rendition of
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● Cannot be expressed as intersection of finite number of linear inequalities for N>3
● For N=4, existence of single nonlinear● non-Shannon inequality(necessary and
sufficient) is known [Liu & Walsh 2014]● Additionally, several hundred linear
non-Shannon inequalities are known[DFZ 2011, Csirmaz 2013]
3-D rendition of
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Outline
● Problem Definition● Computer assisted proofs: General structure● Polyhedral bounds on
– Shannon (Outer) bound
● Polyhedral computation interpretation of rate region computation
● A projection technique for computing achievable rate region
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Shannon Outer Bound
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Shannon Outer Bound
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Shannon Outer Bound
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Outline
● Problem Definition● Computer assisted proofs: General structure● Polyhedral bounds on
– Shannon (Outer) bound
– (Representable) Matroid (Inner) bound(s)
● Polyhedral computation interpretation of rate region computation
● A projection technique for computing achievable rate region
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(Representable) Matroid Inner bound(s)
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(Representable) Matroid Inner bound(s)
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(Representable) Matroid Inner bound(s)
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(Representable) Matroid Inner bound(s)
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(Representable) Matroid Inner bound(s)
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(Representable) Matroid Inner bound(s)
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(Representable) Matroid Inner Bound(s)
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Outline
● Problem Definition● Computer assisted proofs: General structure● Polyhedral bounds on
– Shannon (Outer) bound
– Matroid (Inner) bound(s)
– Subspace (Inner) bounds
● Polyhedral computation interpretation of rate region computation
● A projection technique for computing achievable rate region
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Subspace Inner Bound(s)
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Subspace Inner Bound(s)
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Subspace Inner Bound(s)
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Subspace Inner Bound(s)
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Subspace Inner Bound(s)
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Software for computer assisted proofs
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Polyhedral bounds on rate region
● Using polyhedral inner/outer bound on yields
polyhedral inner/outer bounds on rate region● Lemma 1: Inner bounds on rate region
computed using or are achievable using linear codes
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Outline
● Problem Definition● Computer assisted proofs: General structure● Polyhedral bounds on
– Shannon (Outer) bound
– Matroid (Inner) bound(s)
– Subspace (Inner) bounds
● Polyhedral computation interpretation of rate region computation
● A projection technique for computing achievable rate region
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Network Coding constraints
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Network Coding constraints
● Consider a type 1 or type 2 constraint H● In general, computing extreme rays of given H and
extreme rays of is equivalent to an iteration of Double Description Method of polyhedral representation conversion
● Lemma 2 [Li et al. 2013]: An extreme ray of is an extreme ray of if it is contained in the hyperplane corresponding to H
● Hence, simple membership check suffices to find extreme rays of
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Software for computer assisted proofs
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Rate constraints
Storage Bandwidth
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Rate constraints
Repair Bandwidth
Storage Bandwidth
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A projection technique for computing achievable rate region
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A projection technique for computing achievable rate region
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A projection technique for computing achievable rate region
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Polyhedral projection via chm
● chm is an implementation of polyhedral projection algorithm called Convex Hull Method by Jayant Apte*
● chmlib v0.x is available at:
http://www.ece.drexel.edu/walsh/aspitrg/software.html
● Rational arithmetic using FLINT: Fast Library for Number Theory
● Rational LP solver based on qsopt
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Polyhedral projection via chm
● Has been used for– The current work
– Computer assisted converse proofs of rate regions of Multilevel Diversity Coding Systems(a special case of multi-source network coding)
– Finding non-Shannon Information Inequalities via Generalized Copy Lemma of Csirmaz
● Can be used for – Finding necessary conditions for non-contexuality of small
marginal scenarios(Quantum Information)
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Results
SoftwareNetwork
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Rate region for H(S1)=1 and H(S2)=1
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Rate region for H(S1)=1 and H(S2)=2
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References● X. Yan, R.W. Yeung, and Zhen Zhang. An implicit characterization of the achievable rate region for acyclic
multisource multisink network coding. Information Theory, IEEE Transactions on, 58(9):5625–5639, 2012.● Dougherty, Randall, Chris Freiling, and Kenneth Zeger. "Non-Shannon information inequalities in four
random variables." arXiv preprint arXiv:1104.3602 (2011).● Csirmaz, László. "Information inequalities for four variables." CEU (2013).● Yunshu Liu and John M. Walsh, "Only One Nonlinear Non-Shannon Inequality is Necessary for Four
Variables", submitted to IEEE Int. Symp. Information Theory (ISIT2014)● Congduan Li, J. Apte, J.M. Walsh, and S. Weber. A new computational approach for determining rate regions
and optimal codes for coded networks. In Network Coding (NetCod), 2013 International Symposium on, pages 1–6, 2013.
● Congduan Li, John MacLaren Walsh, Steven Weber. Matroid bounds on the region of entropic vectors. In 51th Annual Allerton Conference on Communication, Control and Computing, October 2013.