Download - Exact Repair problems with multiple sources: CISS 2014

CISS 2014, Princeton NJ 1

Exact Repair Problems with Multiple Sources

Jayant Apte*, Congduan Li, John MacLaren Walsh, Steven Weber

ECE Dept. Drexel University


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


Outline

● Problem Definition● Computer assisted proofs: General structure● Polyhedral bounds on● Polyhedral computation interpretation of rate




(n,k,d) Exact Repair with multiple sources


2-source (3,2,2) exact repair problem



2 sources



3 encoding functions



3 storage random variables



3 decoders with different demands



6 repair encodingfunctions



3 repair decodingfunctions



Total 11 random variables


Implicit characterization of rate region(Yan et al.)


Outline





Motivation

SourcesDecoderDemands

SoftwareNetwork


Motivation

Software

SourcesDecoderDemands

NetworkRate Region

and optimal codes


Software for computer assisted proofs


Computer assisted converse



Inequalities obtained as an implication of linear Shannon-type,non-Shannon-type, non-linear non-Shannon type inequalities andnetwork constraints


Computer assisted achievability


Outline





● Closure of set of all 'entropic' vectors arising from N-variable probability distributions

3-D rendition of



● Each entropic vector is formed by stacking entropies of subsets of N random variables

3-D rendition of



● Each entropic vector is formed by stacking entropies of subsets of N random variables

● Cone:

3-D rendition of


● 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


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


Shannon Outer Bound


Outline



– (Representable) Matroid (Inner) bound(s)




(Representable) Matroid Inner bound(s)


(Representable) Matroid Inner Bound(s)


Outline



– Matroid (Inner) bound(s)

– Subspace (Inner) bounds




Subspace Inner Bound(s)


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


Outline



– Matroid (Inner) bound(s)

– Subspace (Inner) bounds




Network Coding constraints


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


Rate constraints

Storage Bandwidth


Rate constraints

Repair Bandwidth

Storage Bandwidth


A projection technique for computing achievable rate region


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

http://www.ece.drexel.edu/walsh/aspitrg/software.html


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)


Results

SoftwareNetwork


Rate region for H(S1)=1 and H(S2)=1


Rate region for H(S1)=1 and H(S2)=2


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.