AShortIntroductiontoIndexCodingwithSideInformation Dau Son ...ccrg/documents/...Coding.pdf · A...

A Short Introduction to Index Coding with Side Information

Dau Son [email protected]

SPMS, Nanyang Technological University


Basic Definitions

S

R1 R2 R3

Has

Requests

x2 x3 x1, x2

x1 x2 x3

x1

x2x3


Basic Definitions

S

R1 R2 R3

Has

Requests

x2 x3 x1, x2

x1 x2 x3

x1

x2x3

S broadcasts (2 transmissions):

{

x1 + x2

x2 + x3

Trivial solution: 3 transmissions


Basic Definitions

S

R1 R2 R3

Has

Requests

x2 x3 x1, x2

x1 x2 x3

x1

x2x3


{

x1 + x2

x2 + x3


1

32

Digraph of Side Information D


Basic Definitions

S

R1 R2 R3

Has

Requests

x2 x3 x1, x2

x1 x2 x3

x1

x2x3


{

x1 + x2

x2 + x3


1

32


M =

1 1 00 1 11 0 1

has rank two. An n × n matrix Mfits a digraph D of order n if

mi,j =

{

1, j = i

0, (i , j) /∈ E(D)


Basic Definitions

S

R1 R2 R3

Has

Requests

x2 x3 x1, x2

x1 x2 x3

x1

x2x3


{

x1 + x2

x2 + x3


1

32


M =

1 1 00 1 11 0 1


mi,j =

{

1, j = i

0, (i , j) /∈ E(D)

minrkq(D) = min{

rankq(M) : M fits D}


Basic Definitions

S

R1 R2 R3

Has

Requests

x2 x3 x1, x2

x1 x2 x3

x1

x2x3


{

x1 + x2

x2 + x3


1

32


M =

1 1 00 1 11 0 1


mi,j =

{

1, j = i

0, (i , j) /∈ E(D)

minrkq(D) = min{

rankq(M) : M fits D}

Bar-Yossef et al. (2006): 1/minrk2(D) is the best rate for scalar linear index codes(IC)


Bar-Yossef et al.’ Conjecture

Definition (Capacity)




a (t, k)-IC: broadcast k q-ary symbols; xi ∈ Ftq





a (t, k)-IC: t/k is the rate






general capacity: C(D) = sup{t/k : ∃ a (t, k)-IC}







scalar capacity: Cs(D) = sup{1/k : ∃ a (1, k)-IC}








scalar linear capacity: Csl (D) = sup{1/k : ∃ a linear (1, k)-IC}









Bar-Yossef et al. (2006) conjectured: Csl (D) (= 1/minrk2(D)) is the best rate










Lubetzky and Stav (2007): q-ary ICsoutperform binary ICs, mixed-field ICoutperform q-ary ICs (Cs(D) >> Csl (D))

Gn











Gn

minrk2(Gn) ≥ n1−ε











Gn


minrkq(Gn) ≤ nε











Gn


minrkq(Gn) ≤ nε

Hn = Gn ∪ Gn

minrkq(Hn) ≥ 2√n, ∀q











Gn


minrkq(Gn) ≤ nε

Hn = Gn ∪ Gn


minrk2(Gn) + minrk3(Gn) ≤ nε











Gn


minrkq(Gn) ≤ nε

Hn = Gn ∪ Gn



Alon et al. (2008): block nonlinear ICsoutperform scalar nonlinear ICs(C(D) >> Cs(D))











Gn


minrkq(Gn) ≤ nε

Hn = Gn ∪ Gn




An IC instance ↪→ Confusion graph C

Best rate = 1/χ(C)











Gn


minrkq(Gn) ≤ nε

Hn = Gn ∪ Gn




An IC instance ↪→ Confusion graph C

Best rate = 1/χ(C)

El Rouayheb et al. (2008): block nonlinearICs outperform block linear ICs; blocklinear ICs outperform scalar linear ICs


Index Coding, Network Coding, and Matroid



Index Coding is a special case ofNetwork Coding (NC)




1

32





1

32


x1 x2 x3

x1 x2 x3

k

b b b

b b b

Corresponding Network Coding instance




1

32


x1 x2 x3

x1 x2 x3

k

b b b

b b b


NC can be reduced to IC




1

32


x1 x2 x3

x1 x2 x3

k

b b b

b b b



NC instance ↪→ IC instance

∃ a linear block NC ⇐⇒ ∃ a linear block IC

∃ a nonlinear block NC =⇒ ∃ a nonlinear block IC



1

32


x1 x2 x3

x1 x2 x3

k

b b b

b b b






↓

nonlinear vs. linear results in NC

⇓

nonlinear vs. linear results in IC




1

32


x1 x2 x3

x1 x2 x3

k

b b b

b b b






↓


⇓


Matroid can be reduced to IC




1

32


x1 x2 x3

x1 x2 x3

k

b b b

b b b






↓


⇓



matroid ↪→ IC instance

∃ a t-representation ⇐⇒ ∃ a linear t-block IC



1

32


x1 x2 x3

x1 x2 x3

k

b b b

b b b






↓


⇓



matroid ↪→ IC instance

∃ a t-representation ⇐⇒ ∃ a linear t-block IC

↓

matroid: 2-rep. but not 1-rep.

⇓

linear 2-block IC with better rate than linear scalar IC


Capacity: Hardness and Bounds

Hardness



Hardness

1 Csl : hard to compute



Hardness

1 Csl : hard to compute(Peeters 1996): Csl (G) = 1/3? (or minrkq(G) = 3?) is NP-complete



Hardness

1 Csl : hard to compute(Peeters 1996): Csl (G) = 1/3? (or minrkq(G) = 3?) is NP-complete(Dau, Skachek, Chee, 2011): Csl (D) = 1/2? (or minrkq(D) = 2?) is NP-complete



Hardness

1 Csl : hard to compute(Peeters 1996): Csl (G) = 1/3? (or minrkq(G) = 3?) is NP-complete(Dau, Skachek, Chee, 2011): Csl (D) = 1/2? (or minrkq(D) = 2?) is NP-complete(Langberg, Sprintson, 2008): finding an IC with rate > αCsl (D) (α ∈ (0, 1]) isNP-hard



Hardness


2 Cs and C : the hardness is not known



Hardness


2 Cs and C : the hardness is not known(Blasiak et al. 2011): polynomial time algo. to approx. C with ratio O( log n

n log log n)



Hardness



n log log n)

Bounds



Hardness



n log log n)

Bounds

1 α(G) ≤ 1Csl (G)

≤ χ(G): χ - chromatic number, α(G) - independent number



Hardness



n log log n)

Bounds

1 α(G) ≤ 1Csl (G)


2 α(G) ≤ 1Csl (G)

≤ n − µ(G): µ(G) - maximum size of a matching



Hardness



n log log n)

Bounds

1 α(G) ≤ 1Csl (G)


2 α(G) ≤ 1Csl (G)


3 α(D) ≤ 1Csl (D)

≤ cc(D): α(D) - size of maximum acyclic induced subgraph,

cc(D) - clique cover number



Hardness



n log log n)

Bounds

1 α(G) ≤ 1Csl (G)


2 α(G) ≤ 1Csl (G)


3 α(D) ≤ 1Csl (D)



4 α(D) ≤ 1Csl (D)

≤ n − ν(D): ν(D) - maximum number of disjoint circuits



Hardness



n log log n)

Bounds

1 α(G) ≤ 1Csl (G)


2 α(G) ≤ 1Csl (G)


3 α(D) ≤ 1Csl (D)



4 α(D) ≤ 1Csl (D)

≤ n − ν(D): ν(D) - maximum number of disjoint circuits

5 α(G) ≤ b2 ≤ 1C(G)

≤ bn = χf (G): b2, bn - solutions of linear programs


(Di)Graphs with Easy-to-Compute Capacities



Scalar Linear Capacity Csl




(Haemers 1978): perfect graphs (1/α(G))





(Bar-Yossef et al. 2006): odd cycles ( 1bn/2c

) and complements (3), acyclic

digraphs (orders)







digraphs (orders)

(Berliner, Langberg 2011): outerplanar graphs (dynamic programming)







digraphs (orders)


(Dau, Skachek, Chee 2011): Koenig-Egervary graphs (1/α(G)), connectivelyreducible digraphs, line digraphs of partially planar digraphs ( 1

n−ν(D)), graphs

having tree structure of Type I (dynamic programming)







digraphs (orders)



n−ν(D)), graphs


Scalar Capacity Cs and General Capacity C







digraphs (orders)



n−ν(D)), graphs



(Bar-Yossef et al. 2006): Cs = Csl for perfect graphs, odd cycles andcomplements, acyclic digraphs







digraphs (orders)



n−ν(D)), graphs



(Bar-Yossef et al. 2006): Cs = Csl for perfect graphs, odd cycles andcomplements, acyclic digraphs

(Blasiak et al. 2011): n-cycles (C = 2/n), complements of n-cycles (C =bn/2c

n),

3-regular Cayley graphs of Zn (C = 2/n)


Error Correction for Index Coding



xencoding

y = xLClassical ECC:noisy

channely + ε

decodingx



xencoding


channely + ε

decodingx

xencoding

y = xLECIC:

decodingx1y + ε1

decodingx2y + ε2

b

b

b

decodingxny + εn

side info.

b

b

b

b

b

b

noisybroadcast channel

side info.

side info.



xencoding


channely + ε

decodingx

xencoding

y = xLECIC:

decodingx1y + ε1

decodingx2y + ε2

b

b

b

decodingxny + εn

side info.

b

b

b

b

b

b


side info.

side info.

(Dau, Skachek, Chee, 2011): scalar linear error-correcting index code (ECIC)

optimal IC + optimal ECC 6= optimal ECIC for small alphabets



xencoding


channely + ε

decodingx

xencoding

y = xLECIC:

decodingx1y + ε1

decodingx2y + ε2

b

b

b

decodingxny + εn

side info.

b

b

b

b

b

b


side info.

side info.



optimal IC + optimal ECC = optimal ECIC for large alphabets



xencoding


channely + ε

decodingx

xencoding

y = xLECIC:

decodingx1y + ε1

decodingx2y + ε2

b

b

b

decodingxny + εn

side info.

b

b

b

b

b

b


side info.

side info.




develop bounds and constructions for length of an optimal ECIC



xencoding


channely + ε

decodingx

xencoding

y = xLECIC:

decodingx1y + ε1

decodingx2y + ε2

b

b

b

decodingxny + εn

side info.

b

b

b

b

b

b


side info.

side info.




develop bounds and constructions for length of an optimal ECIC

syndrome decoding


Security for Index Coding



A scalar linear IC: x 7→ xL, where L is an n × k matrix; Rate: 1/k




Setting

S

R1

R2

Rn

xLbbb

Adversary A

owns t xj ’slistens to µ transmissions

introduces δ errors

GOAL:Ri can decode xi

adversary has no information about other xj ’s




Setting

S

R1

R2

Rn

xLbbb

Adversary A





Construction

(Dau, Skachek, Chee, 2011):

xindex coding

−→ xL(0)coset codinga

−→ (xL(0)|g)M




Setting

S

R1

R2

Rn

xLbbb

Adversary A





Construction

(Dau, Skachek, Chee, 2011):

xindex coding

−→ xL(0)coset codinga

−→ (xL(0)|g)M

G.M. of

G.M. of an MDS code µ

n

minrk + µ+ 2δ

M =an MDS code

aintroduced by Ozarow and Wyner in “Wiretap Channel II,” 1985


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