A Short Introduction to Index Coding with Side Information
Dau Son [email protected]
SPMS, Nanyang Technological University
A Short Introduction to Index Coding with Side Information
Basic Definitions
S
R1 R2 R3
Has
Requests
x2 x3 x1, x2
x1 x2 x3
x1
x2x3
A Short Introduction to Index Coding with Side Information
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
A Short Introduction to Index Coding with Side Information
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
1
32
Digraph of Side Information D
A Short Introduction to Index Coding with Side Information
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
1
32
Digraph of Side Information D
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)
A Short Introduction to Index Coding with Side Information
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
1
32
Digraph of Side Information D
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)
minrkq(D) = min{
rankq(M) : M fits D}
A Short Introduction to Index Coding with Side Information
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
1
32
Digraph of Side Information D
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)
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)
A Short Introduction to Index Coding with Side Information
Bar-Yossef et al.’ Conjecture
Definition (Capacity)
A Short Introduction to Index Coding with Side Information
Bar-Yossef et al.’ Conjecture
Definition (Capacity)
a (t, k)-IC: broadcast k q-ary symbols; xi ∈ Ftq
A Short Introduction to Index Coding with Side Information
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
A Short Introduction to Index Coding with Side Information
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}
A Short Introduction to Index Coding with Side Information
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}
A Short Introduction to Index Coding with Side Information
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}
A Short Introduction to Index Coding with Side Information
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
A Short Introduction to Index Coding with Side Information
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
A Short Introduction to Index Coding with Side Information
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
minrk2(Gn) ≥ n1−ε
A Short Introduction to Index Coding with Side Information
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
minrk2(Gn) ≥ n1−ε
minrkq(Gn) ≤ nε
A Short Introduction to Index Coding with Side Information
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
minrk2(Gn) ≥ n1−ε
minrkq(Gn) ≤ nε
Hn = Gn ∪ Gn
minrkq(Hn) ≥ 2√n, ∀q
A Short Introduction to Index Coding with Side Information
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
minrk2(Gn) ≥ n1−ε
minrkq(Gn) ≤ nε
Hn = Gn ∪ Gn
minrkq(Hn) ≥ 2√n, ∀q
minrk2(Gn) + minrk3(Gn) ≤ nε
A Short Introduction to Index Coding with Side Information
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
minrk2(Gn) ≥ n1−ε
minrkq(Gn) ≤ nε
Hn = Gn ∪ Gn
minrkq(Hn) ≥ 2√n, ∀q
minrk2(Gn) + minrk3(Gn) ≤ nε
Alon et al. (2008): block nonlinear ICsoutperform scalar nonlinear ICs(C(D) >> Cs(D))
A Short Introduction to Index Coding with Side Information
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
minrk2(Gn) ≥ n1−ε
minrkq(Gn) ≤ nε
Hn = Gn ∪ Gn
minrkq(Hn) ≥ 2√n, ∀q
minrk2(Gn) + minrk3(Gn) ≤ nε
Alon et al. (2008): block nonlinear ICsoutperform scalar nonlinear ICs(C(D) >> Cs(D))
An IC instance ↪→ Confusion graph C
Best rate = 1/χ(C)
A Short Introduction to Index Coding with Side Information
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
minrk2(Gn) ≥ n1−ε
minrkq(Gn) ≤ nε
Hn = Gn ∪ Gn
minrkq(Hn) ≥ 2√n, ∀q
minrk2(Gn) + minrk3(Gn) ≤ nε
Alon et al. (2008): block nonlinear ICsoutperform scalar nonlinear ICs(C(D) >> Cs(D))
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
A Short Introduction to Index Coding with Side Information
Index Coding, Network Coding, and Matroid
A Short Introduction to Index Coding with Side Information
Index Coding, Network Coding, and Matroid
Index Coding is a special case ofNetwork Coding (NC)
A Short Introduction to Index Coding with Side Information
Index Coding, Network Coding, and Matroid
Index Coding is a special case ofNetwork Coding (NC)
1
32
Digraph of Side Information D
A Short Introduction to Index Coding with Side Information
Index Coding, Network Coding, and Matroid
Index Coding is a special case ofNetwork Coding (NC)
1
32
Digraph of Side Information D
x1 x2 x3
x1 x2 x3
k
b b b
b b b
Corresponding Network Coding instance
A Short Introduction to Index Coding with Side Information
Index Coding, Network Coding, and Matroid
Index Coding is a special case ofNetwork Coding (NC)
1
32
Digraph of Side Information D
x1 x2 x3
x1 x2 x3
k
b b b
b b b
Corresponding Network Coding instance
NC can be reduced to IC
A Short Introduction to Index Coding with Side Information
Index Coding, Network Coding, and Matroid
Index Coding is a special case ofNetwork Coding (NC)
1
32
Digraph of Side Information D
x1 x2 x3
x1 x2 x3
k
b b b
b b b
Corresponding Network Coding instance
NC can be reduced to IC
NC instance ↪→ IC instance
∃ a linear block NC ⇐⇒ ∃ a linear block IC
∃ a nonlinear block NC =⇒ ∃ a nonlinear block IC
Index Coding, Network Coding, and Matroid
Index Coding is a special case ofNetwork Coding (NC)
1
32
Digraph of Side Information D
x1 x2 x3
x1 x2 x3
k
b b b
b b b
Corresponding Network Coding instance
NC can be reduced to IC
NC instance ↪→ IC instance
∃ a linear block NC ⇐⇒ ∃ a linear block IC
∃ a nonlinear block NC =⇒ ∃ a nonlinear block IC
↓
nonlinear vs. linear results in NC
⇓
nonlinear vs. linear results in IC
A Short Introduction to Index Coding with Side Information
Index Coding, Network Coding, and Matroid
Index Coding is a special case ofNetwork Coding (NC)
1
32
Digraph of Side Information D
x1 x2 x3
x1 x2 x3
k
b b b
b b b
Corresponding Network Coding instance
NC can be reduced to IC
NC instance ↪→ IC instance
∃ a linear block NC ⇐⇒ ∃ a linear block IC
∃ a nonlinear block NC =⇒ ∃ a nonlinear block IC
↓
nonlinear vs. linear results in NC
⇓
nonlinear vs. linear results in IC
Matroid can be reduced to IC
A Short Introduction to Index Coding with Side Information
Index Coding, Network Coding, and Matroid
Index Coding is a special case ofNetwork Coding (NC)
1
32
Digraph of Side Information D
x1 x2 x3
x1 x2 x3
k
b b b
b b b
Corresponding Network Coding instance
NC can be reduced to IC
NC instance ↪→ IC instance
∃ a linear block NC ⇐⇒ ∃ a linear block IC
∃ a nonlinear block NC =⇒ ∃ a nonlinear block IC
↓
nonlinear vs. linear results in NC
⇓
nonlinear vs. linear results in IC
Matroid can be reduced to IC
matroid ↪→ IC instance
∃ a t-representation ⇐⇒ ∃ a linear t-block IC
Index Coding, Network Coding, and Matroid
Index Coding is a special case ofNetwork Coding (NC)
1
32
Digraph of Side Information D
x1 x2 x3
x1 x2 x3
k
b b b
b b b
Corresponding Network Coding instance
NC can be reduced to IC
NC instance ↪→ IC instance
∃ a linear block NC ⇐⇒ ∃ a linear block IC
∃ a nonlinear block NC =⇒ ∃ a nonlinear block IC
↓
nonlinear vs. linear results in NC
⇓
nonlinear vs. linear results in IC
Matroid can be reduced to IC
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
A Short Introduction to Index Coding with Side Information
Capacity: Hardness and Bounds
Hardness
A Short Introduction to Index Coding with Side Information
Capacity: Hardness and Bounds
Hardness
1 Csl : hard to compute
A Short Introduction to Index Coding with Side Information
Capacity: Hardness and Bounds
Hardness
1 Csl : hard to compute(Peeters 1996): Csl (G) = 1/3? (or minrkq(G) = 3?) is NP-complete
A Short Introduction to Index Coding with Side Information
Capacity: Hardness and Bounds
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
A Short Introduction to Index Coding with Side Information
Capacity: Hardness and Bounds
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
A Short Introduction to Index Coding with Side Information
Capacity: Hardness and Bounds
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
2 Cs and C : the hardness is not known
A Short Introduction to Index Coding with Side Information
Capacity: Hardness and Bounds
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
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)
A Short Introduction to Index Coding with Side Information
Capacity: Hardness and Bounds
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
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)
Bounds
A Short Introduction to Index Coding with Side Information
Capacity: Hardness and Bounds
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
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)
Bounds
1 α(G) ≤ 1Csl (G)
≤ χ(G): χ - chromatic number, α(G) - independent number
A Short Introduction to Index Coding with Side Information
Capacity: Hardness and Bounds
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
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)
Bounds
1 α(G) ≤ 1Csl (G)
≤ χ(G): χ - chromatic number, α(G) - independent number
2 α(G) ≤ 1Csl (G)
≤ n − µ(G): µ(G) - maximum size of a matching
A Short Introduction to Index Coding with Side Information
Capacity: Hardness and Bounds
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
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)
Bounds
1 α(G) ≤ 1Csl (G)
≤ χ(G): χ - chromatic number, α(G) - independent number
2 α(G) ≤ 1Csl (G)
≤ n − µ(G): µ(G) - maximum size of a matching
3 α(D) ≤ 1Csl (D)
≤ cc(D): α(D) - size of maximum acyclic induced subgraph,
cc(D) - clique cover number
A Short Introduction to Index Coding with Side Information
Capacity: Hardness and Bounds
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
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)
Bounds
1 α(G) ≤ 1Csl (G)
≤ χ(G): χ - chromatic number, α(G) - independent number
2 α(G) ≤ 1Csl (G)
≤ n − µ(G): µ(G) - maximum size of a matching
3 α(D) ≤ 1Csl (D)
≤ cc(D): α(D) - size of maximum acyclic induced subgraph,
cc(D) - clique cover number
4 α(D) ≤ 1Csl (D)
≤ n − ν(D): ν(D) - maximum number of disjoint circuits
A Short Introduction to Index Coding with Side Information
Capacity: Hardness and Bounds
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
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)
Bounds
1 α(G) ≤ 1Csl (G)
≤ χ(G): χ - chromatic number, α(G) - independent number
2 α(G) ≤ 1Csl (G)
≤ n − µ(G): µ(G) - maximum size of a matching
3 α(D) ≤ 1Csl (D)
≤ cc(D): α(D) - size of maximum acyclic induced subgraph,
cc(D) - clique cover number
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
A Short Introduction to Index Coding with Side Information
(Di)Graphs with Easy-to-Compute Capacities
A Short Introduction to Index Coding with Side Information
(Di)Graphs with Easy-to-Compute Capacities
Scalar Linear Capacity Csl
A Short Introduction to Index Coding with Side Information
(Di)Graphs with Easy-to-Compute Capacities
Scalar Linear Capacity Csl
(Haemers 1978): perfect graphs (1/α(G))
A Short Introduction to Index Coding with Side Information
(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)
A Short Introduction to Index Coding with Side Information
(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)
(Berliner, Langberg 2011): outerplanar graphs (dynamic programming)
A Short Introduction to Index Coding with Side Information
(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)
(Berliner, Langberg 2011): outerplanar graphs (dynamic programming)
(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)
A Short Introduction to Index Coding with Side Information
(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)
(Berliner, Langberg 2011): outerplanar graphs (dynamic programming)
(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)
Scalar Capacity Cs and General Capacity C
A Short Introduction to Index Coding with Side Information
(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)
(Berliner, Langberg 2011): outerplanar graphs (dynamic programming)
(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)
Scalar Capacity Cs and General Capacity C
(Bar-Yossef et al. 2006): Cs = Csl for perfect graphs, odd cycles andcomplements, acyclic digraphs
A Short Introduction to Index Coding with Side Information
(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)
(Berliner, Langberg 2011): outerplanar graphs (dynamic programming)
(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)
Scalar Capacity Cs and General Capacity C
(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)
A Short Introduction to Index Coding with Side Information
Error Correction for Index Coding
A Short Introduction to Index Coding with Side Information
Error Correction for Index Coding
xencoding
y = xLClassical ECC:noisy
channely + ε
decodingx
A Short Introduction to Index Coding with Side Information
Error Correction for Index Coding
xencoding
y = xLClassical ECC:noisy
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.
A Short Introduction to Index Coding with Side Information
Error Correction for Index Coding
xencoding
y = xLClassical ECC:noisy
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.
(Dau, Skachek, Chee, 2011): scalar linear error-correcting index code (ECIC)
optimal IC + optimal ECC 6= optimal ECIC for small alphabets
A Short Introduction to Index Coding with Side Information
Error Correction for Index Coding
xencoding
y = xLClassical ECC:noisy
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.
(Dau, Skachek, Chee, 2011): scalar linear error-correcting index code (ECIC)
optimal IC + optimal ECC 6= optimal ECIC for small alphabets
optimal IC + optimal ECC = optimal ECIC for large alphabets
A Short Introduction to Index Coding with Side Information
Error Correction for Index Coding
xencoding
y = xLClassical ECC:noisy
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.
(Dau, Skachek, Chee, 2011): scalar linear error-correcting index code (ECIC)
optimal IC + optimal ECC 6= optimal ECIC for small alphabets
optimal IC + optimal ECC = optimal ECIC for large alphabets
develop bounds and constructions for length of an optimal ECIC
A Short Introduction to Index Coding with Side Information
Error Correction for Index Coding
xencoding
y = xLClassical ECC:noisy
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.
(Dau, Skachek, Chee, 2011): scalar linear error-correcting index code (ECIC)
optimal IC + optimal ECC 6= optimal ECIC for small alphabets
optimal IC + optimal ECC = optimal ECIC for large alphabets
develop bounds and constructions for length of an optimal ECIC
syndrome decoding
A Short Introduction to Index Coding with Side Information
Security for Index Coding
A Short Introduction to Index Coding with Side Information
Security for Index Coding
A scalar linear IC: x 7→ xL, where L is an n × k matrix; Rate: 1/k
A Short Introduction to Index Coding with Side Information
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
A Short Introduction to Index Coding with Side Information
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
Construction
(Dau, Skachek, Chee, 2011):
xindex coding
−→ xL(0)coset codinga
−→ (xL(0)|g)M
A Short Introduction to Index Coding with Side Information
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
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
A Short Introduction to Index Coding with Side Information
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