1. Index Coding
Amin Gohari

2. The problem 𝑚 nodes who want to know 𝑚 files Side Information
𝑊 1 , 𝑊 2 , ⋯, 𝑊 𝑚
𝑊 𝑖 ∈ {0,1} 𝑛 𝑖
Side Information
𝑊 1
𝑊 2
𝑊 3
Definition of Index Coding and Side Information Graph. Here node 1 wants W_1, node two wants W_2, etc. Receiver 1 knows both W_2 and W_3. Receiver 2 knows W_3 and vice versa.

3. Example: Coding 1 𝑊 𝑖 ∈{0,1} 𝑊 1 𝑊 5 𝑊 2 𝑊 1 , 𝑊 2 , 𝑊 3 , 𝑊 4 , 𝑊 5
𝑊 1 , 𝑊 2 , 𝑊 3 , 𝑊 4 , 𝑊 5
𝑊 4
𝑊 3

4. Example: Coding 2 𝑊 𝑖 ∈{0,1} 𝑊 1 𝑊 5 𝑊 2 𝑊 1 𝑊 2 ⊕ 𝑊 3 𝑊 4 ⊕ 𝑊 5 𝑊 4

5. Example: Coding 3 𝑊 1 , 𝑊 1 ′ 𝑊 𝑖 , 𝑊 𝑖 ′ ∈{0,1} 𝑊 5 , 𝑊 5 ′ 𝑊 2 ,
𝑊 𝑖 , 𝑊 𝑖 ′ ∈{0,1}
𝑊 5 ,
𝑊 5 ′
𝑊 2 ,
𝑊 2 ′
𝑊 1 ⊕ 𝑊 2 ′
𝑊 2 ⊕ 𝑊 3 ′
𝑊 3 ⊕ 𝑊 4 ′
𝑊 4 ⊕ 𝑊 5 ′
𝑊 4 ,
𝑊 4 ′
𝑊 3 ,
𝑊 3 ′
𝑊 5 ⊕ 𝑊 1 ′

6. Index Coding Rate (Ex. 1) نرخ: 1 5 , 1 5 , 1 5 , 1 5 , 1 5
Node 1: a1, a2, a3, a4, a5, a6, … Node 2: b1, b2, b3, b4, b5, b6, … Node 3: c1, c2, c3, c4, c5, c6, … Node 4: d1, d2, d3, d4, d5, d6, … Node 5: e1, e2, e3, e4, e5, e6, … 5 5 5 5 5 5
نرخ:
1 5 , 1 5 , 1 5 , 1 5 , 1 5

7. Index Coding Rate (Ex. 2) نرخ: 1 3 , 1 3 , 1 3 , 1 3 , 1 3
Node 1: a1, a2, a3, a4, a5, a6, … Node 2: b1, b2, b3, b4, b5, b6, … Node 3: c1, c2, c3, c4, c5, c6, … Node 4: d1, d2, d3, d4, d5, d6, … Node 5: e1, e2, e3, e4, e5, e6, … 3 3 3 3 3 3
نرخ:
1 3 , 1 3 , 1 3 , 1 3 , 1 3

8. Index Coding Rate (Ex. 3) نرخ: 2 5 , 2 5 , 2 5 , 2 5 , 2 5
Node 1: a1, a2, a3, a4, a5, a6, … Node 2: b1, b2, b3, b4, b5, b6, … Node 3: c1, c2, c3, c4, c5, c6, … Node 4: d1, d2, d3, d4, d5, d6, … Node 5: e1, e2, e3, e4, e5, e6, … 5 5 5
نرخ:
2 5 , 2 5 , 2 5 , 2 5 , 2 5

9. Formal Definition of Rate
𝒲 𝑖 = {0,1} 𝑛 𝑖
Public Message: 𝑓: 𝒲 1 × 𝒲 2 ×⋯× 𝒲 𝑚 → {0,1} 𝑛
Rate 𝑟 = 𝑛 1 𝑛 , 𝑛 2 𝑛 , ⋯, 𝑛 𝑚 𝑛
Symmetric case:
𝑟 = 1/𝛽 1 , 1/𝛽 1 , ⋯, 1/𝛽 1 : fixed 𝑛 𝑖
𝑟 = 1/𝛽,1/𝛽, ⋯,1/𝛽 : limit when 𝑛 𝑖 , 𝑛→∞
Definition of Index Codes and their rates. Rate of a user is the ratio of the size of its message divided by the length of the communication; the higher the rate, the better it is. Basically we want to minimize N given message alphabets.
(Optional to mention: Zero-error and asymptotically vanishing errors give the same result.)

10. Types of Index Coding One Shot vs. Streaming Data (Asymptotic) 𝑊 1 𝑊 2
𝑊 3
Definition of the above classifications. One-shot: size of W_i’s are fixed. Asymptotic: size of W_i’s goes to infinity.
Linear: the sets W_i are strings on a field (for instance binary strings), and each symbol of the public message is a linear combination on elements of W_i.

11. Some bounds on 𝛽 Independence number and clique cover 𝑊 1 𝑊 2 𝑊 3
Definition of the above classifications. One-shot: size of W_i’s are fixed. Asymptotic: size of W_i’s goes to infinity.
Linear: the sets W_i are strings on a field (for instance binary strings), and each symbol of the public message is a linear combination on elements of W_i.

12. Some bounds on 𝛽: Clique covering number
Definition of the above classifications. One-shot: size of W_i’s are fixed. Asymptotic: size of W_i’s goes to infinity.
Linear: the sets W_i are strings on a field (for instance binary strings), and each symbol of the public message is a linear combination on elements of W_i.

13. Some bounds on 𝛽: Vector Assignment
Assign a vector to each node
Definition of the above classifications. One-shot: size of W_i’s are fixed. Asymptotic: size of W_i’s goes to infinity.
Linear: the sets W_i are strings on a field (for instance binary strings), and each symbol of the public message is a linear combination on elements of W_i.

14. Some bounds on 𝛽: MinRank

15. New Contributions

16. Critical graphs
Given a set of rates, we are looking for graphs with minimum number of edges that support this rate

17. Result 1: Symmetric Critical Graphs
The unique 𝑚-vertex graph that supports 𝑟 =(𝑟, 𝑟, ⋯, 𝑟) and contains minimum possible number of edges is the union of 1 𝑟 cliques of almost equal sizes.
Statement of Theorem 1 of the paper.
𝐺 1
𝐺 2
𝐺 1 𝑟

18. Result 1: Symmetric Critical Graphs
The unique 𝑚-vertex graph that supports 𝑟 =(𝑟, 𝑟, ⋯, 𝑟) and contains minimum possible number of edges is the union of 1 𝑟 cliques of almost equal sizes. Example. 𝑟= 1 3 , 𝑚=8
Statement of Theorem 1: An example
The key ingredient in the proof is a known result that if a set of nodes do not have a directed cycle, their rates should be less than or equal to one. The rest is simply graph theory.

19. Proof
Lemma. If a set of vertices do not contain a directed cycle, then the sum of the rates of those nodes should be less than or equal to one.

20. every subset of size greater than 1 𝑟 has a cycle
Proof
𝑟 =(𝑟, 𝑟, ⋯, 𝑟)
every subset of size greater than 1 𝑟 has a cycle

21. Result 2
If 𝐺 1 → 𝐺 2 , we can remove the intermediate edges without changing the capacity region in following scenarios:
𝐺 1
𝐺 2
G_1G_2 means that there are only edges from G_1 component to G_2.
Statement of theorem 2, part a of the paper.
In a setup like this, elimination of intermediate edges does not change the capacity region (for three cases our of 4 possible cases). For the non-linear one-shot, we have a counter example.
In other words, for any valid coding function like 𝑓 for 𝐺 1 → 𝐺 2 , there exist two coding functions 𝑓 1 and 𝑓 2 (for 𝐺 1 and 𝐺 2 respectively) whose concatenation is a valid coding function for 𝐺 1 ∪ 𝐺 2 and has equal rate to 𝑓.

22. A Property of Critical Graphs
Edges that do not lie in any cycles can be eliminated. So, Critical Graphs are Union of Strongly Connected Components. (USCS)
As a result, a critical graph will not have edges between its strongly connected components. Thus, critical graphs are USCS.

23. A Property of Critical Graphs
Edges that do not lie in any cycles can be eliminated. So, Critical Graphs are Union of Strongly Connected Components. (USCS)
As a result, a critical graph will not have edges between its strongly connected components. Thus, critical graphs are USCS.