1. NETWORK CODING

2. Routing is concerned with establishing end to end paths between sources and sinks of information. In existing networks each node in a given route acts as a switch.

3. Coding at node is called network coding. Does network coding improve the throughput of the network?

4. Theorem 1:Min-Cut Max-Flow The maximum amount of flow is equal to the capacity of a minimum cut.

5. Network coding increases throughput

6. Theoretically it was proved by Ahlswede et.al. that there exists a network code which achieves the rate region as given by max-flow min-cut. What is the structure of the code that achieves this optimality? Why linear network coding? Under what conditions is a linear network coding problem solvable?

7. Linear Network coding was studied using block codes by Li and Yeung. Conditions for the solvability of a multi-cast network and framework for solving the linear- network problem are addressed by Koetter and Medard.

8. Problem Formulation:

9. Definitions: Let χ(v) = {X(v,1),X(v,2),…..X(v,n(v)} Connection: c is a triple (v,v’,χ(v,v’)) є V X V X P χ(v) Rate: Rate of the connection c is defined as R(c) = where H(X) is the entropy rate of random process X

10. ALGEBRAIC FORMULATION Definition: Let G = (V,E) be a delay-free communication network. We say that G is a - linear network, if for all links, the random processes Y(e) and Z(v,j) are given

11. An Example :

13. Solving for relation between x and z, we get z = xM where x is the vector of input processes, z is the vector of output processes. and M is the transfer matrix which can be split as follows, M = AГB T

14. where

15. What do we have to guarantee for the given network connections to be feasible? Ans: det(M) 0,so that x=z*inv(M),which in turn requires det(Г) 0. One possible solution to the example is,choose and all other β’s as 0.

16. Lemma 1 :Let Ғ [X1,X2,..., Xn] be the ring of polynomials over an infinite field in variables X1,X2,..., Xn. For any non-zero element f ∈ Ғ [X1,X2,..., Xn] there exists an infinite set of n-tuples (x1, x2,..., xn) ∈ Ғ n such that f(x1, x2,..., xn) 0. Hence for each assignment to the parameters β e’,e which leads to det(M) 0,gives a solution to the problem and thus infinite number of solutions are possible.

17. Theorem 2 :Let a linear network be given. The following three statements are equivalent: 1) A point-to-point connection c = (v, v’,χ(v, v’)) is possible. 2) The MIN-CUT MAX-FLOW bound is satisfied for a rate R(c). 3) The determinant of the R(c)×R(c) transfer matrix M is nonzero over the ring Ғ 2 [..., α e,l,..., β e’,e,..., ε e’,j,...].

18. Proof: Equivalence between 1 and 3 The Ford-Fulkerson algorithm implies that a solution to the linear network coding problem exists. Choosing this solution for the parameters of the linear network coding problem yields a solution such that M is the identity matrix and hence the determinant of M over Ғ [..., α e,l,..., β e’,e..., ε e’,j,...] does not vanish identically.

19. Conversely, if the determinant of M is nonzero over Ғ 2 [..., α e,l,..., β e’,e,..., ε e’,j,...] we can invert matrix M by choosing parameters ε e,l accordingly. From Lemma 1 we know that we can choose the parameters as to make this determinant non-zero. Conclusion :Studying the feasibility of connections in a linear network scenario is equivalent to studying the properties of solutions to polynomial equations over the field, called algebraic varieties.

20. Directed labeled line graph (DLLG):

21. Adjacency Matrix: F = Definition:

22. Lemma 2: Let F be the adjacency matrix of the labeled line graph of a cycle-free network G. The matrix I − F has a polynomial inverse in Ғ 2 [..., β e’,e,...]. Proof: F is a strict upper-triangular matrix provided original graph is acyclic and the vertices in DLLG are arranged according to ancestral ordering.

23. Theorem 3 :Let a network be given with matrices A, B and F. The transfer matrix of the network is M = A(I − F) −1 B T where I is the |E| × |E| identity matrix. Proof: The impulse response between an input random process x and an output random process z is obtained by adding all gains along all paths that the x can take to contribute to z which is given by I+F+F 2 +…+F N = (I-F) -1

24. Multicast Networks Theorem 4: Let a delay-free network G and a set of desired connections С = {(v,u 1,χ(v)) …, (v,u N,χ(v))} be given. The network problem (G,C) is solvable if and only if the MIN-CUT MAX-FLOW bound is satisfied for all connections in C.

25. Proof: o Transfer matrix M with dimension |χ(v)| x N|χ(v)| o Each submatrix |χ(v)| x |χ(v)| has non-zero determinant over Ғ 2 [ξ] o Product of N determinant is non-zero polynomial in Ғ 2 [ξ] o Lemma 1 We can find assignment for ξ all N determinant are non-zero in o Choose B so that M is the N-fold multiplication of |χ(v)| x |χ(v)| matrix.

26. Thus we must find a point that does not lie on the algebraic variety cut out by this polynomial in ξ Algorithm to find a such that F(a) ≠ 0 1.Find the maximal degree δ of F in any variable ξ j and let i be the smallest number such that 2 i > δ. 2.Find an element a t in Ғ 2 i such that F(ξ)| ξt=at ≠ 0 and let F ← F(ξ)| ξt=at 3.If t = n then halt, else t ← t + 1, goto 2. Output: (a 1, a 2, …, a n )

27. What should be the upper bound on the degree of extension field ? Corollary 1: –Let a delay-free communication network G and a solvable multicast network problem be given with one source and N receivers. Let R be the rate at which the source generates information. There exists a solution to the network coding problem in a finite field F 2 m with m ≤ log 2 (NR + 1)

28. Why Algorithm 1 will work and is sufficient ? Theorem 5: Let F be the product of the determinants of the transfer matrices for the individual connections and let δ be the maximal degree of F with respect to any variable ξ i. There exists a solution to the multicast network problem in F 2 i, where i is the smallest number such that 2 i > δ. Algorithm 1 finds such a solution.

29. Proof: Consider F as a polynomial in ξ 2,ξ 3 …ξ n such as, F = ξ 1 ξ 2 ξ 6 … ξ n + … +ξ 2 1 + ξ 1 ξ n +ξ 1 with coefficients from Ғ 2 [ξ 1 ]. Consider polynomial in ξ 1,F l = (ξ 2i 1 - ξ 1 ) Then all the non-zero roots of F l forms the field elements of Ғ 2 i Since δ < 2 i polynomial from ring Ғ 2 [ξ 1 ] are not divisible by F l Hence there exist a 1 ∈ Ғ 2 i atleast one of the coefficient evaluates to nonzero element of Ғ 2 i

30. General Network We have to ensure that there is no disturbing interference from other connections.

31. From Theorem 2, the network problem is solvable only if MIN-CUT MAX-FLOW bound is satisfied i.e, the determinant of M 1,1 and M 2,1 is non-zero over Ғ 2 [ξ]. But this network is not solvable. Why ?

32. Generalized MIN CUT-MAX FLOW Condition Theorem 6: L et M = {M ij } be the corresponding transfer matrix relating the set of input nodes to the set of output nodes. The network problem (G,C) is solvable if and only if there exists an assignment of numbers to ξ such that, M i,j = 0 for all pairs (v i, v j ) of vertices such that (v i, v j,X (v i, v j )) C If C contains the connections (v i1,v j, X (v i1, v j )), (v i2, v j,X (v i2, v j )),…, (v il, v j,X (v il, v j )) the submatrix [M T i1,j M T i2,j,...., M T il,j ] is a non singular ν(v j ) × ν(v j ) matrix.

33. Conversely, if 1) is not satisfied, there is an interference which cannot be distinguish by v j Checking this two condition can be a tedious task as we have to find an assignment ξ Can we resolve solvability issue without necessarily finding a solution ? Theory of Grobner bases provides a structured approach which reveals the solvability of the network problem.

34. Robust Networks Can Network Coding be used to protect against link failures in networks ? Can we find a static solution ξ for classes of failure patterns ? Why static solution ? Under which failure pattern a successful network usage is still guaranteed ?

35. Notation: Let e = (u,v) denote the failing link. Let constant 0 be observed on failing link Set β e’,e, β e,e’’ and α e,l to zero for all e’, e’’ and l. Let this set of parameters be denoted as B e = {ξ i : ξ i is identified with β e’,e, β e,e’’ & α e,l } Let M[ξ] be the system matrix for a particular linear network coding problem For particular failure pattern f define, B(f)=

36. Lemma 3: Let M[ξ] be the system matrix of a linear network coding problem and let f be a particular link failure pattern. The system matrix M f [ξ] for the network G f obtained by deleting the failing links is obtained by replacing all ξ i ∈ B(f) with zero, i.e. M f [ξ] = M[ξ]| ξi=0:ξi ∈ B(f)

37. Theorem 7: Let a linear network G and a set of connections C = {(v, u 1,X (v)), (v, u 2,X (v)),..., (v, u N,X (v))} be given. There exists a common static solution to the network problems (G f,C ) for all f in F Proof: Consider the product, Follows from Lemma 1. But what is the trade-off ?

38. Theorem 8: Let a delay-free communication network G and a solvable multicast network problem be given with one source and N receivers. Moreover, let F be the set of failure patterns from which we want to recover. Let R be the rate at which the source generates information. There exists a solution to the network coding problem in a finite field F 2 m with m ≤log 2 (|F|NR + 1).