1. Network Coding and its Applications in Communication Networks
Alex Sprintson
Computer Engineering Group
Department of Electrical and Computer Engineering
Texas A&M University

2. NP-hardness result
Lemma (Rasala et. al. 2003). Deciding whether there exists a linear network code with alphabet size q for a multicast network coding instance is NP-hard when q is a prime power.

3. NP-hardness result qmin= least prime power > X(G)-1
Proof: Reduction from the chromatic number problem s y
x+y S1 S1 y x y s1 s2 s3 s4
x+0*y S2 S2 S3 S3
x+1*y T4 T5 S4 S4 T1 T2 T3
qmin= least prime power > X(G)-1

4. Cyclic networks
In certain settings, cycles are necessary

5. Quiz
Given a network (G,s,T) with minimum cut h, is it possible to transmit h packets to all T terminals if each link can only transmit one packet?
Always yes, if the network is acyclic

6. Quiz
Sometimes, if the network has cycles

7. Quiz
These networks are not really cyclic:

8. Quiz A example of a “truly” cyclic network
We prove: it is impossible to send two packets in one round to all destinations
Proof: By way of contradiction

9. Quiz
Let e be the first link of the cycle v1->v2->v3->v4->v1 that transmits a packet.
Case study. If e=(v1,v2) then t1 gets no information about b

10. Solution
We have shown that it is not possible to send two packets in one round to both terminals
However, it is possible to send 2n packets in n+1 rounds
Asymptotically, the rate is two packets per round
Use convolution codes

11. Convolution Codes
Idea: mix messages from different rounds

12. Convolution Codes
Recovery at terminal t1

13. Algebraic framework Ax=z aij,xi,ziGF(2n)
Observation: Any two rows of matrix A must be linearly independent

14. Algebraic framework (cont.)
A4x=z
Observation: Matrices A1, A2,…,Ak must be of full rank (invertible)

15. Notation Y(e) All operations in the network are linear
- the collection of () packets generated at node 
- Packets transmitted on link e
- Packets received by the destination node
Y(e)
All operations in the network are linear

16. Intermediate source At the intermediate nodes
All operations in the network are linear

17. Receiver node

18. Example 1

19. Example 2

20. Transfer matrix

21. Transfer matrix
Observation: The information propagates throughout networks on paths with different length.
Length of the path - the number of hops between the source and the tail of the edge.
For example, for edge e5 there are two paths - one through path (e2,e5) and one through path (e1,e3,e5)

22. Transfer matrix
Let be the vector of packets transmitted over all links
Let be the contribution (gain) of paths on length i:
Note that

23. Transfer matrix
Contribution of paths of length zero:

24. Contribution of paths of length one

25. Contribution of paths of length two

26. Transfer matrix

27. Transfer matrix

28. Transfer matrix

29. Transfer matrix

30. Transfer matrix

31. Transfer matrix

32. Linear network system

34. Example 3
Point-to-point connections

35. Example 3

36. Point-to-Point connections
Define matrices A, B, and M

37. Point-to-Point connections
Idea: Choose parameters in an extension field such that the determinant of
is nonzero over
Then, we can set A to be the identity matrix and B so that M is the identity matrix

38. Critical property The equation
Admits a choice of variables over so that the polynomial does not evaluate to zero!!

39. Point-to-Point connections
A possible solution:
Set
And set to be equal zero.
Equivalent to finding three disjoint paths between s and t!

40. An algebraic max flow min cut condition
[KM01, 02, 03]

41. Recap (from Wikipedia)
( )

42. Polynomial Ring
Let F be a field. The set of all polynomials with coefficients in the field F, together with the addition and the multiplication forms itself a ring, the polynomial ring over F, which is denoted by F[X].