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

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.

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

Cyclic networks In certain settings, cycles are necessary

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

Quiz Sometimes, if the network has cycles

Quiz These networks are not really cyclic:

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

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

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

Convolution Codes Idea: mix messages from different rounds

Convolution Codes Recovery at terminal t1

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

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

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

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

Receiver node

Example 1

Example 2

Transfer matrix

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)

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

Transfer matrix Contribution of paths of length zero:

Contribution of paths of length one

Contribution of paths of length two

Transfer matrix

Transfer matrix

Transfer matrix

Transfer matrix

Transfer matrix

Transfer matrix

Linear network system

Example 3 Point-to-point connections

Example 3

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

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

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

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

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

Recap (from Wikipedia) ( )

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].