1. Network Coding Project presentation Communication Theory 16:332:545 Amith Vikram Atin Kumar Jasvinder Singh Vinoo Ganesan

2. Outline  Introduction  Network coding concept  Literature Survey  Terminology and Notation  Study and Implementation  Solvability in Multicast Networks  Algorithm and Pseudo-code  Low Complexity Network Codes  Network Recovery and Management  Scope for future work

3. Network Coding Concept Goal Goal: To transfer data at the maximum achievable throughput in a network. Idea Idea: Process incoming data at nodes in the network Introduction V1 V2 V3 V4V5 b1b2 b1 V6 V7 b1 b2 b2 b1 b1 b2 b1+b2

4. Literature Survey Network Information Flow - Ahlswede, Cai, Li, Yeung, 2000 Characterized the admissible coding rate region for multicast networks coding’ Proved that maximum throughput in a network can be achieved using ‘coding’ Linear network Coding – Li, Yeung, Cai, 2003 Coding at nodes treated as linear transformation of incoming data Showed that individual maxflow bounds of each receiver can be achieved but over a time period of the LCM of the maxflow bounds Algebraic Approach – Koetter and Medard, 2002 Proposed algebraic framework to study networks and capacity Necessary and sufficient conditions for coding to be acheivable Necessary and sufficient conditions for robustness to link failures Network Management – Ho, Koetter and Medard,2002 Quantify Network Management information required to affect link failure recovery Low complexity Network Codes – Jaggi, Kamal Jain, Philip Chou,2003 Field size and thus arithmetic complexity is small; link usage is lower Introduction

5. Terminology and Notation Network denoted as a graph G=(V,E) V ----- Set of vertices (nodes) E ----- Set of Edges (line joining pairs of vertices) Input vector at source ’s’ x = [x 1,x 2,…,x n ] Information on each outgoing link ‘e’ of source Introduction

6. Terminology and Notation Information on outgoing link e* on intermediate node where ‘m’ is the number of incoming edges on the node e* ‘y e ’ is the incoming information on the incoming link e Output vector at the destination (sink) node z = [z 1,…,z n ] Introduction

7. Terminology and Notation Output vector ‘z’ is z = x * M where ‘M’ is the system transfer matrix M = A * G * B where A is [α i,j ] is a n * k matrix where ‘k’ is total number of edges in the network. G = (I-F) -1 is the k * k adjacency matrix B is [ε i,j ] is a k * n matrix Introduction

8. Terminology and Notation y t2 s x t1 z w Cut: Cut: A partition of vertex set into 2 classes, S containing source and S’ containing the sink. Value of the cut: where ‘C(e)’ is the rate constraint of each link Min-Cut Max-Flow Lemma: maxflow maxflow(s,t l ) Let ‘G’ be a graph with source node ‘s’ and sink nodes ‘t1’ and ‘t2’, and rate constraints ‘R’.Then for l=1,2, the maxflow from s to t l is the value of the min-cut between s and t l and is denoted by maxflow(s,t l ) Introduction

9. Study and Implementation Finding a network code for a given multicast problem Solvability conditions Single source single sink : det (M) ≠ 0 Single source multiple sink : ∏ det (M i ) ≠ 0 Multiple source multiple sink : det (M ii ) ≠ 0 det (M ii ) = 0 i Study and Implementation

10. Algorithm for finding network codes F(x)a Given polynomial F(x), find a such that F(a) ≠ 0 Find maximal degree ‘∂’ of F in any variable x i and choose smallest ‘i’ such that 2 i > ∂ Study and Implementation

11. Algorithm for finding network codes Find an element ‘a t ’ in F 2 i such that ≠ F(x) ≠ 0 and F F(x) If t = n then halt, else t t+1, goto previous step ‘a’ is the solution to the above problem x t =a t Study and Implementation

12. Bound on Field size There exists a solution to the single source multicast network coding problem in a finite field 2 m with Study and Implementation

13. Simulation Steps Generate a random network (single source multicast) Find the network capacity using maxflow algorithm Generate matrices A,G, B from the network topology Solve for the network parameters Study and Implementation

14. Coding vs Routing Is coding really required? How to check if routing achieves capacity? Routing is a special case of coding with constraints on codes Put constraints on codes and solve to see if routing is feasible Study and Implementation

15. Simulation results b2 V1 V2 V3 V4 V5 b1+b2b2 b1+b2 b1 b2 b2b1+b2 b1 b2 b2 V1 V2 V3 V4 V5 b1b2 b1 b1 b2 b2b1 b1 b2 Study and Implementation

16. Simulation results AT=AT= Study and Implementation

17. Simulation results Study and Implementation

18. Low Complexity Network Codes Gives a solution to the single source multicast network coding problem in a finite field 2 m with Uses only union of edge-disjoint paths to each receiver thus avoiding ‘flooding’ Study and Implementation

19. Network Recovery and Management Nodes need to change their ‘behavior’ for recovery from link failures Network management involves switching between appropriate codes for recovery from link failures Management requirement can be quantified by the number of different codes needed Study and Implementation

20. Network Recovery and Management Two formulations of quantification Centralized formulation Network behavior described by an overall code Network management requirement quantified by logarithm of the number of codes needed Node based formulation Network behavior described by the number of nodes which change behavior Quantified by the sum of the logarithm of the number of different behaviors of each node Study and Implementation

21. Network Recovery and Management Theorem Theorem : For a single receiver network with r processes and a minimum capacity of C, tight bounds on the number of codes needed for the no-failure scenario and all single link failures, assuming they are recoverable are :

22. To be included in the final report Faster implementation of the code- generating algorithm Comparison of Routing vs Coding on large number of random networks

23. Future direction of research Joint source-channel-network coding