Network Information Flow

1. Network Information Flow Nikhil Bhargava (2004MCS2650) Under the guidance of Prof. S.N Maheshwari (Dept. of Computer Science and Engineering) IIT, Delhi

2. Overview of the Presentation • Introduction to the Problem. • Previous work done. • Analysis of Butterfly network. • Results and Conclusions. • Future research focus. • References.

3. Introduction to the problem • Aim is to improve the throughput in single/multiple source network multicast scenario. • Throughput will be low since conventional switching of information will time share the links (details to follow). • Could be improved by doing coding (i.e. XOR operations on the incoming bit streams) at nodes or vertices of the network graph. • Problem is to find the maximum information flow for a point/multi-point to multi-point multicast network.

4. Introduction to the problem (contd.) • Each node in a conventional network functions as a switch which: • Either replicates information received • Forwards info. from input link to output links • Network coding is used to boost throughput. • In this approach, each node receives information from all the input links, encodes it (i.e. combines it by the XOR operations) and sends information to the output link.

5. Previous work • Ahlswede et. al., in their seminal work [1] have introduced a new class of problems of finding maximum admissible coding rate region for a generic communication multicast network. • They gave a upper bound based on max-flow min-cut theorem for the information flow that could be achieved by network coding. • Note that in network coding, the flow is not preserved. • Max-flow min-cut theorem in graph theory literature is for information flow preserving networks.

6. Difference between Fluid flow and information flow

7. Network switching

8. Network coding

9. Previous work (contd.) • Li, Yeung, and Cai [2] showed that the multicast capacity can be achieved by linear network coding for acyclic networks i.e. networks having a graph with no cycles. • Yeung, Li, Cai, and Zhang in [4] have done an extensive survey on the theory of network coding.

10. Motivation for the problem • Liang [6] have given a game theoretic approach to solve single source network switching for a given communication network. • He computed network coding gain from max-flow min-cut bound. • Based upon certain conditions on link capacities, a game matrix is constructed and solved to give the maximum flow using network switching. • The network considered is not general and there are no known ways to provide analytical solutions to maximum flow • It has to be done on a case by case basis.

11. Motivation for the problem • There is no formal way to compute maximum flow for a generic network. • The game theory approach needs computation for each network on a case by case basis. • Need an alternative to determine network switching gain (i.e. maximum information flow by switching) for a single source multicast network. • Need to understand the problem of multi-source multi-sink network coding.

12. Network Coding • Intermediate nodes transmit packets that are functions of the received packets. • Mostly linear functions are used. • Known result that linear functions are enough to achieve the max-flow min-cut bound for both cyclic and acyclic networks. • Can make the network robust to link failures. • Peer-to-peer multicast file sharing network • Wireless Networks, sensor, adhoc, mobile.

13. Work done • Started analyzing butterfly network [1] to find its switching gap. • Switching gap for a network is defined as the ratio of maximum achievable information rate using network coding (NC) to that of network switching (NS). • Enumerated min-cuts and calculated NC using max-flow min-cut theorem

14. Enumerated multicast routes for each sink and created a game matrix for it. • Solved the matrix to get maximum achievable information rate due to network switching. • Calculated switching gap and analyzed it for different cases of link capacities. • Extended the network by taking its dual and triple version and then analyzed each of them.

15. Singular Symmetric Butterfly network

16. Min-cuts for one sink in Singular Symmetric Butterfly network

17. The sub graph for sink t1 has following 7 s-t cuts • {(s,a), (s,b)} = 2w1 • {(s,a), (b,c)} = w1+w2 • {(s,a), (a,c), (d,t1)} = 2w2+w3 • {(a, t1), (a,c), (b,c)} = w1+w2+w5 • {(s,a), (a,c), (c,d)} = w1+w2+w4 • {(a, t1), (c,d)} = w3+w4 • {(a, t1), (d, t1)} = w3+w5

18. The sub graph for sink t2 has following 7 s-t cuts • {(s,a), (s,b)} = 2w1 • {(s,b), (a,c)} = w1+w2 • {(b, t2), (a,c), (b,c)} = 2w2+w3 • {(d, t2), (s,b), (b,c)} = w1+w2+w5 • {(s,b), (b,c), (c,d)} = w1+w2+w4 • {(b, t2), (c,d)} = w3+w4 • {(b, t2), (d, t2)} = w3+w5

19. Max. Information flow due to network coding • Assuming following conditions on link capacities • w1<w2 • w1<w3 • w4<w5 Maximum information flow due to network coding is w1+min(w1, w4)

20. Max. Information flow due to network switching

21. Rows denote edges and columns denote multicast routes. • It has nine edges and 7 multicast routes. • Edges (s,a); (s,a) and (c,d) are dominating edges (a,t1),(a,c); (b,t2),(b,c) and (d, t1), (d, t2) respectively. • Maximum information flow due to network switching is (2w1+min(2w1, w4))/2 • Switching gap comes out to be 2(w1+min(w1, w4))/(2w1+min(2w1, w4)) • In case all edge capacities are equal, switching gap comes out to be 4/3.

22. Dual Symmetric Butterfly network

23. Triple Symmetric Butterfly network

24. Results • I have analyzed special cases of Ahlswede’s butterfly network to find switching gap. • Used game theory to calculate the maximum information flow using switching case. • Based upon observations for singular, dual and triple version of the above network, gave an intuitive result for generic class of above network • Game theory principles fails to solve the matrix for generic case.

25. Conclusions • For the chosen class of networks • The max-flow min-cut bound remains constant. • Information flow achieved by network switching decreases as one increases the size of the network. • Thus overall switching gap increases • Network coding becomes more useful for large graphs.

26. Future Focus • Find network switching gain using min-cut trees for single source and later multi-source networks (open problem). • Find network coding gain for multi-source multicast network (open problem). • Find optimum switching strategy for a specific class of graphs.

27. References [1] Ahlswede, N. Cai, S.-Y. Li, and R. W. Yeung, “Network information flow,” IEEE Trans. Inform. Theory, vol. 46, no. 4, pp. 12041216, July 2000. [2] S.-Y. Li, R. W. Yeung, and N. Cai, “Linear network coding,” IEEE Trans. Inform. Theory, vol. IT-49, no. 2, pp. 371381, Feb. 2003. [3] Christina Fragouli, JeanYves Le Boudec, Jorg Widmer, “Network Coding: An Instant Primer,” LCA-REPORT- 2005-010. [4] R. W. Yeung, S.-Y. Li, N. Cai, and Z.Zhang,“Theory of network coding,” submitted to Foundations and Trends in Commun. and Inform. Theory, preprint, 2005.

28. References [5] C.K Ngai and R. W. Yeung, “Network switching gap of combination networks,” in 2004 IEEE Inform. Theory Workshop, 24-29, Oct 2004, pp.283-287. [6] Xue-Bin Liang, “On the Switching Gap of Ahlswede- Cai-Li-Yeung’s Single-Source Multicast Network,” 2006 IEEE Int. Symp. Inform. Theory, Seattle, Washington, USA, July 2006. [7] Xue-Bin Liang, “Matrix Games in the Multicast Networks: Maximum Information Flows With Network Switching,” IEEE Trans. Inform. Theory, vol. 52, no. 6, June 2006. [8] G. Owen, Game Theory, 3rd edition, San Diego: Academic Press, 1995.