Network Information Flow

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

2. Overview of the Presentation • Introduction to the Problem • Work done • Results and Conclusions • Future work • 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 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. 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.

8. Motivation for the problem • Liang [6] have given a game theoretic approach to solve single source network switching for a given communication network. • 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.

9. 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.

10. Network switching

11. 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 Work • 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).

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.