Network Information Flow Nikhil Bhargava 2004 MCS 2650

Network Information Flow Nikhil Bhargava (2004 MCS 2650) Under the guidance of Prof. Naresh Sharma (Dept. of Electrical Engineering) Prof. S. N Maheshwari (Dept. of Computer Science and Engineering) IIT, Delhi

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

Introduction to the problem • Aim is to improve 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.

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.

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.

Difference between Fluid flow and information flow

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 theory of network coding.

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.

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 multisource multi-sink network coding.

Network switching

Network coding

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.

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

• 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.

Singular Symmetric Butterfly network

Min-cuts for one sink in Singular Symmetric Butterfly network

• The sub graph for sink t 1 has following 7 s-t cuts 1. {(s, a), (s, b)} = 2 w 1 2. {(s, a), (b, c)} = w 1+w 2 3. {(s, a), (a, c), (d, t 1)} = 2 w 2+w 3 4. {(a, t 1), (a, c), (b, c)} = w 1+w 2+w 5 5. {(s, a), (a, c), (c, d)} = w 1+w 2+w 4 6. {(a, t 1), (c, d)} = w 3+w 4 7. {(a, t 1), (d, t 1)} = w 3+w 5

• The sub graph for sink t 2 has following 7 s-t cuts 1. {(s, a), (s, b)} = 2 w 1 2. {(s, b), (a, c)} = w 1+w 2 3. {(b, t 2), (a, c), (b, c)} = 2 w 2+w 3 4. {(d, t 2), (s, b), (b, c)} = w 1+w 2+w 5 5. {(s, b), (b, c), (c, d)} = w 1+w 2+w 4 6. {(b, t 2), (c, d)} = w 3+w 4 7. {(b, t 2), (d, t 2)} = w 3+w 5

Max. Information flow due to network coding • Assuming following conditions on link capacities 1. w 1<w 2 2. w 1<w 3 3. w 4<w 5 Maximum information flow due to network coding is w 1+min(w 1, w 4)

Max. Information flow due to network switching

• 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, t 1), (a, c); (b, t 2), (b, c) and (d, t 1), (d, t 2) respectively. • Maximum information flow due to network switching is (2 w 1+min(2 w 1, w 4))/2 • Switching gap comes out to be 2(w 1+min(w 1, w 4))/(2 w 1+min(2 w 1, w 4)) • In case all edge capacities are equal, switching gap comes out to be 4/3

Dual Symmetric Butterfly network

Triple Symmetric Butterfly network

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.

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.

Future Work • Find network switching gain using mincut trees for single source and later multi-source networks (open problem). • Find network coding gain for multisource multicast network (open problem).

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, Jean. Yves 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.

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, 3 rd edition, San Diego: Academic Press, 1995.