1. 1 Nick Harvey (MIT) Kamal Jain (MSR) Lap Chi Lau (U. Toronto) Chandra Nair (MSR) Yunnan Wu (MSR) Conservative Network Coding

2. 2 Outline Motivation Acyclic networks Cyclic networks Conclusion

3. 3 The General Multi-Session Network Coding Problem Given a network coding problem: Directed graph G = (V,E) k “commodities” (streams of information) Sources: s 1, …, s k Receiver set for each source T 1, …, T k At what rate can the sources transmit? This is very general and very hard…

4. 4 “Conservative Network Coding” Given a network coding problem: Directed graph G = (V,E) k “commodities” (streams of information) Sources: s 1, …, s k Receiver set for each source T 1, …, T k At what rate can the sources transmit? Consider solutions where intermediate nodes are conservative i.e., a node rejects anything it does not want. i.e., commodity i is not allowed to leave the set T i  {s i }

5. 5 Motivations Practical motivation In peer-to-peer networks, a node may not have incentive to relay traffic for others a node may be concerned about security troubles Theoretical motivation In the special case when there is a single commodity, there are elegant results.

6. 6 Single Session Conservative Networking (Broadcasting) Integer Routing Rate Fractional Routing Rate Network Coding Rate = = = Cut Bound Edmonds’ Theorem (1972): Given a directed graph and a source node s, the maximum number of edge disjoint spanning trees rooted at s is equal to the minimum s- cut capacity.

7. 7 Example t3t3 t1t1 t2t2 s t4t4 “As long as we can route information to each node individually at rate C, we can route information simultaneously to all destinations at rate C.”

8. 8 Generalization? For conservative networking, Integer Routing Rate Fractional Routing Rate Network Coding Rate ? ?

9. 9 Outline Motivation Acyclic networks Cyclic networks Conclusion Integer Routing Rate Fractional Routing Rate Network Coding Rate = =

10. 10 Colored Cut Condition srsr sbsb

11. 11 Colored Cut Condition Colors on nodes  colors on edges srsr sbsb

12. 12 Colored Cut Condition Blue and Red need to cross the cut We have a {red, blue} edge, a red edge and a blue edge So okay! srsr sbsb

13. 13 Colored Cut Condition Blue and Red need to cross the cut We have a {red, blue} edge and a blue edge So okay! srsr sbsb

14. 14 Colored Cut Condition Generally, for each node-set cut, the set of edges across the cut must enable that the colors that need to cross the cut indeed can cross. A bi-partite matching condition srsr sbsb trtr tbtb

15. 15 Proof that Colored Cut Bound is Achievable by Routing Visit the nodes in the topological order, v 1,…,v n By inductive hypothesis, the previous nodes v 1,…,v k can indeed recover the messages they want. Consider node v k+1 Colored cut condition must hold; Conversely, if it holds, there exists an integer routing solution. t r,g t g,b t r,b

16. 16 Outline Motivation Acyclic networks Cyclic networks Conclusion Integer Routing Rate Fractional Routing Rate Network Coding Rate = = Integer Routing Rate Fractional Routing Rate Network Coding Rate ? ?

17. 17 Outline Motivation Acyclic networks Cyclic networks Conclusion Integer Routing Rate Fractional Routing Rate Network Coding Rate = = Integer Routing Rate Fractional Routing Rate Network Coding Rate < <

18. 18 Proof by Reduction A k-pairs problem G  A conservative network problem G’ Find k-pairs problems such that

19. 19 Therefore

20. 20 Reduction k-pairs  conservative networking s1s2 t1t2 s1 s2 t1t2 v1 v2 T1 T2 Vertex Set V Sources s 1,s 2 Sinks t 1,t 2 Add vertices v 1, v 2 Add edges t i -v i Add edges v i -u ∀ u ∈ V – t i Set T i = V + v i G G’

21. 21 Step 1: s1 s2 t1t2 v1 v2 T1 T2 Easy

22. 22 Step 2: s1 s2 t1t2 v1 v2 T1 T2 Disjoint trees  Disjoint paths

23. 23 Reduction does not preserve rates for coding A k-pairs problem G  A conservative network problem G’ “three butterflies flying together”

24. 24 Proof by Reduction A k-pairs problem G  A conservative network problem G’ Find k-pairs problems such that

25. 25 s1s1 s2s2 c u t2t2 t1t1

26. 26 s1 t1 s2 t2

27. 27 Results for Cyclic Networks Integer Routing Rate Fractional Routing Rate Network Coding Rate << “Buy one get one free”: Integer Routing Solution is NP-hard

28. 28 A Simpler Example 1 2 3 4 5 6 8 7

29. 29 A Simpler Example 1 2 3 4 5 6 8 7

30. 30 Conclusion Conservative networking model, motivated by practice and theory Neat result for acyclic networks that generalize Edmonds’ Theorem Counter examples for cyclic networks Even if nodes are conservative, network coding can help “Cycles are tricky!” Bound obtained by examining nodes in isolation is loose Bound obtained by examining node-set cuts in isolation is loose Generally require entropy arguments