1. Packing Multicast Trees Philip A. Chou Sudipta Sengupta Minghua Chen Jin Li Kobayashi Workshop on Modeling and Analysis of Computer and Communication Systems, Princeton University, May 9, 2008

2. Problem What is the capacity region What is a transmission scheme that achieves a given transmission rate vector Given Directed graph (V,E) with edge capacities Multiple multicast sessions {(s i,T i )}, each with Sender s i Receiver set T i Transmission rate r i i i 2 2 1 1 11 1

3. Butterfly Network with Two Unicast Sessions sender s 1 rate r 1 sender s 2 rate r 2 receiver t 2 receiver t 1 r2r2 r1r1 Capacity Region ?

4. Network Coding vs Routing y1y1 y2y2 y3y3 f 1 (y 1,y 2,y 3 ) f 2 (y 1,y 2,y 3 )

5. Butterfly Network with Two Unicast Sessions sender s 1 rate r 1 sender s 2 rate r 2 receiver t 2 receiver t 1 a b b a XOR a+b r2r2 r1r1 Capacity Region

6. Characterization of Capacity for Acyclic Graphs [Yan, Yeung, Zhang; ISIT 2007]

7. Single Unicast Session r ≤ MinCut(s,t) MinCut(s,t) is achievable, i.e., MaxFlow(s,t) = MinCut(s,t), by packing edge-disjoint directed paths s t Value of s-t cut [Menger; 1927] Given Directed graph (V,E) with edge capacities Single unicast session with Sender s Receiver t Transmission rate r

8. Single Broadcast Session r ≤ min v Є V MinCut(s,v) min v Є V MinCut(s,v) is achievable (“broadcast capacity”) by packing edge-disjoint directed spanning trees [Edmonds; 1972] Given Directed graph (V,E) with edge capacities Single broadcast session with Sender s Receiver set T=V Transmission rate r

9. Single Broadcast Session r ≤ min v Є V MinCut(s,v) min v Є V MinCut(s,v) is achievable (“broadcast capacity”) by packing edge-disjoint directed spanning trees [Edmonds; 1972] Given Directed graph (V,E) with edge capacities Single broadcast session with Sender s Receiver set T=V Transmission rate r

10. Single Multicast Session r ≤ min t Є T MinCut(s,t) min t Є T MinCut(s,t) is NOT always achievable by packing edge-disjoint multicast (Steiner) trees Given Directed graph (V,E) with edge capacities Single multicast session with Sender s Receiver set Transmission rate r

11. Packing Multicast Trees Insufficient to achieve MinCut optimal routing throughput = 1 network coding throughput = 2 a,b a a a a b b b b a+b,a,b [Alswede, Cai, Li, Yeung ; 2000] min t Є T MinCut(s,t) is always achievable by network coding

12. Linear Network Coding Sufficient to achieve MinCut Linear network coding is sufficient to achieve min t Є T MinCut(s,t) Polynomial time algorithm for finding coefficients y1y1 y2y2 y3y3 α 1 y 1 + α 2 y 2 + α 3 y 3 β 1 y 1 + β 2 y 2 + β 3 y 3 [Jaggi, Chou, Jain, Effros; Sanders, et al.; 2003] [Erez, Feder; 2005] [Cai, Li, Yeung; 2003] [Koetter and Médard; 2003]

13. Linear Network Coding Sufficient to achieve MinCut Packing a maximum-rate set of multicast trees is NP hard Rate gap to min t Є T MinCut(s,t) can be a factor of log n [Jaggi, Chou, Jain, Effros; Sanders, et al.; 2003] [Jain, Mahdian, Salavatipour; 2003]

14. Linear Network Coding NOT Sufficient for Multiple Sessions Linear network coding is NOT generally sufficient to achieve capacity for multiple sessions [Dougherty, Freiling, Zeger; 2005]

15. P2P Networks Upload bandwidth is bottleneck Completely connected overlay

16. P2P Network as Special Case Networks w/edge capacities P2P Networks v c out (v)

17. P2P Network as Set of Networks Corollary to Edmonds’ Theorem: Given a P2P network with a single broadcast session (i.e., a single sender and all other nodes as receivers), the maximum throughput is achievable by routing over a set of directed spanning trees c out (v) v c(e)c(e)

18. Throughput: Mutualcast [Li, Chou, Zhang; 2005] Cutset analysis: Throughput achieves mincut bound t1t1 s tNtN … t2t2

19. Throughput: Mutualcast with Helpers [Li, Chou, Zhang; 2005] t1t1 s tNtN Cutset analysis: Throughput achieves mincut bound … hnhn

20. Given P2P network Source s, receiver set T, helper set H = V  T  s Then any rate r  C can be achieved by routing over at most |V| depth-1 and depth-2 multicast trees Capacity is given by Remarks: Despite existence of helpers, no network coding needed! |V| trees instead of O(|V| |V| ) Simple, short trees Mutualcast Theorem

21. Given P2P network Multiple sources s i and receiver sets T i, i=1,…,|S|, s.t. sets (s i U T i ) are identical or disjoint; no connections between the disjoint receiver sets; H = V  U (s i U T i ) Then any rate vector can be achieved by routing over at most |V|  |S| depth-1 and depth-2 multicast trees Multi-session Mutualcast Theorem [Sengupta, Chen, Chou, Li; ISIT 2008] H

22. Capacity region is given by set of linear inequalities: Remark: Despite multiple sessions (and helpers), still no inter- session or intra-session network coding is needed! Still |V| trees instead of O(|V| |V| ); still simple, short Multi-session Mutualcast Theorem [Sengupta, Chen, Chou, Li; ISIT 2008]

23. Application to Video Conferencing Maximizing over all can be found by maximizing over all tree rates x ij, i=1,…,|S|, j=1,…,|V|, s.t., i=1,…,|S| Network Utility Maximization problem U1(r1)U1(r1) U2(r2)U2(r2) U4(r4)U4(r4) U5(r5)U5(r5) U3(r3)U3(r3)

24. For general multi-session multicast over directed graphs with edge capacities Capacity region is stated in terms of  * Hard to know if a rate vector r is in capacity region For restricted case of multi-session multicast with identical or disjoint receiver sets over P2P networks Capacity region is given by |V| inequalities over at most |V|  |S| variables (rates on each tree) Any point in this region is achievable by routing over at most |V|  |S| depth-1 and depth-2 trees Summary