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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

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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

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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 ?

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

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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

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

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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

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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

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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

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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

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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

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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]

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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]

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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]

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P2P Networks Upload bandwidth is bottleneck Completely connected overlay

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

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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)

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

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

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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

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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

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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]

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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)

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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

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