Optimal Multicast Algorithms Sidharth Jaggi Michelle Effros Philip A. Chou Kamal Jain

Menger’s Theorem C Min-cut Max-flow Theorem Ford-Fulkerson Algorithm S R

Network Coding S R1R1 R2R2 b1b1 b2b2 b2b2 b2b2 b1b1 b1b1 (b 1,b 2 ) b 1 +b 2 (b 1,b 2 ) Example due to Cai (2000)

Network Information Flow Network Coding (Single-sender Multiple-receivers, directed graphs)  Ahlswede/Cai/Li/Yeung, 2000, (Random coding argument)  Li/Yeung, 2003, (Linear network coding) (high complexity)  Koetter/Medard, 2002, (Algebraic network coding) Network Coding (Multiple-senders Multiple-receivers)  Li/Yeung (loose bounds)  Koetter/Medard, 2002, (Algebraic codes, NP-hard) Network Coding (Other cases)  Link failures (Koetter/Medard, 2002)  Wireless networks (Jaggi, unpublished)  Delays, cyclic graphs...

Multicast algorithms Assumptions Directed, acyclic graph. Each link has unit capacity. Links have zero delay. Upper bound for multicast capacity C, C ≤ min{C i } S R1R1 R2R2 RrRr CrCr C1C1 C2C2 Network

Multicast algorithms b1b1 b2b2 bmbm β1β1 β2β2 βkβk F(2 m )-linear network can achieve multicast capacity C! F(2 m )-linear network (Koetter/Medard) Source:- Group together `m’ bits, Any node:- Perform linear combinations over finite field F(2 m )

Multicast algorithms Caveats to Koetter/Medard algorithm  May “flood” the network unnecessarily  Field size may need to be “large” (2 m > rC)  Design complexity may be “large” (related to flooding) Our algorithm – you can have your cake and eat it too.  No “flooding”  Field size “small” (2 m > r-1)  Design complexity smaller

Encoding/Decoding Decoding: If decoder R i receives symbols [y 1...y k ], output [x 1...x k ]=[M i ] -1 [y 1...y k ] T β1β1 β2β2 βkβk VcVc v1v1 v2v2 vkvk Encoding: Required β's provided by coefficients of linear combinations of v's

Minimum Field Size... This class of networks, for q(q+1)/2 receivers, minimum field size = q

Minimum Field Size Open Questions  Either q-1 or (q(q+1)-2)/2 tight?  What, in general, is the smallest q for a particular network?

Almost-optimal Random Binary Linear Codes (ARBLCs) b1b1 b2b2 bmbm If m(C-R) > log(V.r), ARBLCs can achieve multicast rate R with zero error! (V = |Vertex-set|) Random, distributed, extremely low complexity design. Can even build in very strong robustness properties... Source:- Group together `m’ bits, Any node:- Perform arbitrary linear combinations over finite field F(2) =

Robustness of ARBLCs S R1R1 R2R2 RrRr CrCr C1C1 C2C2 Original Network C = min{C i }

Robustness of ARBLCs S R1R1 R2R2 RrRr Cr'Cr' C1'C1' C2'C2' Faulty Network C' = min{C i '} If value of C' known to S, same code can achieve C' rate! (interior nodes oblivious)

Loopy Graphs/Graphs with delays "Loopy" graphs? Graphs with delays? S Ahlswede et al. R1R1 R1R1 R1R1 R2R2 R2R2 R2R2 RrRr RrRr RrRr

Future work... Only some nodes can encode Practical implementation  Synchronicity/delays  Unknown topology  Packet losses Issues related to next-generation network protocols (FAST)

... Utility of WAN in Lab Access to any subset of routers Practical testing  Can introduce arbitrary delays patterns  Topology under our control  Have greater handle on packet loss statistics (needed to develop theoretical models) Examine behaviour of network codes with FAST