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Optimal Multicast Algorithms

Optimal Multicast Algorithms. Sidharth Jaggi Michelle Effros Philip A. Chou Kamal Jain. Menger’s Theorem. Min-cut Max-flow Theorem. Ford-Fulkerson Algorithm. C. R. S. Network Coding. S. b 1. b 2. b 1. b 2. b 1 +b 2. b 1. b 2. b 1 +b 2. b 1 +b 2. R 1. R 2.

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Optimal Multicast Algorithms

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

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

  3. Network Coding S b1 b2 b1 b2 b1+b2 b1 b2 b1+b2 b1+b2 R1 R2 Example due to Cai (2000) (b1,b2) (b1,b2)

  4. Multicast algorithms Assumptions Directed, acyclic graph. Each link has unit capacity. Links have zero delay. Upper bound for multicast capacity C, C ≤ min{Ci} R1 C1 C2 R2 S Network Cr Rr

  5. Multicast algorithms F(2m)-linear network (Koetter/Medard) b1 b2 bm Source:- Group together `m’ bits, Any node:- Perform linear combinations over finite field F(2m) β1 β2 F(2m)-linear network can achieve multicast capacity C! βk

  6. Multicast algorithms • Caveats to Koetter/Medard algorithm • May “flood” the network unnecessarily • Field size may need to be “large” (2m > 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” (2m > r-1) • Design complexity smaller

  7. Encoding/Decoding v1 β1 Decoding: If decoder Ri receives symbols [y1...yk], output [x1...xk]=[Mi]-1[y1 ...yk]T Encoding: Required β's provided by coefficients of linear combinations of v's v2 Vc β2 βk vk

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

  9. 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?

  10. Almost-optimal Random Binary Linear Codes (ARBLCs) = b1 b2 bm Source:- Group together `m’ bits, Any node:- Perform arbitrary linear combinations over finite field F(2) 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...

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

  12. ... 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

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