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Broadcast Erasure Channel with Feedback: the Two Multicast Case — Algorithms and Bounds

Broadcast Erasure Channel with Feedback: the Two Multicast Case — Algorithms and Bounds . Efe Onaran 1 , Marios Gatzianas 2 and Christina Fragouli 2. 1 Department of Electrical and Electronics Engineering, Bilkent Univ., Turkey.

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Broadcast Erasure Channel with Feedback: the Two Multicast Case — Algorithms and Bounds

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  1. Broadcast Erasure Channel with Feedback: the Two Multicast Case — Algorithms and Bounds Efe Onaran1, Marios Gatzianas2 and Christina Fragouli2 1 Department of Electrical and Electronics Engineering, Bilkent Univ., Turkey. 2 School of Computer and Communication Sciences, EPFL, Switzerland.

  2. Outline Role of feedback on unicast and multicast traffic over BEC  Motivation for multiple multicast sessions. A capacity outer bound and a conjecture. Exploiting feedback: performance/complexity tradeoff • Optimal algorithm for 2 users per group. • An algorithm that is asymptotically better than timesharing (as erasure prob 0) Conclusions.

  3. Effect of feedback on BEC unicast capacity • Fact: FB increases capacity region of BEC with multiple unicast sessions Optimal FB scheme: W/o FB (TS) SRC/TX With FB Transmit to D1. There exist packets missed by D1 and seen by D2. Transmit to D2. There exist packets missed by D2 and seen by D1. Transmit . Also works with linear combs. D1 D2 Benefit of FB-based scheme over TS (equal rates) for N users= increasing w.r.t. N Not only does FB help for multiple unicast traffic, it also yields increasing benefits vs number of users (especially for high )

  4. Effect of feedback on BEC multicast capacity • Fact: For a single multicast session of users, FB offers no benefit. • Topic of this paper: what happens for multiple multicast sessions? Does FB offer any benefit? How does this benefit scale with number of users per group? SRC/TX achievable through network coding SRC/TX Useful tools for 2 multicast problem: Inner bound (TS) Outer bound 1. A multicast capacity outer bound  use 2-unicast bound 2. An achievable scheme (benchmark): use TS between sessions ?

  5. Our contribution • For , outer bound is tight. • For any , we propose a very simple FB-based algorithm that outperforms TS (especially for small ). Timesharing is asymptotically optimal as . • All schemes we have tried so far approach TS performance-wise as . This suggests the following conjecture: • We provide some evidence for this conjecture via a special case proof, specifically timesharing is asymptotically optimal if .

  6. Optimal scheme for N=2 (outline) 1. Create virtual queues indexed by sets where or The basic idea is taken from the 2 unicast case, i.e. keep track (via virtual queues) of overheard packets and suitably combine packets (in the spirit of ) SRC/TX 2. Transmit linear combinations of all packets stored in queues. {1,2} {1,2,4} 3. Depending on FB, we may move a packet into another queue (left-to-right manner) and use counters to keep track of how many packets a user wants from a queue. {3,4} • The trick to achieve optimality is to combine more than one queues. • At present, it is not clear how to extend optimal scheme to higher N. • We need an exponential number of queues. {3,4,2} {1,2,3} {1,2,3,4} {3,4,1}

  7. A (suboptimal) extension to arbitrary N Scheme EXTN: Instead of combining queues, process each queue individually. Benefit over TS Outer bound: TS inner bound: ΕΧΤΝ: but at the cost of exponential number of queues Benefit over TS decreases vs N. The exact opposite of the multiple unicast case! Q: can we retain the benefit over TS with a polynomial number of queues? (A: yes, with just 3 queues for any N)

  8. Simplest scheme with 1-O(2) performance Rule for packet movement: If transmitted packet is erased (*) by at least one user in and received by all users in  move packet from to SRC/TX

  9. Is timesharing optimal as N? We need to show that for any coding scheme it holds: Let be the feedback ACK/NACK sequence for user at transmission . : feedback ACK/NACK sequence for user Define pseudo-distance: : feedback ACK/NACK sequence for all users CR + memoryless + erasure: Fano: Distance:

  10. Conclusions • For small number of users (say, <10) per group, feedback offers significant benefit, especially for high . • We can construct schemes with exponential or polynomial number of queues which outperform timesharing for any finite N. • For large number of users (say, >100) per group, all proposed schemes perform very close to timesharing  feedback is not beneficial. • There exists partial evidence that timesharing is asymptotically optimal as .

  11. Thank you!

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