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Instantly Decodable Network Codes for Real-Time Applications

Instantly Decodable Network Codes for Real-Time Applications. Presented by Marios Gkatzianas. Anh Le, Arash Tehrani , Alex Dimakis , Athina Markopoulou UC Irvine, USC, UT Austin. Real-Time Applications with Wireless Broadcast. Real-time applications that use wireless broadcast:

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Instantly Decodable Network Codes for Real-Time Applications

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  1. Instantly Decodable Network Codes for Real-Time Applications Presented by MariosGkatzianas AnhLe, ArashTehrani, Alex Dimakis, AthinaMarkopoulou UC Irvine, USC, UT Austin

  2. Real-Time Applications with Wireless Broadcast Real-time applications that use wireless broadcast: • Live video streaming • Multiplayer games Unique characteristics: • Strict deadlines • Loss tolerant • Problem: Retransmissions in the presence of loss

  3. Broadcast Loss Recovery for Real-Time Applications Strict deadlines: • Instantly decodable network codes (IDNC) Loss tolerant: • We formulate Real-Time IDNC • What is a coded packet that is instantly decodable and innovative to the maximum number of users?

  4. Related Work • Instantly decodable, opportunistic codes [Katti’08] [Keller ’08] [Sadeghi ’09] [Athanasiadou’12] • IDNC focuses on minimizing the completion delay[Sorour ’10, 11, 12] • Index coding and data exchange problems[Birk ’06] [El Rouayheb ’10]

  5. Our Contributions • We show Real-Time IDNC is NP-Hard • Equivalent to finding a Max Clique in an IDNC graph • Provide a reduction from Exact Cover by 3-Sets • Analysis of instances with random loss probabilities • Provide a polynomial time solutionfor Random Max Clique and Random Real-Time IDNC

  6. Outline • Real-Time IDNC • Equivalence to Max Clique • NP-Hardness • Random Real-Time IDNC • Optimal coded packet (clique number) • Coding algorithm

  7. Real-Time IDNC: Problem Formulation • A set ofm packets, broadcast by a source • nusers, interested in all packets • Each user received only a subset of packets • To recover loss: What is a coded packet that is instantly decodableand innovative to the maximum number of users?

  8. Real-Time IDNC: Example • 6 packets: p1 , … , p6 • 3 users: u1 , u2 , u3p1 p2 p3 p4 p5 p6 • u1 has p1 , p20 0 1 1 1 1 • u2 has p3 , p51 1 0 1 0 1 • u3 has p3 , p61 1 0 1 1 0 p5+ p6is instantly decodable and innovative to u2 and u3 p2 + p3 is instantly decodable and innovative to all users

  9. Mapping: Real-Time IDNC to Max Clique Real-Time IDNC ≡ Max Clique in IDNC graph

  10. Real-Time IDNC is NP-Hard • Real-Time IDNC ≡ Integer Quadratic Programming (IQP) • IQP is NP-Hard: reduction from Exact Cover by 3-Sets Real-Time IDNC ≡ Max Clique ≡ IQPNP-Hard

  11. Outline • Real-Time IDNC • Equivalence to Max Clique • NP-Hardness • Random Real-Time IDNC • Optimal coded packet (clique number) • Coding algorithm

  12. Random Real-Time IDNC • Setup: iid loss probability p • aij = 1 with probability p • Analysis sketch: • Fix a set of jcolumns • Define a good row as having one 1 among these j columns • A row is good with probability f(j) = j p (1-p) j-1 • Number of good rows has Binomial distribution: Bin(n, f(j))

  13. Analysis of Random Real-Time IDNC • Lemma 5 (sketch):The size of the maximum clique that touches j columns is close to the number of good rows w.r.t. these j columns • Lemma 6 (sketch):The size of the maximum clique that touches j columns concentrates around n f(j) • Theorem 7 (sketch): Maximum clique, that touches any j columns, has size concentrating around n f(j)

  14. Analysis of Random Real-Time IDNC (cont.) • Corollary 8:The maximum clique touches j* columns, where j* = argmax f(j) • Observations: • j*is a constant for a fixed loss rate p • j* increases as loss rate p decreases (more packets should be coded together)

  15. Max-Clique Algorithm • Observations: • The maximum clique concentrates around n f(j*) • j* is a constant for a fixed loss rate p • Polynomial-time algorithm to find the maximum clique and optimal coded packet: • Examine all cliques that touch j columns for all jδ-close to j* • Complexity: O (n m j* + δ )

  16. Evaluation Results • Simulation with random loss rate (20 users, 20 packets) Max-Clique outperforms all other algorithms at any loss rate

  17. Conclusion • We formulate Real-Time IDNC • Equivalent to Max Clique in IDNC graph • NP-Hard proof • Analysis of Random Real-Time IDNC • Polynomial time solution to findmax clique and optimal coded packet

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