1. Instantly Decodable Network Codes for Real-Time Applications Anh Le, Arash Tehrani, Alex Dimakis, Athina Markopoulou UC Irvine, USC, UT Austin Presented by Marios Gatzianas

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

3. Broadcast Loss Recovery for Real-Time Applications Strict deadlines: Instantly decodable network codes (IDNC) Loss tolerant: We formulate Real-Time IDNC 3 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] 4

5. Our Contributions 5 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 solution for Random Max Clique and Random Real-Time IDNC

6. Outline 6 1.Real-Time IDNC Equivalence to Max Clique NP-Hardness 2.Random Real-Time IDNC Optimal coded packet (clique number) Coding algorithm

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

8. Real-Time IDNC: Example -6 packets: p 1, …, p 6 -3 users: u 1, u 2, u 3 p 1 p 2 p 3 p 4 p 5 p 6 u 1 has p 1, p 2 0 0 1 1 1 1 u 2 has p 3, p 5 1 1 0 1 0 1 u 3 has p 3, p 6 1 1 0 1 1 0 p 5 + p 6 is instantly decodable and innovative to u 2 and u 3 p 2 + p 3 is instantly decodable and innovative to all users 8

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

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

11. Outline 11 1.Real-Time IDNC Equivalence to Max Clique NP-Hardness 2.Random Real-Time IDNC Optimal coded packet (clique number) Coding algorithm

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

13. Analysis of Random Real-Time IDNC 13 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.) 14 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 15 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 j δ-close to j* Complexity: O (n m j* + δ )

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

17. Conclusion 17 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 find max clique and optimal coded packet