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

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

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?

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

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

Outline

6

1. Real-Time IDNC• Equivalence to Max Clique

• NP-Hardness

2. Random Real-Time IDNC• Optimal coded packet (clique number)

• Coding algorithm

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

Real-Time IDNC: Example- 6 packets: p1 , … , p6

- 3 users: u1 , u2 , u3

p1 p2 p3 p4 p5 p6

• u1 has p1 , p2 0 0 1 1 1 1

• u2 has p3 , p5 1 1 0 1 0 1

• u3 has p3 , p6 1 1 0 1 1 0

p5 + p6 is instantly decodable and innovative to u2 and u3

p2 + p3 is instantly decodable and innovative to all users

8

Mapping: Real-Time IDNC to Max Clique

9

Real-Time IDNC ≡ Max Clique in IDNC graph

Real-Time IDNC is NP-Hard

10

Real-Time IDNC ≡ Max Clique ≡ IQPNP-Hard

IQP is NP-Hard: reduction from Exact Cover by 3-Sets

Real-Time IDNC ≡ Integer Quadratic Programming (IQP)

Outline

11

1. Real-Time IDNC• Equivalence to Max Clique

• NP-Hardness

2. Random Real-Time IDNC• Optimal coded packet (clique number)

• Coding algorithm

Random Real-Time IDNC

12

Setup: iid loss probability p• aij = 1 with probability p

Analysis sketch:• Fix a set of j columns

• Define a good row as having one 1 among these j columnso 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))

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)

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)

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 )

Evaluation Results

16

Simulation with random loss rate (20 users, 20 packets)

Max-Clique outperforms all other algorithms at any loss rate

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