Instantly Decodable Network Codes
for Real-Time Applications
Presented by Marios Gatzianas
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:
• Live video streaming
• Multiplayer games
Unique characteristics:
• Strict deadlines
• Loss tolerant
Problem: Retransmissions in the presence of loss
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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?
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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]
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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 solution for
Random Max Clique and Random Real-Time IDNC
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Outline
1. Real-Time IDNC
• Equivalence to Max Clique
• NP-Hardness
2. Random Real-Time IDNC
• Optimal coded packet (clique number)
• Coding algorithm
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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?
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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
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Mapping: Real-Time IDNC to Max Clique
Real-Time IDNC ≡ Max Clique in IDNC graph
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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 ≡ IQP
NP-Hard
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Outline
1. Real-Time IDNC
• Equivalence to Max Clique
• NP-Hardness
2. Random Real-Time IDNC
• Optimal coded packet (clique number)
• Coding algorithm
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Random Real-Time IDNC
 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 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))
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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)
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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)
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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 j δ-close to j*
• Complexity: O (n m j* + δ )
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Evaluation Results
 Simulation with random loss rate (20 users, 20 packets)
Max-Clique outperforms all other algorithms at any loss rate
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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 find
max clique and optimal coded packet
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