1. Decentralized Coded Caching Attains Order-Optimal Memory-Rate Tradeoff
Mohammad Ali Maddah-Ali
Bell Labs, Alcatel-Lucent
joint work with
Urs Niesen
Allerton
October 2013

2. Video on Demand High temporal traffic variability
Caching (prefetching) can help to smooth traffic

3. Caching (Prefetching)
Server
Placement phase: populate caches
Demands are not known yet
Delivery phase: reveal request, deliver content

4. Problem Setting
N Files
Server
Shared Link
K Users
Cache
Contents
Size M
Question: Smallest worst-case rate R(M) needed in delivery phase?
How to choose (1) caching functions, (2) delivery functions
Placement: - Cache arbitrary function of the files (linear, nonlinear, …)
Delivery:
-Requests are revealed to the server
- Server sends a function of the files

5. Coded Caching Uncoded Caching Coded Caching [Maddah-Ali, Niesen 2012]
N Files, K Users, Cache Size M
Uncoded Caching
Caches used to deliver content locally
Local cache size matters
Coded Caching [Maddah-Ali, Niesen 2012]
The main gain in caching is global
Global cache size matters (even though caches are isolated)

6. Centralized Coded Caching
N=3 Files, K=3 Users, Cache Size M=2
Maddah-Ali, Niesen, 2012
A12
A13
A23
B12
B13
B23
C12
C13
C23
Approximately Optimum
A23
B13
C12
A23⊕B13⊕C12
1/3
A12
A13
B12
B13
B23
C12
C13
C23
A23
Multicasting Opportunity between three users with different demands

7. Centralized Coded Caching
N=3 Files, K=3 Users, Cache Size M=2
A12
A13
A23
B12
B13
B23
C12
C13
C23
Centralized caching needs
Number and identity of the users in advance
In practice, it is not the case,
Users may turn off
Users may be asynchronous
Topology may time-varying (wireless)
A12
A13
B12
B13
B23
C12
C13
C23
A23
Question: Can we achieve similar gain without such knowledge?

8. Decentralized Proposed Scheme
N=3 Files, K=3 Users, Cache Size M=2 1 2 3 12 13 23
123 1 2 3 12 13 23
123 1 2 3 12 13 23
123
Delivery: Greedy linear encoding
Prefetching: Each user caches 2/3 of the bits of each file
- randomly,
- uniformly,
- independently. 2 1 ⊕ 23 13 12 ⊕ 3 2 ⊕ 3 1 ⊕ 1 12 13
123 2 12 23
123 3 13 23
123 1 12 13
123 2 12 23
123 3 13 23
123 1 12 13
123 2 12 23
123 3 13 23
123

9. Decentralized Caching

10. Decentralized Caching
Centralized Prefetching: 12 13 23 12 13 23 12 13 23
Decentralized Prefetching: 1 2 3 12 13 23
123 1 2 3 12 13 23
123 1 2 3 12 13 23
123

11. Comparison Uncoded Local Cache Gain: Proportional to local cache size
N Files, K Users, Cache Size M
Uncoded
Local Cache Gain:
Proportional to local cache size
Offers minor gain
Coded (Centralized): [Maddah-Ali, Niesen, 2012]
Global Cache Gain:
Proportional to global cache size
Offers gain in the order of number of users
Coded (Decentralized)

12. The proposed scheme is optimum within a constant factor in rate.
Can We Do Better?
Theorem:
The proposed scheme is optimum within a constant factor in rate.
Information-theoretic bound
The constant gap is uniform in the problem parameters
No significant gains beside local and global

13. Asynchronous Delivery
Segment 1
Segment 2
Segment 3
Segment 1
Segment 2
Segment 3
Segment 1
Segment 2
Segment 3

14. Conclusion
We can achieve within a constant factor of the optimum caching performance through
Decentralized and uncoded prefetching
Greedy and linearly coded delivery
Significant improvement over uncoded caching schemes
Reduction in rate up to order of number of users
Papers available on arXiv:
Maddah-Ali and Niesen: Fundamental Limits of Caching (Sept. 2012)
Maddah-Ali and Niesen: Decentralized Coded Caching Attains Order-Optimal Memory-Rate Tradeoff ( Jan. 2013)
Niesen and Maddah-Ali: Coded Caching with Nonuniform Demands (Aug. 2013)