Decentralized Coded Caching Attains Order-Optimal Memory-Rate Tradeoff

Mohammad Ali Maddah-AliBell Labs, Alcatel-Lucent

joint work with

Urs Niesen

Allerton October 2013

Video on Demand

0 2 4 6 8 10 12 14 16 18 20 220

20

40

60

80

100

Time

Nor

mal

ized

Dem

and

• High temporal traffic variability

• Caching (prefetching) can help to smooth traffic

Caching (Prefetching)

• Placement phase: populate caches

• Demands are not known yet

• Delivery phase: reveal request, deliver content

Server

Server

K Users

Cache Contents

Shared Link

Problem Setting

N Files

Size M

Placement: - Cache arbitrary function of the files (linear, nonlinear, …)Delivery: -Requests are revealed to the server

- Server sends a function of the files Question: Smallest worst-case rate R(M) needed in delivery phase?How to choose (1) caching functions, (2) delivery functions

Coded CachingN 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)

Centralized Coded CachingN=3 Files, K=3 Users, Cache Size M=2

A12 A13 A23

B12 B13 B23

C12 C13 C23

A23⊕B13⊕C12

Multicasting Opportunity between three users with different demands

A12 A12

A13

A13

A23 A23

B12

B13

B12

B23

B13

B23

C12

C13

C12

C23

C13

C23

1/3

A23

B13

C12

Maddah-Ali, Niesen, 2012

Approximately Optimum

Centralized Coded CachingN=3 Files, K=3 Users, Cache Size M=2

A12 A13 A23

B12 B13 B23

C12 C13 C23

A12 A12

A13

A13

A23 A23

B12

B13

B12

B23

B13

B23

C12

C13

C12

C23

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)

Question: Can we achieve similar gain without such knowledge?

Decentralized Proposed Scheme1 2 3 12 13 23 12301 2 3 12 13 23 1230

1 2 3 12 13 23 1230

1 12 13 123

1 12 13 123

1 12 13 123

2 12 23 123

2 12 23 123

2 12 23 123

3 13 23 123

3 13 23 123

3 13 23 123

32 ⊕

Prefetching: Each user caches 2/3 of the bits of each file - randomly, - uniformly, - independently.

Delivery: Greedy linear encoding

0 0 0

21 ⊕

31 ⊕

23 13 12⊕ ⊕

N=3 Files, K=3 Users, Cache Size M=2

Decentralized Caching

Decentralized Caching

1 2 3 12 13 23 12301 2 3 12 13 23 1230

1 2 3 12 13 23 1230

12 13 23

12 13 23

12 13 23

Centralized Prefetching:

Decentralized Prefetching:

Coded (Centralized): [Maddah-Ali, Niesen, 2012]

Coded (Decentralized)

Uncoded Local Cache Gain:

• Proportional to local cache size

• Offers minor gain

Global Cache Gain:

• Proportional to global cache size

• Offers gain in the order of number of users

ComparisonN Files, K Users, Cache Size M

Can We Do Better?

The proposed scheme is optimum within a constant factor in rate.

Theorem:

• Information-theoretic bound

• The constant gap is uniform in the problem parameters

• No significant gains beside local and global

Asynchronous Delivery

Segment 1 Segment 2 Segment 3

Segment 1 Segment 2 Segment 3

Segment 1 Segment 2 Segment 3

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)