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An Optimal Linear Error Correcting Scheme for Shared Caching with Small Cache Sizes

Sonu Rathi, Anoop Thomas and Monolina Dutta

Indian Institute of Technology Bhubaneswar, India

IEEE International Symposium on Information Theory (ISIT) Los Angeles, California, USA June 2020

Outline

◮ Introduction:

◮ Review of Coded Caching ◮ Shared Caching Problem

◮ Optimality of Caching Scheme for Shared Caching Problem ◮ Error Correcting Delivery Scheme ◮ Conclusion ◮ Future Research

Review of Coded Caching

K users caches size M server N files shared link

◮ Placement Phase:

◮ Cache contents are filled during off peak hours.

• M. A. Maddah-Ali and U. Niesen, “Fundamental limits of caching,” IEEE Trans. Inf. Theory, vol. 60, no. 5, pp.

2856–2867, May 2014.

Review of Coded Caching

K users caches size M server N files shared link

◮ Delivery Phase:

◮ The users reveal their demands to the server, d = (d1, d2, . . . , dK). ◮ Server has to develop encoding schemes to satisfy the user demands by making use of the caches.

Review of Coded Caching

K users caches size M server N files shared link

◮ Coded caching problem:

◮ Design of placement and delivery phase to reduce the traffic rate of network.

Shared Caching (SC) Problem

Server N Files Shared link K Users Cache 1 Cache Λ Size M Size M

◮ N files, K users and Λ helper nodes. ◮ Each user has access to a helper cache. ◮ Each helper cache accessed by multiple users.

• E. Parrinello, A. ¨

Unsal and P. Elia, ”Fundamental Limits of Coded Caching With Multiple Antennas, Shared Caches and Uncoded Prefetching,” in IEEE Transactions on Information Theory, vol. 66, no. 4, pp. 2252-2268, April 2020.

Shared Caching (SC) Problem

Server N Files Shared link K Users Cache 1 Cache Λ Size M Size M

◮ SC Scheme involves three steps:

• 1. Placement phase.
• 2. User-to-cache association phase: L = (L1, L2, . . . , LΛ) fully describes

the number of users associated to each cache.

• 3. Delivery phase.

Contribution of Paper

◮ For a certain cache memory size and for the case where number of users is equal to the number of files, the optimality of the peak rate is proved by using cut-set bound techniques. ◮ The optimality of the delivery scheme is proved by using techniques from index coding. ◮ An optimal linear error correcting scheme is proposed for a special case

• f the shared caching problem. The expressions for the average rate

and the peak rate of the error correcting delivery scheme are obtained.

Rate of the Shared Caching Scheme for N = K, M = 1

Λ

◮ For the shared cache scheme using coded placement, the delivery rate for N = K and M = 1

Λ is given as

R(L, ZSC, d, δ = 0) =

• N − ML0,

if Ne(d) = N. Ne(d), if Ne(d) < N. ◮ ZSC is the placement scheme of shared cache scheme. ◮ Ne(d) is the number of distinct demands of users.

• H. Xu, C. Gong and X. Wang, ”Efficient File Delivery for Coded Prefetching in Shared Cache Networks With Multiple

Requests Per User,” in Proc. 2019 IEEE Transactions on Communications, vol. 67, no. 4, April 2019, pp. 2849-2865.

Optimality of the Peak Rate

◮ For N ∈ N files, N users each with cache memory size M, and cache association profile L, the peak rate R∗

peak(L) ≥ N − ML0

. ◮ By the cut-set bound, we have R∗

peak(L)F + L0MF ≥ NF.

◮ On rearranging, we obtain R∗

peak(L) ≥ N − ML0

. ◮ Lower bound is coincide with the rate achieved.

Optimality of Non-distinct Demands

◮ Optimality is proved by using techniques from Index Coding Problem (ICP). ◮ ICP has a central server possessing n messages and m receivers. ◮ Side information: each node possess a subset of files present at server. ◮ The server transmits a message so that each node recover its demanded files by using side information and transmitted message. ◮ The aim is to characterize the minimum number of transmissions for any given instance of the ICP.

• Y. Birk and T. Kol, “Coding-on-demand by an informed source (ISCOD) for efficient broadcast of different

supplemental data to caching clients,” IEEE Trans. Inf. Theory, vol. 52, no. 6, pp. 2825-2830, Jun. 2006.

Optimality of Non-distinct Demands

◮ Delivery phase of every coded caching can be represented as ICP. ◮ Delivery phase of a (K, N, M) caching scheme corresponds to NK index coding problems. ◮ Since the placement is coded, the delivery phase corresponds to a Generalized Index Coding Problem (GICP). ◮ The generalized index coding problem corresponding to the shared caching problem, using placement scheme ZSC, user-to-cache association profile L, and demand d is denoted by I(ZSC, L, d).

Optimality of Non-distinct Demands

◮ There are two important parameters for a generalized index coding problem.

◮ κ(I)- min-rank of the generalized index coding problem I ◮ α(I)- generalized independence number.

◮ Upper bound on the length of an optimal linear index code for the index coding problem I(ZSC, L, d) κ(I(ZSC, L, d)) ≤

• NΛ − L0,

if Ne(d) = N. Ne(d)Λ, if Ne(d) < N. ◮ Lower bound on the length of an optimal linear index code for the index coding problem I(ZSC, L, d) α(I(ZSC, L, d)) =

• NΛ − L0,

if Ne(d) = N. Ne(d)Λ, if Ne(d) < N.

Example of Shared Caching Problem Showing Optimality

◮ Consider a shared caching problem with N = K = 4, and Λ = 3. Each cache is of size M = 1

3. Cache association profile L1 = (2, 1, 1) L1 = (3, 1, 0) L2 = (4, 0, 0) d Ne(d) α(I1) κ(I1) α(I2) κ(I2) α(I3) κ(I3) (1, 2, 3, 4) 4 9 9 10 10 11 11 (1, 2, 3, 1) 3 9 9 9 9 9 9 (1, 2, 1, 2) 2 6 6 6 6 6 6 (1, 1, 1, 1) 1 3 3 3 3 3 3

◮ The generalized independence number is equal to the min-rank for each of the different demands d. ◮ This proves the optimality of the delivery scheme of shared caching scheme for all values of d.

Problem Statement

Server N Files Shared link K Users Cache 1 Cache Λ Size M Size M Error prone

◮ The links between server and the end users may be error prone. ◮ Finite number of transmission errors can occur during delivery phase. ◮ Placement phase is assumed to be error free. ◮ Users must be able to decode their demands in the presence of errors.

• N. S. Karat, S. Dey, A. Thomas, and B. S. Rajan, “An optimal linear error correcting delivery scheme for coded caching

with shared caches,” in Proc. 2019 IEEE International Symposium on Information Theory (ISIT), Jul. 2019, pp. 1217–1221.

Error Correcting Delivery Scheme

◮ An optimal error correcting delivery scheme can be obtained by concatenating a classical error correcting code capable of correcting δ errors and an optimal delivery scheme. ◮ The rate of an optimal delivery error correcting scheme is obtained from a optimal error correcting delivery scheme for a corresponding generalized index coding problem. ◮ For a shared caching problem, with N = K and M = 1/Λ the optimal rate of δ-error correcting delivery scheme is R∗(L, ZSC, d, δ) = 1 ΛNq[κ(I(ZSC, L, d)), 2δ + 1] .

Error Correcting Delivery Scheme

◮ Optimal scheme obtained by using α-bound and κ-bound Nq[α(I), 2δ + 1]

• α-bound

≤ Nq[I, δ] ≤ Nq[κ(I), 2δ + 1]

• κ-bound

◮ For a shared caching problem with N = K and M = 1/Λ, the average rate of an optimal error correcting delivery scheme with δ-error correcting capability is 1 ΛEd

• Nq[κ(I(ZSC, L, d)), 2δ + 1]
• .

Conclusions

◮ In shared cache problem optimality is proved for the case where number of files is equal to number of users and when the cache size M = 1

Λ using cut set bound techniques.

◮ In shared cache problem optimality is proved for the case where number of files is equal to number of users and when the cache size M = 1

Λ using index coding.

◮ Expressions for the peak and average rate for linear error correcting delivery scheme is obtained.

Future Research

◮ Extension to unequal cache size ◮ Extension to other memory values ◮ Extension to multi-level caching.

Thank You