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Combinatorial Multi-Access Coded Caching with Private Caches

Dhruv Pratap Singh, Anjana A. Mahesh, B. Sundar Rajan

TL;DR

This work introduces the combinatorial multi-access with private (CMAP) coded caching model, where each user accesses a distinct r-subset of Λ access caches and also holds a private cache. It develops a centralized, uncoded-placement achievability that generalizes MAN and CMACC, and derives a complementary Alpha bound and a cut-set bound to characterize the optimal worst-case rate under general and uncoded placements. The results show the proposed scheme approaches the lower bound as the memory accessed by a user grows, with special optimality established for the Λ=4 case under certain parameters. The combination of index-coding based lower bounds, memory-sharing analysis, and numerical validation provides a thorough understanding of CMAP performance and benchmarks for future caching system designs.

Abstract

We consider a variant of the coded caching problem where users connect to two types of caches, called private and access caches. The problem setting consists of a server with a library of files and a set of access caches. Each user, equipped with a private cache, connects to a distinct $r-$subset of the access caches. The server populates both types of caches with files in uncoded format. For this setting, we provide an achievable scheme and derive a lower bound on the number of transmissions for this scheme. We also present a lower and upper bound for the optimal worst-case rate under uncoded placement for this setting using the rates of the Maddah-Ali--Niesen scheme for dedicated and combinatorial multi-access coded caching settings, respectively. Further, we derive a lower bound on the optimal worst-case rate for any general placement policy using cut-set arguments. We also provide numerical plots comparing the rate of the proposed achievability scheme with the above bounds, from which it can be observed that the proposed scheme approaches the lower bound when the amount of memory accessed by a user is large. Finally, we discuss the optimality w.r.t worst-case rate when the system has four access caches.

Combinatorial Multi-Access Coded Caching with Private Caches

TL;DR

This work introduces the combinatorial multi-access with private (CMAP) coded caching model, where each user accesses a distinct r-subset of Λ access caches and also holds a private cache. It develops a centralized, uncoded-placement achievability that generalizes MAN and CMACC, and derives a complementary Alpha bound and a cut-set bound to characterize the optimal worst-case rate under general and uncoded placements. The results show the proposed scheme approaches the lower bound as the memory accessed by a user grows, with special optimality established for the Λ=4 case under certain parameters. The combination of index-coding based lower bounds, memory-sharing analysis, and numerical validation provides a thorough understanding of CMAP performance and benchmarks for future caching system designs.

Abstract

We consider a variant of the coded caching problem where users connect to two types of caches, called private and access caches. The problem setting consists of a server with a library of files and a set of access caches. Each user, equipped with a private cache, connects to a distinct subset of the access caches. The server populates both types of caches with files in uncoded format. For this setting, we provide an achievable scheme and derive a lower bound on the number of transmissions for this scheme. We also present a lower and upper bound for the optimal worst-case rate under uncoded placement for this setting using the rates of the Maddah-Ali--Niesen scheme for dedicated and combinatorial multi-access coded caching settings, respectively. Further, we derive a lower bound on the optimal worst-case rate for any general placement policy using cut-set arguments. We also provide numerical plots comparing the rate of the proposed achievability scheme with the above bounds, from which it can be observed that the proposed scheme approaches the lower bound when the amount of memory accessed by a user is large. Finally, we discuss the optimality w.r.t worst-case rate when the system has four access caches.
Paper Structure (17 sections, 4 theorems, 35 equations, 6 figures)

This paper contains 17 sections, 4 theorems, 35 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. System Model and Preliminaries
  4. System Model
  5. MAN Scheme
  6. MAN Scheme for Combinatorial Multi-Access Coded Caching (CMACC) Network
  7. Index Coding Preliminaries
  8. Main Results
  9. Achievability and Lower Bound
  10. Achievability Scheme
  11. Alpha Bound
  12. Numerical Comparison
  13. Optimality for $\Lambda=4$ Case
  14. Optimality for $(\Lambda=4, N \geq K, r=2, t_a=1, t_p=1)$
  15. Optimality for $(\Lambda=4, N \geq K, r=2, t_a=1, t_p=2)$.
  16. ...and 2 more sections

Key Result

Proposition 1

For a $(\Lambda,r, M_a, M_p, N)-$CMAP coded caching system, the optimal worst-case rate $R_{UC}^{\textasteriskcentered}(M_a, M_p)$ under uncoded placement is bounded as: where, $R^{\textasteriskcentered}_{D}(M)$ is the rate achieved by MAN scheme MAN and $R^{\textasteriskcentered}_{CMACC}(M)$ is the rate achieved by MAN scheme for CMACC networkPD.

Figures (6)

  • Figure 1: The $(\Lambda,r,M_a,M_p,N)-$CMAP Coded Caching System.
  • Figure 2: Rate vs. $r$ and $t$ for $M_p = \frac{N}{K}$.
  • Figure 3: Rate vs. $t$ for $r=2$ and $M_p = \frac{N}{K}$.
  • Figure 4: Rate vs. $t$ for $r=3$ and $M_p = \frac{N}{K}$.
  • Figure 5: Rate vs. $r$ for $t=1$ and $M_p = \frac{N}{K}$.
  • ...and 1 more figures

Theorems & Definitions (21)

  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2: Achievability
  • proof
  • Theorem 3: Alpha Bound
  • proof
  • Example 1
  • Example 2
  • ...and 11 more