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PDA Construction via Union of Cartesian Product Cache Configurations for Coded Caching

Jinyu Wang, Minquan Cheng, Kai Wan, Giuseppe Caire

TL;DR

By applying the proposed construction of a PDA construction to existing base PDAs, three new coded caching schemes are obtained, which cover some existing schemes as special cases and can achieve lower load with simultaneously lower subpacketization for some memory ratios.

Abstract

Caching is an efficient technique to reduce peak traffic by storing popular content in local caches. Placement delivery array (PDA) proposed by Yan et al. is a combinatorial structure to design coded caching schemes with uncoded placement and one-shot linear delivery. By taking the $m$-fold Cartesian product of a small base PDA, Wang et al. constructed a big PDA while maintaining the memory ratio and transmission load unchanged, which achieves linear growth in both the number of users and coded caching gain. In order to achieve exponential growth in both the number of users and coded caching gain, in this paper we propose a PDA construction by taking the union operation of the cache configurations from the $m$-fold Cartesian product of a base PDA. The resulting PDA leads to a coded caching scheme with subpacketization increasing sub-exponentially with the number of users while keeping the load constant for fixed memory ratio. By applying the proposed construction to existing base PDAs, three new coded caching schemes are obtained, which cover some existing schemes as special cases and can achieve lower load with simultaneously lower subpacketization for some memory ratios.

PDA Construction via Union of Cartesian Product Cache Configurations for Coded Caching

TL;DR

By applying the proposed construction of a PDA construction to existing base PDAs, three new coded caching schemes are obtained, which cover some existing schemes as special cases and can achieve lower load with simultaneously lower subpacketization for some memory ratios.

Abstract

Caching is an efficient technique to reduce peak traffic by storing popular content in local caches. Placement delivery array (PDA) proposed by Yan et al. is a combinatorial structure to design coded caching schemes with uncoded placement and one-shot linear delivery. By taking the -fold Cartesian product of a small base PDA, Wang et al. constructed a big PDA while maintaining the memory ratio and transmission load unchanged, which achieves linear growth in both the number of users and coded caching gain. In order to achieve exponential growth in both the number of users and coded caching gain, in this paper we propose a PDA construction by taking the union operation of the cache configurations from the -fold Cartesian product of a base PDA. The resulting PDA leads to a coded caching scheme with subpacketization increasing sub-exponentially with the number of users while keeping the load constant for fixed memory ratio. By applying the proposed construction to existing base PDAs, three new coded caching schemes are obtained, which cover some existing schemes as special cases and can achieve lower load with simultaneously lower subpacketization for some memory ratios.
Paper Structure (12 sections, 8 theorems, 78 equations, 4 figures, 2 tables)

This paper contains 12 sections, 8 theorems, 78 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. System model and related works
  4. System Model
  5. Placement Delivery Array
  6. PDA construction via Cartesian product
  7. Main results
  8. The PDA construction in Theorem \ref{['thmulpmt']}
  9. Performance Analysis
  10. Comparison to Existing Shared-link Coded Caching Schemes
  11. conclusion
  12. Proof of Theorem \ref{['thmulpmt']}

Key Result

Lemma 1

(YCTC) If there exists a $(K,F,Z,S)$ PDA, there always exists a $(K,M,N)$ coded caching scheme with the memory ratio $\frac{M}{N}=\frac{Z}{F}$, subpacketization $F$ and load $R=\frac{S}{F}$.

Figures (4)

  • Figure 1: The steps of generating the $12$-user PDA $\mathbf{P}_{3}$ from the base PDA $\mathbf{P}$ in WCWC.
  • Figure 2: The first subarray of the placement array $\mathbf{P}_{3,2}^*$ and the corresponding subarray of the PDA $\mathbf{P}_{3,2}$.
  • Figure 3: The general steps of generating a PDA $\mathbf{P}_{m,t}$ from a base PDA $\mathbf{P}$, where $\mathbf{A}_1$ is the first subarray of the base PDA.
  • Figure 4: The memory-load and memory-subpacketization tradeoffs of Scheme A in Corollary \ref{['MNbs']} and the schemes in MNWCLCSJTLDYTCCCKSM.

Theorems & Definitions (15)

  • Definition 1
  • Lemma 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • Corollary 1
  • ...and 5 more