[Paper Review] Constructions of Coded Caching Schemes with Flexible Memory Size

Minquan Cheng, Jing Jiang+2 more|arXiv (Cornell University)|Aug 22, 2017

Caching and Content Delivery25 references4 citations

TL;DR

This paper proposes four new coded caching schemes using placement delivery arrays (PDAs) to reduce subpacketization F while maintaining low delivery rate R. By constructing novel PDAs based on combinatorial designs and modular arithmetic, the authors achieve sub-exponential growth of F with user count K, significantly reducing computational complexity compared to the Maddah-Ali-Niesen scheme, with a tunable tradeoff between R and F.

ABSTRACT

Coded caching scheme recently has become quite popular in the wireless network due to its effectively reducing the transmission amount (denote such an amount by $R$) during peak traffic times. However to realize a coded caching scheme, each file must be divided into $F$ packets which usually increases the computation complexity of a coded caching scheme. So we prefer to construct a caching scheme that decreases the order of $F$ for practical implementations. In this paper, we construct four classes of new schemes where two classes can significantly reduce the value of $F$ by increasing a little $R$ comparing with the well known scheme proposed by Maddah-Ali and Niesen, and $F$ in the other two classes grows sub-exponentially with $K$ by sacrificing more $R$. It is worth noting that a tradeoff between $R$ and $F$, which is a hot topic in the field of caching scheme, is proposed by our constructions. In addition, our constructions include all the results constructed by Yan et al., (IEEE Trans. Inf. Theory 63, 5821-5833, 2017) and some main results obtained by Shangguan et al., (arXiv preprint arXiv:1608.03989v1) as the special cases.

Motivation & Objective

• Reduce the subpacketization level F in coded caching schemes to improve practical feasibility for large-scale systems.
• Address the high computational complexity caused by exponential growth of F in existing schemes like the Maddah-Ali-Niesen (MN) scheme.
• Establish a flexible tradeoff between delivery rate R and subpacketization F to balance performance and implementation cost.
• Generalize and unify prior constructions by showing they are special cases of the proposed framework.
• Develop new PDA-based constructions that allow for arbitrary memory sizes M/N and scalable user counts K.

Proposed method

• Propose two new classes of PDAs (Theorems 2 and 3) using combinatorial structures over finite fields and modular arithmetic to control F and R.
• Construct extended schemes (Theorems 4 and 5) by combining existing PDA structures with additional indexing and replacement rules to further reduce F.
• Define user and file indices using tuples over Z_q and subsets of indices, enabling structured placement and delivery.
• Use a mapping rule (Eq. 55) to assign coded delivery symbols to user-cache pairs based on index alignment and modular operations.
• Apply a replacement strategy to introduce *-entries in PDAs to simulate uncoded placement while preserving coded delivery efficiency.
• Prove correctness via construction and verification of PDA properties: each file symbol appears exactly once per user, and all demands are satisfied.

Experimental results

Research questions

• RQ1 Can we construct coded caching schemes with sub-exponential F growth in K while keeping R close to the MN scheme?
• RQ2 Is it possible to design PDAs that allow flexible memory ratios M/N and reduce F without sacrificing delivery rate beyond a fixed multiplicative factor?
• RQ3 How can existing constructions (e.g., Yan et al., Shangguan et al.) be unified and generalized under a single PDA-based framework?
• RQ4 What tradeoff between R and F can be achieved through structured PDA design, and how does it compare to known bounds?
• RQ5 Can new PDA constructions be systematically derived using algebraic and combinatorial methods like resolvable designs and Ruzsa-Szemerédi graphs?

Key findings

• Theorem 2 constructs a (K, F, Z, S) PDA with K = (m choose t) q^t, F = floor((q-1)/(q-z))^t q^m, Z/F = 1 - ((q-z)/q)^t, and S/F = ((q-z)/floor((q-1)/(q-z)))^t, achieving sub-exponential F growth with K.
• Theorem 3 provides a scheme with K = (m+1)q, F = floor((q-1)/(q-z)) q^m, M/N = z/q, and R = (q-z)/floor((q-1)/(q-z)), reducing F significantly with only a small increase in R compared to MN.
• Theorem 4 achieves F = q^m and R = (q-z)^t for K = (m choose t) floor((q-1)/(q-z))^t q^t, demonstrating a substantial reduction in F at the cost of higher R.
• Theorem 5 yields F = q^m and R = q-z for K = (m floor((q-1)/(q-z)) + 1)q, showing that F remains polynomial in m while R increases linearly with z.
• All four constructions generalize prior results: they include the Yan et al. and Shangguan et al. schemes as special cases.
• The proposed schemes establish a new, exact tradeoff between R and F, with F growing sub-exponentially in K for two classes and significantly reduced F for the other two, even at the cost of moderate R increase.