[Paper Review] Centralized coded caching schemes: A hypergraph theoretical approach

Chong Shangguan, Yiwei Zhang+1 more|arXiv (Cornell University)|Aug 13, 2016

Caching and Content Delivery9 references12 citations

TL;DR

This paper introduces a hypergraph-theoretic framework to design centralized coded caching schemes with constant transmission rate R and sub-exponential file partitioning F. By modeling placement delivery arrays (PDAs) as linear, (6,3)-free 3-uniform 3-partite hypergraphs, the authors prove that F cannot grow linearly with K for constant R, but construct two infinite families of schemes where F grows sub-exponentially, significantly reducing complexity compared to prior exponential constructions.

ABSTRACT

The centralized coded caching scheme is a technique proposed by Maddah-Ali and Niesen as a solution to reduce the network burden in peak times in a wireless system. Later Yan et al. reformulated the problem as designing a corresponding placement delivery array, and proposed two new schemes from this perspective. These schemes above significantly reduce the transmission rate $R$, compared with the uncoded caching scheme. However, to implement the new schemes, each file should be cut into $F$ pieces, where $F$ increases exponentially with the number of users $K$. Such constraint is obviously infeasible in the practical setting, especially when $K$ is large. Thus it is desirable to design caching schemes with constant rate $R$ (independent of $K$) as well as small $F$. In this paper we view the centralized coded caching problem in a hypergraph perspective and show that designing a feasible placement delivery array is equivalent to constructing a linear and (6, 3)-free 3-uniform 3-partite hypergraph. Several new results and constructions arise from our novel point of view. First, by using the famous (6, 3)-theorem in extremal combinatorics, we show that constant rate caching schemes with $F$ growing linearly with $K$ do not exist. Second, we present two infinite classes of centralized coded caching schemes, which include the schemes of Ali-Niesen and Yan et al. as special cases, respectively. Moreover, our constructions show that constant rate caching schemes with $F$ growing sub-exponentially with $K$ do exist.

Motivation & Objective

• To address the impractical exponential file partitioning F in existing constant-rate coded caching schemes.
• To establish a theoretical foundation linking coded caching design to extremal combinatorics via hypergraph structures.
• To construct new caching schemes with constant rate R and F growing sub-exponentially in K, improving on Ali-Niesen and Yan et al. schemes.
• To determine the fundamental limits of F for constant-rate centralized coded caching, particularly whether polynomial F suffices.

Proposed method

• Model the placement delivery array (PDA) design as constructing a linear, (6,3)-free 3-uniform 3-partite hypergraph.
• Apply the (6,3)-theorem from extremal combinatorics to prove that F cannot grow linearly with K for constant R.
• Construct two infinite families of caching schemes generalizing Ali-Niesen and Yan et al. schemes, parameterized by q.
• Use combinatorial designs and asymptotic analysis to bound F and R, showing sub-exponential growth of F is achievable.
• Establish equivalence between PDA design and regular partial Latin squares with the Blackburn property.
• Reformulate the problem using strong edge coloring in bipartite graphs, linking PDA existence to the strong chromatic index.

Experimental results

Research questions

• RQ1 Can centralized coded caching schemes with constant transmission rate R exist where the file partitioning F grows linearly with the number of users K?
• RQ2 Do there exist constant-rate caching schemes with F growing sub-exponentially in K, and if so, how small can F be?
• RQ3 Is it possible to construct caching schemes with F growing polynomially in K while maintaining constant R?
• RQ4 What is the minimal F required for a constant-rate scheme when M/N and R are fixed independently of K?

Key findings

• Constant-rate caching schemes with F growing linearly in K do not exist, as proven via the (6,3)-theorem in extremal combinatorics.
• Two new infinite families of centralized coded caching schemes are constructed, generalizing both the Ali-Niesen and Yan et al. schemes.
• For M/N = 1/q, the file partitioning F in the new constructions is reduced from Ω(q^{K/q}) to O(q^{√(8K)/q}), achieving sub-exponential growth.
• For M/N = (q-1)/q, F is reduced from Ω(q^{K/q}) to O(q^{√(K/2q)}), demonstrating significant improvement in complexity.
• The constructions show that constant-rate schemes with F growing sub-exponentially in K are achievable, resolving a key practical limitation of prior schemes.
• The equivalence between PDA design and regular partial Latin squares with the Blackburn property is formally established, offering a new combinatorial perspective.