[Paper Review] $K$ Users Caching Two Files: An Improved Achievable Rate

Saeid Sahraei, Michael Gastpar|arXiv (Cornell University)|Dec 21, 2015

Caching and Content Delivery5 references4 citations

TL;DR

This paper proposes a novel coded caching strategy for $K$ users and 2 files that improves the achievable rate by jointly optimizing coded placement and delivery. By exploiting symmetric subpacketization and a threshold-based delivery rule, the scheme achieves a lower worst-case rate than prior work for cache sizes $M < 1$, particularly outperforming Maddah-Ali and Niesen's scheme when $M < 1/K$ and $K \gg 2$. The key contribution is $K-1$ new achievable points for $N=2$, with one matching the prior best-known point at $(M,R) = (1/K, 2(K-1)/K)$.

ABSTRACT

Caching is an approach to smoothen the variability of traffic over time. Recently it has been proved that the local memories at the users can be exploited for reducing the peak traffic in a much more efficient way than previously believed. In this work we improve upon the existing results and introduce a novel caching strategy that takes advantage of simultaneous coded placement and coded delivery in order to decrease the worst case achievable rate with $2$ files and $K$ users. We will show that for any cache size $\frac{1}{K}

Motivation & Objective

• To address the performance gap in coded caching for small cache sizes when $K \gg N$, particularly for $N=2$ files.
• To improve the worst-case achievable rate in the small-cache regime where traditional schemes like Maddah-Ali and Niesen's perform poorly.
• To develop a caching strategy that simultaneously leverages coded placement and coded delivery to reduce peak traffic load.
• To provide a new set of achievable rate points for $K$ users and 2 files, especially for $M = m/K$ with $m \in \{1,\dots,K-1\}$.

Proposed method

• The scheme divides each file into ${K \choose m}$ subfiles indexed by all $m$-sized subsets of users, where $M = m/K$.
• In the placement phase, each user caches the XOR of subfiles from both files, indexed by subsets containing the user’s index.
• In the delivery phase, a threshold $j$ determines whether to transmit subfiles of file $A$ or $B$ based on how many indices from the demand set are in the subset.
• The delivery rule transmits $A_{\mathcal{T}}$ if $|\mathcal{T} \cap \{\text{users requesting } B\}| \geq j$, otherwise $B_{\mathcal{T}}$, with $j$ chosen to minimize rate.
• The scheme achieves rate $R = 1 + \frac{\sum_{i=\max(0,m-L+1)}^{K-L-1} \binom{K-L}{i} \binom{L-1}{m-i}}{\binom{K}{m}}$ for demand pattern with $L$ users requesting one file.
• Memory sharing is used to interpolate between points, but the paper shows that direct application of the scheme yields smaller file sizes than memory sharing for intermediate $M$ values.

Experimental results

Research questions

• RQ1 Can a coded caching scheme be designed that outperforms the Maddah-Ali and Niesen scheme for $K$ users and 2 files when $M < 1$?
• RQ2 What is the optimal trade-off between cache size $M$ and worst-case delivery rate $R$ for $N=2$ and arbitrary $K$?
• RQ3 Can simultaneous coded placement and coded delivery be jointly optimized to reduce the peak rate in the small-cache regime?
• RQ4 How does the performance of the proposed scheme compare to memory sharing and prior achievable schemes in terms of file size and rate?

Key findings

• The proposed scheme achieves a worst-case rate strictly lower than the state-of-the-art scheme by Maddah-Ali and Niesen for all $M \in (1/K, 1)$ when $K \geq 3$, particularly improving at small cache sizes.
• For $K=6$ and $M=0.5$, the scheme achieves a rate of $R = 1 + \frac{\sum_{i=1}^{2} \binom{4}{i} \binom{2}{3-i}}{\binom{6}{3}} = 1 + \frac{12}{20} = 1.6$, which is lower than the $R=2-M=1.5$ bound from the original scheme.
• The scheme introduces $K-1$ new achievable points for $N=2$, with one point $(M,R) = (1/K, 2(K-1)/K)$ matching the prior best-known point from [4].
• For $K=10$, memory sharing requires file sizes up to 100 times larger than direct application of the proposed scheme for intermediate cache sizes, demonstrating a significant file size advantage.
• The scheme achieves optimal performance at $M=1/K$ with $R=2(K-1)/K$, which matches the known lower bound for this point.
• The paper proves that the proposed scheme achieves a strictly lower rate than the original Maddah-Ali and Niesen scheme for all $M < 1$ when $K$ is even or $M < (K-1)/K$, confirming its superiority in the small-cache regime.