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Coded Caching with Linear Subpacketization is Possible using Ruzsa-Szem\'eredi Graphs
Karthikeyan Shanmugam, Antonia M. Tulino, Alexandros G. Dimakis
TL;DR
Prior coded-caching benefits appeared only when files were split into exponentially many packets as the number of users grew. This paper connects Ruzsa-Szeméredi graphs to centralized caching and achieves linear file packetization with transmissions K^δ for arbitrarily small δ.
Problem
Prior coded-caching benefits appeared only when file packetization F scaled exponentially with the number of users K, making those schemes non-practical.
Source: §1 · p. 2Method
A novel connection between Ruzsa-Szeméredi graphs and coded caching turns dense graph constructions into centralized caching schemes.
Source: §1 · p. 2Results
For any arbitrarily small constant δ, R=K^δ transmissions and F=K packets suffice for the stated centralized scheme.
Source: §1 · p. 2Takeaways & Limitations
The construction settles an outstanding existence question for schemes with linear file size and very small rate, but requires very large K and may be impractical.
Source: §3 · p. 7The informal scaling-law discussion assumes constant M/N and K growing large, although the mathematical claims also cover more general parameter scalings.
Source: §1 · p. 2
Abstract
from arXiv · showhide
Coded caching is a problem where encoded broadcasts are used to satisfy users requesting popular files and having caching capabilities. Recent work by Maddah-Ali and Niesen showed that it is possible to satisfy a scaling number of users with only a constant number of broadcast transmissions by exploiting coding and caching. Unfortunately, all previous schemes required the splitting of files into an exponential number of packets before the significant coding gains of caching appeared. The question of what can be achieved with polynomial subpacketization (in the number of users) has been a central open problem in this area. We resolve this problem and present the first coded caching scheme with polynomial (in fact, linear) subpacketization. We obtain a number of transmissions that is not constant, but can be any polynomial in the number of users with an exponent arbitrarily close to zero. Our central technical tool is a novel connection between Ruzsa-Szem\'eredi graphs and coded caching.
1 Introduction
Coded caching uses user caches as designed side information for broadcasts, and earlier work showed it could serve requests with a constant number of transmissions. This paper connects coded caching to Ruzsa-Szeméredi graphs to achieve linear subpacketization, while noting that the graph construction requires very large user populations.
1 Introduction
- Earlier coded caching work showed that coding and caching can serve requests with a constant number of transmissions, unlike conventional caching’s linear scaling in K.The introduction describes the constant-transmission result of Maddah-Ali and Niesen.Source: §1 · p. 2
- Exponential subpacketization was a central barrier: previous schemes needed files split into exponentially many packets before coding gains appeared.For decentralized schemes, the cited analysis required F→∞, and a later result gave a linear-transmission lower bound for F≤exp(KM/N).Source: §1 · p. 2
Our Contribution
- With F=K packets, the proposed scheme serves K users using R=K^δ transmissions for arbitrarily small constant δ.The introduction states this result for a constant cache-to-library ratio and gives a construction with M/N≥K^-ε.Source: §1 · p. 2
- The scheme’s key technical link is that dense Ruzsa-Szeméredi graphs yield coded caching with linear file size and constant or sub-constant cache-to-library ratio.The introduction also identifies Ruzsa-Szeméredi graphs as the basis for the construction.Source: §1 · p. 2
- The graph constructions used require very large K and do not seem directly applicable to real caching systems.The authors nevertheless describe linear packet size with near-optimal transmission count as a surprising result.Source: §1 · p. 3
2 Ruzsa-Szeméredi Graphs and Coded Caching
The section connects coded caching schemes to Ruzsa-Szeméredi graphs, using graph structure to guide cache placement and delivery. An Alon–Moitra–Sudakov construction yields linear subpacketization with a number of transmissions that can be nearly constant, including for sub-constant cache-to-library ratios.
2 Ruzsa-Szeméredi Graphs and Coded Caching
- The graph-to-caching connection is stated directly, with the scheme's feasibility established for any set of user demands.The proof checks cache constraints and shows that every demanded packet not already cached belongs to an edge covered by an induced matching.Source: §2 · pp. 4–5
- Each non-edge determines cache placement, while each induced matching determines an XOR transmission for delivery.The placement stores packets according to the endpoints of non-edges; the matching-based XOR lets users recover demanded packets.Source: §2 · pp. 4–5
- A (K, t, r, M/N) Ruzsa-Szeméredi scheme uses F = K packets per file, with delivery based on t induced-matchings.The delivery sends one packet transmission for each matching, corresponding to t/F file transmissions.Source: §2 · p. 4
2.1 The Ruzsa-Szeméredi Construction by Alon, Moitra and Sudakov
- The construction by Alon, Moitra and Sudakov supplies a dense Ruzsa-Szeméredi graph that can be used to obtain the scheme's transmission and cache parameters.The section describes the construction as having induced matchings and a lower bound on minimum degree, then applies the graph-to-caching result.Source: §2.1 · p. 6
- For any 0 < δ < 1 and sufficiently large K, the construction gives R = K^δ transmissions with F = K packets.The section also states that, for sub-constant M/N, file size remains linear in K while the number of transmissions is almost constant.Source: §2.1 · p. 6
3 Conclusion
The paper connects Ruzsa-Szeméredi graphs to coded caching and uses a dense construction to obtain schemes with linear file size and almost constant transmission rate. The construction requires very large user populations, motivating the search for dense graphs with fewer vertices.
3 Conclusion
- The paper establishes a connection between Ruzsa-Szeméredi graphs from extremal combinatorics and coded caching.Source: §3 · p. 7
- A dense Ruzsa-Szeméredi construction gives linear file size, sub-constant cache-to-library ratio, and almost constant transmission rate for any demand when the user population is large.The result settles an outstanding existence question for schemes with linear file size and very small rate.Source: §3 · p. 7
- The construction requires a very large number of users, so finding dense Ruzsa-Szeméredi graphs with fewer vertices remains an open direction.Source: §3 · p. 7