Coded Caching via Line Graphs of Bipartite Graphs

Abstract

We present a coded caching framework using line graphs of bipartite graphs. A clique cover of the line graph describes the uncached subfiles at users. A clique cover of the complement of the square of the line graph gives a transmission scheme that satisfies user demands. We then define a specific class of such caching line graphs, for which the subpacketization, rate, and uncached fraction of the coded caching problem can be captured via its graph theoretic parameters. We present a construction of such caching line graphs using projective geometry. The presented scheme has a rate bounded from above by a constant with subpacketization level q O((logq K) 2 ) and uncached fraction Θ( 1 √ K ), where K is the number of users and q is a prime power. We also present a subpacketizationdependent lower bound on the rate of coded caching schemes for a given broadcast setup.

Improving Placement Delivery Array Coded Caching Schemes With Coded Placement

“…It implies that this scheme maybe the best and the tradeoff is tight for some parameters. There are many other constructions focusing on further reducing the subpacketization by increasing the transmission rate, such as by the special (6, 3)-free hypergraphs [12], the resolvable combinatorial design and linear block codes [14], the (r, t) Ruzsa-Szeméredi graphs [7], [13], the strong edge coloring of bipartite graphs [9], projective space [16], etc. In [7], Shanmugam et al discovered that all deterministic F -division coded caching schemes can be recasted into a PDA when K ≤ N .…”

Some New Coded Caching Schemes With Smaller Subpacketization via Some Known Results

“…Obviously, the packet number F = K KM /N increases too quickly with K to be used in practice when K is large. There are some works paid attention to coded caching schemes with lower subpacketization level, for instances, [5], [11], [18], [20], [21], [26], [27] etc.…”

Coded Caching based on Combinatorial Designs

“…All of these schemes offered reductions in subpacketization as compared to [1], at the cost of some increase in the rate, for constant memory fraction M N . However, to the best of our knowledge, most of the schemes (for reasonable values of K) available in literature require subpacketization exponential in K. A subpacketization subexponential in K has been obtained in [5] using a line graph model for coded caching along with a projective geometry based scheme. For constant rate, the scheme in [5] achieves a subpacketization level of O(q (logq K) 2 ), however demanding that the uncached fraction, (1 − M N ) = Θ( 1 √ K ).…”

“…However, to the best of our knowledge, most of the schemes (for reasonable values of K) available in literature require subpacketization exponential in K. A subpacketization subexponential in K has been obtained in [5] using a line graph model for coded caching along with a projective geometry based scheme. For constant rate, the scheme in [5] achieves a subpacketization level of O(q (logq K) 2 ), however demanding that the uncached fraction, (1 − M N ) = Θ( 1 √ K ). The first contribution of this work is to present a new binary matrix model for coded caching.…”