Coded Caching via Projective Geometry: A new low subpacketization scheme
aa r X i v : . [ c s . I T ] F e b Coded Caching via Projective Geometry:A new low subpacketization scheme
Hari Hara Suthan C, Bhavana M, Prasad KrishnanSignal Processing and Communications Research Center,International Institute of Information Technology, Hyderabad.Email: { hari.hara@research., bhavana.mvn@research., prasad.krishnan@ } iiit.ac.in Abstract —Coded Caching is a promising solution to reducethe peak traffic in broadcast networks by prefetching the pop-ular content close to end users and using coded transmissions.One of the chief issues of most coded caching schemes inliterature is the issue of large subpacketization , i.e., they requireeach file to be divided into a large number of subfiles. In thiswork, we present a coded caching scheme using line graphs ofbipartite graphs in conjunction with projective geometries overfinite fields. The presented scheme achieves a rate Θ( K log q K ) ( K being the number of users, q is some prime power)with subexponential subpacketization q O ((log q K ) ) when cachedfraction is upper bounded by a constant ( MN ≤ q α ) for somepositive integer α ). Compared to earlier schemes, the presentedscheme has a lower subpacketization (albeit possessing a higherrate). We also present a new subpacketization dependent lowerbound on the rate for caching schemes in which each subfile iscached in the same number of users. Compared to the previouslyknown bounds, this bound seems to perform better for a rangeof parameters of the caching system. I. I
NTRODUCTION
The key performance challenges that next generation wire-less networks (5G) face are low latency, high throughput andenergy efficiency [1]. Content delivery networks have beenestimated to carry of the global internet traffic by [2].
Coded caching was proposed recently in a landmarkpaper by Maddah-Ali and Niesen [3] and has emerged asan important tool to address major challenges of future com-munication networks. Since its inception coded caching hasproved as an efficient tool to trade-off expensive bandwidthwith abundantly available and cost-effective memory at theuser/network nodes.In [3], the setup consists of a single server with N equi-popular files of same size (divided into F subfiles each ofthe same size, where F is known as the subpacketization parameter), and K users(clients) each having a local memorycalled cache that can store M F subfiles. The centralizedcoded caching scheme of [3] works in two phases. In the caching phase (which occurs during off peak times) the cacheof each client is populated with some MN fraction of each filein the server. In the delivery phase (which happens duringpeak traffic times), the clients demand one file each from theserver, to satisfy which the server sends coded transmissions.The rate ( R ) of such a coded caching scheme is defined asthe ratio of the number of bits transmitted to the size of eachfile, which can be calculated asRate R = Number of transmissions in the delivery phaseNumber of subfiles in a file , when each transmission is of the same size as the subfiles.The Ali-Niesen scheme in [3] achieves R = K (1 − MN ) γ ,where γ = 1 + KMN is the global caching gain , i.e., thenumber of users served by each transmission in the deliveryscheme. This rate was shown to be optimal for uncoded cacheplacement [4]. Further, the subpacketization level used by theAli-Niesen scheme to achieve this rate is F = (cid:0) K KMN (cid:1) . Notethat as K grows large, F ≈ KH ( M/N ) , (for constant MN , H ( . ) being the binary entropy). This means that the fileshave to be extremely large for even - clients, makingthe Ali-Niesen scheme impractical for applications.Since then several new coded caching schemes with lowersubpacketization have been constructed at the cost of increasein rate, or cache requirement, or the number of users [5]–[8]. Among these, an important construction was reportedin [6], via a combinatorial object that the authors defined,known as Placement Delivery Arrays (PDAs). The PDAconstructed in [6] achieved a global caching gain of
MKN (oneless than that of [3]), while improving the subpacketizationby an exponential factor compared to [3]. However, in thisconstruction, as well as in most others in literature, thesubpacketization required for the caching schemes continuesto be exponential in K r (for some positive integer r ) to thebest of our knowledge.Recently, a line graph based approach to coded cachingwas introduced in [9]. Using this framework, a constructionfor a caching scheme was given via a projective geometryover a finite field. The scheme presented in [9] achievesa constant rate with subpacketization subexponential in K (cid:16) F = q O (( log q K ) ) (cid:17) for some prime power q ). However thedrawback of this scheme is that the uncached fraction of eachfile has to be large (cid:16) (1 − MN ) = Θ( √ K ) (cid:17) . We remedy thisdrawback (to some extent) in this work.The contributions and organization of this paper are asfollows. In Section II, we review the line graph based codedcaching scheme proposed in [9], while refining it slightlyfor our purposes. In Section III, we propose a new lowerbound for the optimal rate R ∗ given parameters K, F, and MN , for the caching schemes in which each subfile is cachedin the same number of users, which is a property satisfiedby all known centralized caching designs in literature (to thebest of our knowledge). Using some numerical examples,we see that this lower bound performs better (for a rangeof parameters) compared to the previously known boundsin [4], [7]. In Section IV, we present a new coded cachingcheme using projective geometries over finite fields in theline graph framework of [9]. In Section V we give theasymptotic analysis of the scheme proposed. We show thatthe scheme achieves a rate R = Θ (cid:16) Klog q K (cid:17) ( q being aprime power) for a constant cache requirement, which canbe extremely small ( MN ≤ q α , for some constant positiveinteger α ). The subpacketization achieved is subexponentialin K , F = q O ((log q K ) ) . We provide a table in Section Vwhich compares the parameters of our scheme to that of [6]. Notations and Terminology: Z + denotes the set of positiveintegers. We denote the set { , . . . , n } by [ n ] where n ∈ Z + .We give the basic definitions in graph theory. The sets V ( G ) , E ( G ) denote vertex set and edge set of a graph G respectively, where E ( G ) ⊆ {{ u, v } : u, v ∈ V ( G ) } . Allgraphs considered in this paper are undirected graphs with noself loops. The neighbourhood of a vertex u ∈ V ( G ) is givenas N ( u ) = { v ∈ V ( G ) : { u, v } ∈ E ( G ) } . The square of agraph G is a graph G having V ( G ) = V ( G ) and an edge { u, v } ∈ E ( G ) if and only if either { u, v } ∈ E ( G ) or thereexists some v ∈ V ( G ) such that { u, v } , { v , v } ∈ E ( G ) .The complement of a graph G is denoted as G . A set H ⊆ V ( G ) is called a clique of G if every two distinctvertices in H are adjacent to each other. A single vertex isalso considered as a clique by definition. A clique cover of G is a collection of disjoint cliques such that each vertexappears in precisely one clique. A bipartite graph B is agraph whose vertices can be partitioned into two independentsets (called left and right vertices of B ) such that edges existonly between left and right vertices. A bipartite graph is (leftor right) regular if the degree of each vertex on (left or right)is same throughout (left or right) partition. A bipartite graphis bi-regular if it is both left regular and right regular. Formore information on graph theory the reader is referred to[10]. II. R EVIEW OF L INE G RAPH SCHEME IN [9]We now review the basic framework and some results of[9]. Consider a coded caching system consisting of a serverwith files { W i : i ∈ [ N ] } . Let K be any set such that |K| = K . We shall use K to indicate the set of K users. Let F be any set such that |F| = F . The subfiles of a file W i aredenoted by W i,f where f ∈ F and W i,f takes values insome Abelian group. Here we consider symmetric caching ,i.e., for any f ∈ F , either a user caches W i,f , ∀ i ∈ [ N ] orthe user does not cache W i,f for any i ∈ [ N ] . Any symmetriccaching scheme can be represented as an equivalent D -leftregular bipartite caching graph B ( K, D, F ) with left verticesbeing K and right vertices being F , and the uncached fraction − MN = DF . The uncached subfiles are identified by theedges of B i.e, for k ∈ K , f ∈ F an edge { k, f } ∈ E ( B ) ifand only if the subfiles W i,f , ∀ i ∈ [ N ] are not present in thecache of user k . This bipartite coded caching setup was givenin [8]. In [9], a line graph based framework was proposedto study the coded caching problem. The line graph L ( G ) (or simply, L ) of an undirected graph G is a graph in whichthe vertex set V ( L ( G )) is the edge set E ( G ) of G , and twovertices of V ( L ( G )) are adjacent if and only if they share acommon vertex in G . The caching scheme was captured via a line graph L of a bipartite caching graph and the deliveryscheme was obtained as a clique cover of complement of thesquare of line graph denoted by L . The following lemmaproved in [9] presents the conditions under which an arbitrarygraph is a line graph of a left regular bipartite graph. Thisenables us to construct a line graph which corresponds to acoded caching scheme. Lemma 1. [9] A graph L containing KD vertices is the linegraph of a D -left-regular bipartite graph B ( K, D, F ) if andonly if the following conditions are satisfied. • The vertices of L can be partitioned into K disjointcliques containing D vertices each. We denote thesecliques by U k : k ∈ K and call them as the user-cliques. • Consider distinct k , k ∈ K . For any vertex v ∈ U k ,there exists at most one vertex w ∈ U k such that { v, w } ∈ E ( L ) . • For any k ∈ K and any vertex v ∈ U k , the set { v } ∪N ( v ) \ U k (containing v and all adjacent vertices of v except those in U k ), forms a clique. We refer to thesecliques as the subfile-cliques. • Let r be the number of subfile-cliques in L and thesubfile-cliques be denoted as S i : i ∈ [ r ] . Then thenumber of right vertices of B is F = r . Any graph L that satisfies the above conditions for some K and D is called as a caching line graph . Since there are r = F subfile cliques, we can denote the subfile-cliques as S f : f ∈ F . It also holds by the construction of L that there isat most one vertex in the intersection of any given user-clique U k and a subfile-clique S f . Further, note that the subfile-cliques also partition the vertices of L . Thus each vertex of L lies precisely in one user-clique and one subfile-clique.Therefore the vertices of L can be indexed using a subsetof K × F , i.e., V ( L ) = { ( k, f ) ∈ K × F : U k ∩ S f = φ } .With this notation, we have U k = { ( k, f ) ∈ V ( L ) : f ∈ F} and S f = { ( k, f ) ∈ V ( L ) : k ∈ K} . Furthermore, it followsthat E ( L ) = {{ ( k, f ) , ( k ′ , f ′ ) } ⊂ V ( L ) : k = k ′ or f = f ′ but not both } .Following [9], the delivery scheme follows according to aclique cover of L . In order to find the cliques in L (called transmission cliques ), first we need to identify the structureof L . It is easy to see that V ( L ) = V ( L ) . In Lemma2 we present the conditions under which an edge exist in L . We will use this lemma in Section IV to identify such transmission cliques in the construction we give. Lemma 2.
Let ( k , f ) , ( k , f ) ∈ V ( L ) . The edge { ( k , f ) , ( k , f ) } ∈ E ( L ) if and only if k = k , f = f and ( k , f ) , ( k , f ) / ∈ V ( L ) .Proof: The If part of the lemma follows from thedefinition of L . We prove the only if part here. Let { ( k , f ) , ( k , f ) } ∈ E ( L ) . Suppose k = k . Then bythe construction of L we have { ( k , f ) , ( k , f ) } ∈ E ( L ) .Therefore { ( k , f ) , ( k , f ) } / ∈ E ( L ) which is a con-tradiction. Hence k = k . Similarly f = f . Suppose ( k , f ) ∈ V ( L ) . Since ( k , f ) , ( k , f ) ∈ V ( L ) . By theconstruction of L we have { ( k , f ) , ( k , f ) } ∈ E ( L ) and { ( k , f ) , ( k , f ) } ∈ E ( L ) . By definition of L , we have ( k , f ) , ( k , f ) } ∈ E ( L ) which is a contradiction. Hence ( k , f ) / ∈ V ( L ) . Similarly ( k , f ) / ∈ V ( L ) .In [9] a particular class of caching line graphs called ( c, d ) - caching line graphs was considered. The advantage ofthese line graphs is that the corresponding caching schemeparameters are obtained naturally in a simple fashion. Thisis captured in the following definition and theorem. Ourconstruction in Section IV is also based on such ( c, d ) -caching line graphs. Definition 1. [9] A caching line graph L such that L has aclique cover consisting of c -sized disjoint subfile cliques and L has a clique cover consisting of d -sized disjoint cliques,for some c, d ∈ Z + , is called a ( c, d ) -caching line graph. Theorem 1. [9] Consider a ( c, d ) -caching line graph L .Then there is a coded caching scheme consisting of thecaching scheme given by L with F = KDc (and thus MN = 1 − cK ), and there is an associated transmission schemebased on the clique cover of L having rate R = cd . III. A
NEW LOWER BOUND ON THE RATE
In this section, we propose a lower bound on rate ofthe delivery scheme for symmetric caching schemes whereeach subfile is stored in equal number of users. Most knownschemes in literature satisfy this property to best of ourknowledge. From Section II, we know that any symmetriccaching scheme can be represented by a left-regular bipartitecaching graph B ( K, D, F ) where − MN = DF . As eachsubfile is not cached at equal number of users, the equivalentbipartite graph with KD edges will be right regular as well,with right degree being K (1 − MN ) for every subfile in B .We first recall a generic lower bound given in [9] basedon structure of B ( K, D, F ) which is used in Theorem 2 toarrive at our new bound. Let H be the subgraph of B inducedby the vertices K ′ ∪ F ′ where K ′ ⊆ K and F ′ ⊆ F . Let N ′ = min ( |K ′ | , K (1 − MN )) . Let U = { k j : j ∈ [ N ′ ] } be a subset of N ′ vertices of K ′ taken in some order. For j ∈ [ N ′ ] , let ρ j be the set of right vertices (subfiles) in H which are adjacent to { k i : i ∈ [ j ] } . Let R ∗ be the infimumof all achievable rates for coded caching problem defined by B . Then, from Theorem 2 of [9], R ∗ F ≥ N ′ X j =1 ρ j . (1) We now obtain a lower bound for the case of symmet-ric caching schemes defined by biregular bipartite cachinggraphs. The proof for this bound is based on a similar ‘nested’bound shown in [7].
Theorem 2.
Let R ∗ be the infimum of all achievable rates forthe coded caching problem defined by a bi-regular bipartitegraph B . Then R ∗ F ≥ D + & D ( K (1 − MN ) − K − ' + · · ·· · · + & KMN + 1 & KMN + 2 & · · · & D ( K (1 − MN ) − K − ' · · · ''' . Proof:
Every user vertex has degree D = F (1 − MN ) in B . Consider a user vertex in B and call it as k . So,by notations of (1), | ρ | = D . Consider the graph inducedby K ∪ N ( k ) vertices of B . Call it G ′ . Since B is abi-regular graph, degree of each subfile vertex f ∈ F ∈ B is exactly K (1 − MN ) . So, by pigeon-holing argument, it isnot difficult to see that there exists a user with degree atleast l ( K (1 − MN ) − DK − m in G ′ . Consider such user vertex andcall it as k . Then | ρ | ≥ l ( K (1 − MN ) − DK − m . In general, foreach j ∈ { , · · · , K (1 − MN ) − } , for the graph inducedby K ∪ N ( k ) ∪ N ( k ) · · · ∪ N ( k j − ) vertices of B , bypigeon-holing argument there exists a user with degreeat least l ( K (1 − MN ) − ( j − K − ( j − l · · · l ( K (1 − MN ) − DK − m · · · mm .Call such user vertex as k j . Then, | ρ j | ≥ l ( K (1 − MN ) − ( j − K − ( j − l · · · l D ( K (1 − MN ) − K − m · · · mm . Runningover all j , and using (1), we therefore get the bound in thetheorem.For a number of parameters we now compare (in Table I)the above new bound on the number of transmissions (column4 of Table I) in an optimal scheme, with the lower boundgiven in [7] (given in column 5, which holds for PDA basedschemes), as well as the lower bound (column 6) based onthe Ali-Niesen rate ( R ∗ ≥ K (1 − MN )1+ MKN ) as shown in [4]. Thebound given in [7] used in Table I is as follows.
Theorem 3. [7] R ∗ F ≥ (cid:6) DKF (cid:7) + l D − F − (cid:6) DKF (cid:7)m + · · ·· · · + l FMN +1 l FMN +2 (cid:6) · · · (cid:6) DKF (cid:7) · · · (cid:7)mm . It can be seen that for many of the parameters, our boundis better than those in [7], [4]. Further, the last column ofTable I denotes the rate achieved by the scheme in SectionIV in this work, for whichever parameters are applicable.
K F D [this work] [7] [4] Scheme[Sec IV] R ∗ F ≥ R ∗ F ≥ R ∗ F ≥ RF
15 50 30 71 54 65 NA24 54 36 109 90 96 NA15 20 12 30 31 26 NA7 42 24 43 33 42 5615 210 168 637 444 630 84013 156 108 285 193 280 468
TABLE I:
For some values of
K, F, D , we compare the lowerbound of this work with that of [7], [4]. The last column gives thenumber of transmissions in the scheme constructed in this paper forwhatever values are applicable.
IV. A
NEW PROJECTIVE GEOMETRY BASED SCHEME
In this section we present a new coded caching schemeusing projective geometries over finite fields. We first reviewsome basic concepts.
A. Review of projective geometries over finite fields [11]
Let k, q ∈ Z + such that q is a prime power. Consider a k -dim (we use “dim” for dimensional) vector space F kq overa finite field F q . Consider an equivalence relation on F kq \{ } (where represents the zero vector) whose equivalenceclasses are -dim subspaces(without ) of F kq . The set ofhese equivalence classes is called the ( k − -dim projectivespace over F q and is denoted by P G q ( k − . For m ∈ [ k ] , let P G q ( k − , m − denote the set of all m -dim subspaces of F kq . It is known that (Chapter in [11]) | P G q ( k − , m − | is equal to the q-binomial coefficient (cid:20) km (cid:21) q , where (cid:20) km (cid:21) q = ( q k − ... ( q k − m +1 − q m − ... ( q − . The following result is known from [11].
Lemma 3 (Chapter 3 in [11]) . Consider a k -dim vector space F kq . Let ≤ r, s, l < k . Then the number of r -dim subspacesintersecting a fixed s -dim subspace in a fixed l -dim subspaceis q ( r − l )( s − l ) (cid:20) k − sr − l (cid:21) q . We now proceed to construct a caching line graph usingprojective geometry.
B. A new caching line graph using projective geometry
Consider k, m, t ∈ Z + such that m + t ≤ k . Let W be afixed ( t − -dim subspace of the vector space F kq .Let V , { V ∈ P G q ( k − , t −
1) : W ⊆ V } . P , { P ∈ P G q ( k − , m + t −
1) : W ⊆ P } . X , ( { V , V , · · · , V m +1 } : ∀ V i ∈ V , m +1 X i =1 V i ∈ P ) . We first initialize L by its user-cliques. The user-cliques areindexed by t -dim subspaces in V . For each V ∈ V createthe vertices corresponding to the user-clique indexed by V as C V , ( ( V, X ) : X ∈ X , V * P V i ∈ X V i ) . Now, for each X ∈ X we construct the subfile clique of L associated with X as C X , ( ( V, X ) : V ∈ V , V * P V i ∈ X V i ) . By definition,these subfile cliques partition the set of vertices in L (theunion of all the user cliques). By invoking the notations fromSection II (Lemma 1), we have K = | V | (number of user-cliques), and subpacketization F = | X | (the number of subfilecliques). We now find the values of K , and the size of thesubfile cliques and the user cliques. Lemma 4. K = (cid:20) k − t + 11 (cid:21) q . | C X | = q m +1 (cid:20) k − m − t (cid:21) q ( for any X ∈ X ) . | C V | = (cid:20) k − tm + 1 (cid:21) q q m +1 m Q i =0 ( q m +1 − q i )( q − m +1 ( m + 1)! ( for any V ∈ V ) . Proof: K = | V | is the number of t -dim subspacesintersecting the fixed ( t − -dim subspace W in the fixed ( t − -dim subspace W . By Lemma 3, we have K = q ( t − t +1)( t − − t +1) (cid:20) k − t + 1 t − t + 1 (cid:21) q = (cid:20) k − t + 11 (cid:21) q . | C X | ( for any X ∈ X ) is the number of t -dim subspacesintersecting the fixed ( m + t ) -dim subspace P V i ∈ X V i in the fixed ( t − -dim subspace W . By Lemma 3, we have | C X | = q ( t − t +1)( m + t − t +1) (cid:20) k − m − tt − t + 1 (cid:21) q = q m +1 (cid:20) k − m − t (cid:21) q .We now obtain | C V | for any V ∈ V . To do this, wewill first find the number (say h ) of ( m + t ) -dim sub-spaces P ∈ P intersecting the fixed t -dim subspace V in the fixed ( t − -dim space W . By using Lemma 3,we thus have h = q ( m + t − t +1)( t − t +1) (cid:20) k − tm + t − t + 1 (cid:21) q = q m +1 (cid:20) k − tm + 1 (cid:21) q . Now, we find the number(say g ) of X ∈ X such that P V i ∈ X V i = P for some P ∈ P . Then it follows that | C V | = hg .Now, to find g , we first prove a few smaller claims. Claim 1:
The number of one dimensional spaces A suchthat W ⊕ A = V ( V being a fixed t -dimensional subspace, ⊕ representing direct sum) is q t − . Proof of Claim 1:
To see this, observe that to satisfy W ⊕ A = V , we must have A = span ( a ) for some a ∈ V \ W. Thusthere are q t − q t − = q t − ( q − choices for a . Howeverthere are precisely q − vectors a whose span is the sameone-dimensional subspace A . Hence we have that the numberof one dimensional spaces A such that W ⊕ A = V is q t − . Claim 2:
Let X = { V , . . . , V m +1 } be some fixedelement in X . The number N of ( m + 1) -sized sets { A , A , · · · , A m +1 } (where A i , ∀ i ∈ [ m +1] are -dim sub-spaces) such that { W ⊕ A , W ⊕ A , · · · , W ⊕ A m +1 } = X is q ( t − m +1) . Proof of Claim 2 : By Claim 1, the number of A i suchthat W ⊕ A i = V i is q t − . As A i s can be independentlychosen to get the corresponding V i s, we thus have that N is q ( t − m +1) . Claim 3: The number g ′ of ( m + 1) -sized sets { A , A , · · · , A m +1 } (where A i , ∀ i ∈ [ m + 1] are -dim subspaces) such that W ⊕ A ⊕ A ⊕ · · · A m +1 = P (for some fixed P ∈ P ) is m Q i =0 ( q m + t − q t − i )( q − m +1 ( m + 1)! . Proof of Claim 3:
Firstly, we note that there exists a set { A , ..., A m +1 } of -dim subspaces such that W ⊕ m +1 L i =1 A i = P , if and only if there exists a (not necessarily unique) setof vectors a i : i ∈ [ m + 1] such that A i = span ( { a i } ) and a i ∈ P \ ( W ⊕ i − L j =1 A j ) , for all i ∈ [ m + 1] . We call sucha set { a i : i = 1 , .., m + 1 } as a generating set of the set { A i : i = 1 , .., m + 1 } .Note that given a set of -dim subspaces { A i : i =1 , .., m +1 } , we can get a generating set { a i : i = 1 , .., m +1 } by choosing any a i ∈ A i \{ } for each i ∈ [ m + 1] . Now,suppose the set { a i : i ∈ [ m + 1] } is a generating set for { A i : i ∈ [ m + 1] } , then so is { c i a i : i ∈ [ m + 1] } , for any c i ∈ F q \{ } . Thus, the number of such distinct generatingsets for any given set of -dim subspaces { A i : i ∈ [ m + 1] } is ( q − m +1 .Now the number of ways to choose an ordered sets ofvectors { a i : i ∈ [ m + 1] } such that a i ∈ P \ ( W ⊕ i − L j =1 A j ) ,or all i ∈ [ m +1] , is m Q i =0 ( q m + t − q t − i ) . Thus, the number of(unordered) such generating sets { a i : i ∈ [ m + 1] } is then m Q i =0 ( q m + t − q t − i )( m +1)! . By arguments in the previous paragraph,it can be seen that these unordered generating sets can bepartitioned into groups of ( q − m +1 , such that each suchgroup includes precisely the set of all generating sets for aparticular set of -dim spaces { A i : i ∈ [ m + 1] } such that W ⊕ m +1 L i =1 A i = P .Thus the number g ′ we are looking for is precisely m Q i =0 ( q m + t − q t − i )( q − m +1 ( m + 1)! . This proves Claim 3.We now prove that the number g of X ∈ X such that P V i ∈ X V i = P for some P ∈ P is m Q i =0 ( q m +1 − q i )( q − m +1 ( m +1)! . To seethis, note that for each X = { V , ..., V m +1 } ∈ X such that P V i ∈ X V i = P , there exists precisely q ( t − m +1) sets of -dimsubspaces { A , .., A m +1 } such that { W ⊕ A i : i ∈ [ m +1] } = { V i : i ∈ [ m + 1] } . By Claim 3, the total number of sets of -dim subspaces { A , .., A m +1 } such that W ⊕ m +1 L i =1 A i = P is g ′ , and these can be partitioned into groups each of size q ( t − m +1) , such that for each set { A , .., A m +1 } in anyparticular group, the set X = { V i = W ⊕ A i : i ∈ [ m +1] } ∈ X is the same. Thus we have that g = g ′ q ( t − m +1) = m Q i =0 ( q m +1 − q i )( q − m +1 ( m + 1)! . Finally, we see that the expression for | C V | = hg matchesthe lemma statement, which proves the lemma.Note that by Lemma 4, we have the size of the subfilecliques of L as | C X | (for any X ∈ X ). We now showthat L has a clique cover with d -sized disjoint cliques forsome d . Therefore L is in fact a ( c = | C X | , d ) -caching linegraph, giving raise to the main result in this section whichis Theorem 4. C. Delivery Scheme from a clique cover of L We first describe a clique of L and show that such equal-sized cliques partition V ( L ) = V ( L ) . This will suffice toshow the delivery scheme as per Theorem 1.Let Y = n { V , V , · · · , V m +2 } : V i ∈ V , ∀ i ∈ [ m + 2] such that m +2 P i =1 V i ∈ P G q ( k − , m + t ) o . We now present aclique of size ( m + 2) in L . Lemma 5.
Consider Y = { V , V , · · · , V m +2 } ∈ Y . Then C Y = { ( V i , Y \ V i ) , ∀ V i ∈ Y } ⊂ V ( L ) is a clique in L .Proof: First note that C Y is well defined as V i * m +2 P l =1 l = i V l (otherwise m +2 P i =1 V i will not be a ( m + t + 1) -dim space) andhence ( V i , Y \ V i ) ∈ V ( L ) . Consider two distinct vertices ( V i , Y \ V i ) , ( V j , Y \ V j ) ∈ C Y . It is clear that V i = V j and Y \ V i = Y \ V j . By the construction of C Y we have V i ⊆ m +2 P l =1 l = j V l and V j ⊆ m +2 P l =1 l = i V l . Therefore we have ( V i , Y \ V j ) , ( V j , Y \ V i ) / ∈ V ( L ) . By invoking Lemma 2, { ( V i , Y \ V i ) , ( V j , Y \ V j ) } ∈ E ( L ) . Hence proved.Now we show that the cliques { C Y : Y ∈ Y } partition V ( L ) . Lemma 6. S Y ∈ Y C Y = V ( L ) , where the union is a disjointunion.Proof: Consider
Y, Y ′ ∈ Y such that Y = Y ′ . Bydefinition of C Y , we have C Y ∩ C Y ′ = φ . Now consider anarbitrary vertex ( V , { V , V , · · · , V m +2 } ) ∈ V ( L ) . By theconstruction of L , m +2 P i =1 V i ∈ P G q ( k − , m + t ) . Therefore ( V , { V , V , · · · , V m +2 } ) lies in the clique, C { V ,V , ··· ,V m +2 } (defined as in Lemma 5). Hence proved.Finally we present our coded caching scheme using thecaching line graph constructed above. Theorem 4.
The caching line graph L constructed above is a c = q m +1 (cid:20) k − m − t (cid:21) q , d = m + 2 ! -caching line graphand defines a coded caching scheme with K = (cid:20) k − t + 11 (cid:21) q , F = (cid:20) k − t + 1 m + 1 (cid:21) q m Q i =0 ( q m +1 − q i )( q − m +1 ( m + 1)! , MN = 1 − q m +1 (cid:20) k − tm + 1 (cid:21) q (cid:20) k − t + 1 m + 1 (cid:21) q , and R = q m +1 (cid:20) k − m − t (cid:21) q m + 2 .Proof: From Lemma 4 and the notations in Lemma 1,we get the expression of K and D = | C V | . Further we seethat the subfile cliques partition the vertices of L by definitionand also c = | C X | for any X ∈ X (the size of each subfileclique). By Lemma 5 and Lemma 6, the size of the cliquesof L is ( m + 2) and they partition the vertices. Hence L is a c = q m +1 (cid:20) k − m − t (cid:21) q , d = m + 2 ! -caching line graph.Thus, we have by Theorem 1, F = KDc = (cid:20) k − t + 1 m + 1 (cid:21) q m Q i =0 ( q m +1 − q i )( q − m +1 ( m + 1)! .MN = 1 − cK = 1 − q m +1 (cid:20) k − tm + 1 (cid:21) q (cid:20) k − t + 1 m + 1 (cid:21) q . Since (cid:20) k − t + 11 (cid:21) q (cid:20) k − m − t (cid:21) q = (cid:20) k − t + 1 m + 1 (cid:21) q (cid:20) k − tm + 1 (cid:21) q . = cd = q m +1 (cid:20) k − m − t (cid:21) q m + 2 . This completes the proof.V. A
SYMPTOTIC ANALYSIS OF THE PROPOSED SCHEME
In this section, we analyse the behaviour of
R, F for thecoded caching scheme proposed in Section IV as − MN islower bounded by a constant and K → ∞ . We show that F = q O (( log q K ) ) , while R = Θ( Klog q K ) . Towards this end,we first give some bounds on q -binomial coefficients. Thesecan be easily derived, however a proof is available in [9]. Lemma 7. [9] For non-negative integers a, b, f , for q beingsome prime power, q ( a − b ) b ≤ (cid:20) ab (cid:21) q ≤ q ( a − b +1) b q ( a − f − b − δ ≤ (cid:20) ab (cid:21) q (cid:20) af (cid:21) q ≤ q ( a − f − b +1) δ , where δ = | b − f | . We now proceed to analyse the asymptotics of the scheme.Throughout our analysis we assume q is constant. Consider, − MN = q m +1 ( q k − t − m − q k − t +1 − ≥ q ( m − k + t ) ( q k − t − m − . To lower bound − MN by a constant, let k − m − t = α ,where α is a constant. Note that α ≥ as m + t ≤ k . Wehave K = (cid:20) k − t + 11 (cid:21) q . We analyse our scheme as ( k − t ) grows large. By Lemma 7, we have q k − t ≤ K ≤ q k − t +1 .Hence we have log q K − ≤ ( k − t ) ≤ log q K. (2)The rate expression in Theorem 4 can be written as R = K (1 − MN ) m + 2 = K (1 − MN ) k − t − α + 2 .Therefore by using (2) we have K (1 − MN )log q K − α +2 ≤ R ≤ K (1 − MN )log q K − − α +2 , Consider, R ≤ K (1 − MN )log q K − − α +2 = K (1 − MN )log q K (1 − α − q K ) − .Now by Taylor’s series expansion we have ≤ (1 − α − q K ) − ≤ as K → ∞ . Therefore we have R ≤ K (1 − MN )log q K . Hence R = O ( K log q K ) .Consider, R ≥ K (1 − MN )log q K − α +2 = K (1 − MN )log q K (1 − α − q K ) − .Now if α ≥ , by Taylor’s series expansion we have ≤ (1 − α − K ) − ≤ as K → ∞ . Therefore we have R ≥ K log q K . If α ∈ { , } , then R ≥ K (1 − MN )log q K − α +2 ≥ K (1 − MN )2 log q K .Hence R = Ω( K log q K ) . Therefore R = Θ (cid:18) K log q K (cid:19) .We now obtain the asymptotics for subpacketization F . ByLemma 7 and from the expression of F in Theorem 4, wehave ( k, m, t, q ) ( m ′ , q ′ ) K K U U F F γ γ [6] [6] [6] [6] (10 , , ,
2) (6 ,
511 511 0.98 0.98 (9 , , ,
2) (14 ,
255 255 0.94 0.94 (8 , , ,
2) (13 ,
127 126 0.88 0.88 (9 , , ,
2) (31 ,
127 128 0.76 0.75 (7 , , ,
2) (15 ,
63 64 0.76 0.75 (7 , , , (39,3)121 120 0.67 0.66 (6 , , ,
2) (14 ,
31 30 0.51 0.50 TABLE II: For some specific values of
K, U = 1 − MN , wecompare the results of [6] with this work. F = K. (cid:20) k − t + 1 m + 1 (cid:21) q K . m Q i =0 ( q m +1 − q i )( q − m +1 ( m + 1)! ≤ Kq ( k − t − m ) m m Q i =0 q i ( q m +1 − i − q − m +1 ( m + 1)!= Kq αm ( m + 1)! m Y i =0 q i (cid:20) m + 1 − i (cid:21) q . Once again, by applying Lemma 7 to the above expression,we have, F ≤ Kq αm ( m +1)! m Q i =0 q i q m +1 − i = q log q K + αm +( m +1)2 ( m +1)! . As m + t ≤ k , thus q m ≤ q k − t ≤ K (by Lemma 7). Hence m ≤ log q K . Also by (2) we have, log q K − ≤ ( m + α ) , which can be written as m +1)! ≤ ⌊ log q K − α ⌋ ! . UsingStirling’s approximation for y ! as √ πx ( ye ) x for large y , andafter some simplifications we see that F = q O ( (log q K ) ) .Finally in Table II, we compare the scheme in Theorem 4with the scheme in [6] for some choices of K, U = 1 − MN , F and γ (the global caching gain, i.e., K (1 − MN ) R , where R is therate achieved by the scheme). We label the parameters of ourscheme in Theorem 4 as K , U , F , γ where γ = d . Theparameters of the scheme presented in [6] are K = q ′ ( m ′ +1) , U = 1 − q ′ , F = ( q ′ ) ( m ′ ) , γ = K (1 − MN ) q ′ − where q ′ , m ′ ∈ Z + . The first column lists ( k, m, t, q ) , K according to theTheorem 4. The second column lists ( m ′ , q ′ ) , K parametersof the scheme in [6]. Since we can not match the parametersfrom this work and [6] exactly, we choose approximatelyequal values. We see from the table that our scheme performsmuch better than [6] in terms of the subpacketization, butpays a price in terms of the rate.R EFERENCES[1] Y. Fadlallah, A. M. Tulino, D. Barone, G. Vettigli, J. Llorca, andJ. Gorce, “Coding for caching in 5g networks,”
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