Maddah-Ali-Niesen Scheme for Multi-access Coded Caching
aa r X i v : . [ c s . I T ] J a n Maddah-Ali-Niesen Scheme for Multi-access CodedCaching
Pooja Nayak Muralidhar, Digvijay Katyal and B. Sundar RajanDepartment of Electrical Communication Engineering, Indian Institute of Science, Bengaluru 560012, KA, IndiaE-mail: {poojam,digvijayk,bsrajan}@iisc.ac.in
Abstract —The well known Maddah-Ali-Niesen (MAN) codedcaching scheme for users with dedicated cache is extended for usein multi-access coded cache scheme where the number of usersneed not be same as the number of caches in the system. The wellknown MAN scheme is recoverable as a special case of the multi-access system considered. The performance of this scheme iscompared with the existing works on multi-access coded caching.To be able to compare the performance of different multi-accessschemes with different number of users for the same numberof caches, the terminology of per user rate (rate divided by thenumber of users) introduced in [17] is used.
I. I
NTRODUCTION
Due to the increasing number of multimedia applicationslike video on demand, tremendous growth in the consumptionof data has been observed in recent years. The seminal workof [1] showed that jointly designing content placement anddelivery, also known as coded caching significantly improvescontent delivery rate requirements. However in reality, dueto practical constraints the subpacketization levels in [1] isnot feasible due to its exponentially increasing nature, withrespect to the increasing number of users. The quest for codedcaching schemes with practical subpacketization levels started,and a wide variety of schemes using different constructions ofplacement delivery arrays [2], line graphs of bipartite graphs[3], linear block codes [4], block designs [5] etc, were found.Parallelly, this technique of jointly designing placement anddelivery were explored in other type of networks like deviceto device networks (D2D) [6], Combination networks [7],networks with shared cache [8], [9], [10] etc. The case ofnetworks with shared caches is particularly interesting sincein these types of networks users can share the caches andhence in a practical perspective it helps in efficient utilizationof memory. Coded caching in networks where demands arenon uniform i.e., files with different popularities [11], [12] hasalso been well studied. However we will be limiting ourselvesto the cases where all the files are equipopular and demandsare distinct.An interesting and practical type of network scenario is amulti-access network where multiple users can access the samecache and multiple caches can be accessed by the same user[12], [15], [16], [17]. The scenario considered in [12], [15]and [16] has K users and K caches and a “sliding window”approach, where user k accesses caches k, k +1 , . . . , k + r − for some r ∈ , , ..., K , using a cyclic wraparound to preservesymmetry is used. Here, r is called cache access degree i.e.the number of caches a user has access to. In the following subsection we provide a brief survey ofthe schemes that are known in the literature of multi-accessnetworks. Known schemes in multi-access1) Hachem-Karamchandani-Diggavi (HKD) Scheme [12]:
Multi-access setups were introduced in the work of [12] wherecaching and delivery in a decentralized setting with multi-access and multi-level popularities under consideration.Theauthors give rate memory trade offs for a multi-level accessmodel (content is divided into discrete levels based on pop-ularity and users are required to connect to a certain numberof access points based on the popularity of the file they haverequested) with multi user setups (user can access multiplecaches). In the case of a multi-user, multi-access model witha single-level caching system, with N files, K caches and K users grouped into U users in a group with access degree r ,such that N ≥ KU and r divides K , and a cache memoryof M ∈ [0 , Nr ] , in the decentralized assumption, considered inthe paper, the achievable rate is given by R SL ( M, K, N, U, r ) = U. min n NM , K o(cid:16) − rMN (cid:17) . Also when r does not divide K , four times the aboveexpression can be achieved and R SL ( M, K, N, U, r ) = 0 if M > Nr .
In [12], the scheme proposed is decentralized. However thescheme can be extended to get a centralized scheme (alsoindicated in [15]) with the rate achieved given by R ( M, K, N, r ) = K − KrMN
KMN . (1)Henceforth we refer to this scheme as the HKD scheme.
2) Reddy-Karamchandani (RK) Scheme [15] :
The schemeproposed in [15] supports a multi-access setup with K usersand K caches with each user connected to r consecutivecaches in a cyclic manner. The rate for this scheme for M = i NK where i ∈ ∪ [ ⌈ Kr ⌉ ] is given by the expression R = K (cid:16) − r MN (cid:17) for i ∈ ∪ hj Kr ki and R = 0 for i = l Kr m generic lower bound on the optimal rate for any suchmulti-access setup with r ≥ K , under the restriction ofuncoded placement is derived as R lb ( M ) = K − h K − ( K − r )( K − r +1)2 K i MKN , if ≤ M ≤ NK ( K − r )( K − r +1)2 K (cid:16) − MKN (cid:17) , if NK ≤ M ≤ NK , if M ≥ NK . The rate achieved by this scheme is proved to be order optimalwith the multiplicative gap between the achievable rate andlower bound, at most for r ≥ K/ . The scheme is also foundto be optimal for the special cases, r = K − , r = K − , r = K − with K even and r = K − s + 1 for some positiveinteger s . Hereafter, we refer to this scheme as the RK scheme.
3) Serbetci-Parinello-Elia (SPE) Scheme [16]:
The workin [16] also deals with the same problem setup as in theprevious cases and provides two new schemes which canserve, on average, more than Kγ + 1 users at a time andfor the special case of r = K − Kγ , the achieved gain isproved to be optimal under uncoded cache placement where γ = MN , γ ∈ { K , K , . . . , } . The general scheme is proposedfor the case Kγ = 2 . The subpacketization for this schemeis given by F = K ( K − r +2)4 and it can be noted that thenumerator should be divisible by and r < ( K +2)2 for thenumber of subpackets to be a positive integer greater than 0.We refer to this scheme as the SPE scheme.
4) Scheme using Cross Resolvable Designs (CRD) [17]:
In [17] the authors develop a multi-access coded cachingscheme from a specific type of resolvable designs calledcross resolvable designs. The number of users supported inthis scheme is higher than the other existing schemes forthe same number of caches and for practically realizablesubpacketization levels. To compare the performance withother schemes, the authors introduce the notion of per userrate or rate per user obtained by normalizing the rate R withthe number of users K supported, i.e., rate per user is RK . Werefer to this scheme as the CRD scheme.
5) Cheng-Liang-Wan-Zhang-Caire (CLWZC) Scheme [18]:
The work of [18] proposes a novel transformation approachto extend the Maddah Ali Niesen scheme to the multiaccesscaching systems, such that the load expression of the schemeby [12] remains achievable even when r does not divide K .This work considers only the multi-access setup with cyclicwraparound where each user has access to r neighboringcache-nodes with a cyclic wraparound topology. The rateexpression for this scheme is same as that for the centralizedHKD scheme. The subpacketization required is K (cid:0) K − t ( r − t (cid:1) ,where t = KMN . A. Our Contributions
In this work a coded caching scheme for multi-access net-works is proposed with the number of users much larger thanthe number of caches. The proposed scheme is extension ofthe MAN scheme and the MAN scheme is can be obtained as a U U U K Z C Z Z Shared Link S UsersCachesServer
Figure 1: Problem setupspecial case. The performance of this scheme is compared withthe existing works. To be able to compare the performance ofdifferent multi-access schemes with different number of usersfor the same number of caches, the terminology of per userrate RK introduced in [17] is used. Notations:
The set { , , ...n } is denoted as [ n ] |X | denotescardinality of the set X .II. T HE P ROPOSED S CHEME
Our problem set up is as as shown in Fig. 1. Let K denotethe set of K users in the network connected via an error freeshared link to a server S storing N files ( N ≥ K ) denoted as W , W , W , . . . , W N each of unit size. Each user can access aunique set of r caches (cache access degree) out of C caches,each capable of storing M files. Z k denotes the content incache k , and we assume that each user has an unlimitedcapacity link to the caches that it is connected to. Let C denotethe set of C caches. Since a user has access to a unique subsetof r caches, every user can be uniquely represented with a r sized subset of C . From now on, we denote the user by the r sized subset of caches it is connected to. Let d U , where U ⊂ C ; |U| = r denote the demand of the user connectedto the set of caches represented by U . Hence, the maximumnumber of users K in this scheme is (cid:0) Cr (cid:1) . Our scheme worksfor any number of users as long as a user is associated witha unique subset of r caches, i.e., no two users are accessingthe same set of r caches. For concreteness we assume that K = (cid:0) Cr (cid:1) . The scheme works in two phases described below:
Placement Phase:
Let t = CMN be an integer. The serverdivides each file W i into (cid:0) Ct (cid:1) subfiles and in the k th cacheplaces the content given by Z k = { W i, T : k ∈ T , T ⊂ C , |T | = t, ∀ i ∈ [ N ] } . It is seen that after the placement, the size of the content storedin the cache is equal to ( C − t − )( Ct ) N = tNC = M .Note that a user may find the same subfiles of a file in morethan one cache it has access to. Let M ′ denote the size of thefiles that a user has access to. We have ′ N = r X i =1 |Z i | (cid:0) Ct (cid:1) − r X ≤ i
Note that the placement is same as the placementof Maddah Ali Niesen scheme in shared link networks [1]with the difference that in [1], each user is equipped with adedicated cache.
Delivery Phase:
For each such subset S ⊂ C of cardinality | S | = t + r , the server transmits M U ⊂ S |U| = r W d U , S \U Now we prove that with the above delivery scheme everyuser will be able to decode the file it wants to.
Proof:
First, we note that when
U ∩ T = φ , then theuser U does not have access to any cache c where, c ∈ T and c / ∈ U . A user U has access to a subfile indexed by T iff U ∩ T 6 = φ . This is because, when U ∩ T 6 = φ , there existsa cache c ∈ U ∩ T , through which U can access the subfileindexed by T .Consider the delivery algorithm mentioned above. We nowargue that each user can successfully recover its requestedmessage. Consider the transmission by the server M U ⊂ S |U| = r W d U , S \U corresponding to the subset of caches, S ⊂ C with | S | = t + r .Consider a user U ⊂ S ; |U| = r . The user U already hasaccess to the subfiles W d U , S \V for any other user indexed by V ⊂ {S : V 6 = U} ; |V| = r since, { S \ V} ∩ U 6 = φ . Henceit can retrieve the subfile W d U , S \{U} from the transmissioncorresponding to S . Likewise, from all such transmissionscorresponding to S , with U ⊂ S , user U gets the missingsubfiles of W d U sent over the shared link. Since this is truefor every such subset S , any user U ⊂ C ; |U| = r can recoverall missing subfiles. Theorem 1:
For every integer t = CMN , the above placementand delivery results in a multi-access scheme with C caches,access degree r , (cid:0) Cr (cid:1) users, subpacketization F = (cid:0) Ct (cid:1) , codinggain g = (cid:0) t + rr (cid:1) , and rate R = ( Ct + r )( Ct ) . Proof:
According to the proposed scheme, the size ofeach subpacket is ( Ct ) . The total number of transmissionsis the total number of choices of S = (cid:0) Ct + r (cid:1) . The codinggain, defined as the total number of users benefited fromeach transmission is thus, the number of r sized subsets of C (number of users) we can choose from a particular S . Thusrate R = ( Ct + r )( Ct ) and the coding gain in this scheme is (cid:0) t + rr (cid:1) Remark 2:
The proposed scheme can be designed for anyinteger t = CMN , and the rate points in between can beachieved through memory sharing. So the proposed schemeexists for any multi-access network with N files, C cachesequipped with memories of size M file units each and accessdegree r . Remark 3:
Note that in this setup, since every user isassociated with a unique subset of caches, even if some usersleave the network, the transmissions involving the existingusers continue to hold. The only constraint is that, any userin the network should be uniquely associated with an r subsetof caches. So our setup is dynamic in the sense that userscan join and leave the system whenever they want, providedthe association with the caches is unique. (cid:0) Cr (cid:1) is simply themaximum number of users associated with distinct r subsetsof caches that can be supported by the described the placementand delivery scheme. Remark 4:
In all the multi-access schemes in literatureexcept the one derived from cross resolvable designs, the r caches connected to any user, store disjoint contents. As aconsequence, it is seen that in [12], [16] and [15], the rate R becomes when MN ≥ r , since the user gets all the N filesfrom the r caches connected to it. However, in these schemes,it is possible to enforce the constraint of making r caches storedisjoint contents since the number of users in the network isonly C . In the setup considered here, since the number of usersis more than C , the same subfiles can be stored in multiplecaches connected to a user and hence rate does not become , when MN ≥ r . Though this redundancy does appear like awaste of memory, since the number of users supported is largein the network, multi-casting opportunities increase resultingin low per user rates. Remark 5:
In the proposed scheme, the contents placed inthe caches connected to any user in the setup are disjoint when t = 1 , since the contents placed in the caches itself is disjointin this special case. Z i ∩ Z j = φ, ∀ i, j ∈ C It can be noted that t = 1 , is the only case when con-tents connected to a user become disjoint unlike the existingschemes in literature where caches connected to a user alwayshold disjoint contents, with the exception of the CRD codedcaching scheme. . Examples In the examples below, we assume that the users arenumbered by lexicographically ordered r subsets. For exampleif r = 2 , C = 4 , then User 1 corresponds to the subset { , } , User 2 to { , } , User 3 to { , } , User 4 to { , } ,User 5 to { , } , and User 6 to { , } . The request vector ( d , d , . . . , d K ) ∈ N K denotes the tuple containing demandsof the users , , .., K . The following three examples corre-spond to the cases t = 1 , t = r, and t = r respectively. Example 1:
Consider a multi-access setup with C = 4 caches, and cache access degree r = 2 . We allow all possiblecombinations of access to r caches. So number of users K = (cid:0) Cr (cid:1) = (cid:0) (cid:1) = 6 . For t = 1 the subpacketization is F = (cid:0) Ct (cid:1) = (cid:0) (cid:1) = 4 . The subfiles are W i, , W i, , W i, , W i, , ∀ i ∈ [ N ] . The cache placement is: Z = { W , , W , , W , , W , , W , , W , }Z = { W , , W , , W , , W , , W , , W , }Z = { W , , W , , W , , W , , W , , W , }Z = { W , , W , , W , , W , , W , , W , } Let the request vector be (1 , , , , , . The transmissionsare: Y { , , } = W , ⊕ W , ⊕ W , Y { , , } = W , ⊕ W , ⊕ W , Y { , , } = W , ⊕ W , ⊕ W , Y { , , } = W , ⊕ W , ⊕ W , Example 2:
Consider a multi-access setup with C = 5 caches, and cache access degree r = 3 , t = 2 . Numberof users K = (cid:0) Cr (cid:1) = (cid:0) (cid:1) = and subpacketization is F = (cid:0) Ct (cid:1) = (cid:0) (cid:1) = . The subfiles are W i, { , } , W i, { , } , W i, { , } , W i, { , } , W i, { , } , W i, { , } , W i, { , } , W i, { , } , W i, { , } , W i, { , } , ∀ i ∈ [ N ] . The cache placement is: Z = { W i, { , } , W i, { , } , W i, { , } , W i, { , } , ∀ i ∈ [ N ] }Z = { W i, { , } , W i, { , } , W i, { , } , W i, { , } , ∀ i ∈ [ N ] }Z = { W i, { , } , W i, { , } , W i, { , } , W i, { , } , ∀ i ∈ [ N ] }Z = { W i, { , } , W i, { , } , W i, { , } , W i, { , } , ∀ i ∈ [ N ] }Z = { W i, { , } , W i, { , } , W i, { , } , W i, { , } , ∀ i ∈ [ N ] } Let the request vector be (1 , , , , , , , , , . The trans-missions are: Y { , , , , } = W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } (a) MN < r (b) MN ≥ r Figure 2: Comparison with the HKD scheme for the number ofcaches, C = 24 . The two schemes are compared for differentvalues of r i.e r ∈ { , , , , } . Example 3:
Consider a multi-access setup with C = 5 caches , and cache access degree r = 2 , t = 2 . Numberof users K = (cid:0) Cr (cid:1) = (cid:0) (cid:1) = and subpacketization is F = (cid:0) Ct (cid:1) = (cid:0) (cid:1) = . The subfiles are W i, { , } , W i, { , } , W i, { , } , W i, { , } , W i, { , } , W i, { , } , W i, { , } , W i, { , } , W i, { , } , W i, { , } , ∀ i ∈ [ N ] . The cache placement is: Z = { W i, { , } , W i, { , } , W i, { , } , W i, { , } , ∀ i ∈ [ N ] }Z = { W i, { , } , W i, { , } , W i, { , } , W i, { , } , ∀ i ∈ [ N ] }Z = { W i, { , } , W i, { , } , W i, { , } , W i, { , } , ∀ i ∈ [ N ] }Z = { W i, { , } , W i, { , } , W i, { , } , W i, { , } , ∀ i ∈ [ N ] }Z = { W i, { , } , W i, { , } , W i, { , } , W i, { , } , ∀ i ∈ [ N ] } a) In the plot shown above, the two schemes are compared fordifferent values of t = CMN i.e t ∈ { , , , } , r = 2 .(b) In the plot shown above, the two schemes are compared fordifferent values of r i.e r ∈ { , , , } , t = CMN = 2 . Figure 3: Comparison with the HKD scheme in terms ofsubpacketization.Let the request vector be (1 , , , , , , , , , . Thetransmissions are: Y { , , , } = W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } Y { , , , } = W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } Y { , , , } = W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } Y { , , , } = W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } Y { , , , } = W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } ⊕ W , { , } III. P
ERFORMANCE ANALYSIS
In this section we compare the performance of the proposedscheme with the schemes available in the literature for multi-access coded caching.
A. Comparison with the HKD Scheme
In Fig. 2a, the proposed scheme and the HKD scheme arecompared in terms of per user rate RK with respect to MN , for MN < r , and for different values of r . The expression for R of centralized equivalent of the HKD scheme is given in (1).Note that r should divide C for the centralized HKD schemeto exist. The number of users K = C in the HKD scheme.Fig. 2b depicts variation of per user rate RK with respect to MN for the case when MN ≥ r . From Fig. 2a, it is seen that theproposed scheme achieves lower per user rate than the HKDscheme when MN < r . For the case, MN ≥ r , in Fig. 2b it is isseen that the rate of the HKD scheme becomes , while the peruser rate of proposed scheme is slightly higher. However thisis expected due to the cyclic wraparound topology consideredin the HKD scheme and the placement in the caches such thatthe user gets all the N files when MN ≥ r .For the centralized equivalent of the HKD scheme, F isgiven by r (cid:0) Cr t (cid:1) . In Fig. 3a, the variation of subpacketization F , with respectto K is studied, keeping access degree r constant for bothschemes and the comparison is made for different values of t .In Fig. 3b, the variation of subpacketization F , with respectto K is studied, keeping t = CMN constant for both schemesand the comparison is made for different values of r . It isseen that the subpacketization levels of proposed scheme issignificantly lower than that of the HKD scheme.Figure 4: Comparison with the SPE scheme for numberof caches, C = 18 for different values of r i.e r ∈{ , , , , , , } .igure 5: Comparison with the SPE scheme for different valuesof r i.e r ∈ { , , , } for the fraction of each file stored ateach cache the same in both the schemes i.e. MN = C . (a) MN < r (b) MN ≥ r Figure 6: Comparison with the RK scheme for the numberof caches, C = 12 for different values of r i.e r ∈{ , , , , , } . B. Comparison with the SPE Scheme
In Fig. 4, the proposed scheme and the SPE scheme arestudied in terms of per user rate RK with respect to MN , bykeeping access degree, r = 2 for both the schemes. Theexpression for rate of the SPE scheme is given in Theorem1 of [16]. For per user rate, it is normalized by number ofusers (For the SPE scheme K = C ). Since the SPE schemeexists only for KMN = 2 , only these specific points are plottedfor comparison. From Fig. 4, it is seen that proposed schemesupports lower per user rates.The next comparison is in terms of subpacketization. In Fig. 5,variation of F for the two schemes with respect to number ofusers K is studied, for different values of r . It is seen that apartfrom supporting large number of users, lower subpacketizationlevels are obtained from the proposed scheme when comparedto the SPE scheme. (a) In the plot shown above, the two schemes are compared fordifferent values of r i.e r ∈ { , , , } , t = CMN = 2 .(b) In the plot shown above, the two schemes are compared fordifferent values of t = CMN i.e t ∈ { , , , } , r = 2 . Figure 7: Comparison with the RK scheme in terms ofsubpacketization level ( F ). C. Comparison with the RK Scheme
In this subsection, the proposed scheme is compared withthe RK scheme scheme by varying MN and noticing its effectigure 8: Comparison with the CRD scheme derived fromaffine planes.on per user rate. For the RK scheme the normalized lowerbound of rate R lb K is plotted. The expression for R lb is takenfrom Theorem 3 in [15]. Since this lower bound is valid for r ≥ C , the comparison in Fig. 6 holds only for r ≥ C .Also K = C in the RK scheme. The schemes are comparedfor different values of r . The per user rate for the proposedscheme is found to be better than that of the RK scheme forthe range MN < r from Fig. 6a.In Fig. 7a, the two schemes are compared keeping M/Nconstant i.e. MN = C . The two schemes have been comparedfor different values of r . In Fig. 7b, the two schemes arecompared by keeping the access degree same in both theschemes i.e. r = 2 . The two schemes have been compared fordifferent values of t . From these plots it can be concluded thatsignificantly low subpacketization levels can be attained withthe proposed scheme when compared with the RK scheme. D. Comparison with the CRD Scheme
The scheme from [17] derived from the cross resolvable de-signs from affine planes is compared with proposed scheme inthis section. The rate per user for the CRD scheme from affineplanes is RK = ( n − n where n is a prime or prime power. Itcan be seen from Fig. 8, that the rate per user supported, in theproposed scheme is significantly less compared to the schemederived from CRD.From [17], MN = n , C = n ( n + 1) , F = n and K = n ( n +1)2 for the multi-access scheme from CRDs derived fromaffine planes, where n is a prime or prime power. In Fig. 9a,the number of caches C and memory fraction MN is kept samefor both schemes, and the two schemes are compared in termsof subpacketization levels F . Since C and MN are dependentonly on the parameter n , for the CRD scheme from affineplanes, the subpacketization values are plotted with respectto n . In Fig. 9b, the subpacketization levels are plotted with (a) Variation of Subpacketization with respect to n.(b) Variation of Subpacketization with respect to number of usersK. Figure 9: Comparison with the CRD scheme in terms ofSubpacketization.respect to number of users K . From Fig. 9a and Fig. 9b, it canbe concluded that the proposed scheme does not perform wellin terms of subpacketization levels when compared with theCRD scheme. But this is explained by the fact that the CRDscheme loses out in rate and gains in terms of subpacketization.Also, the number of users supported in the proposed schemeis more than that of the CRD scheme. E. Comparison with the Cheng-Liang-Wan-Zhang-Caire(CLWZC) Scheme [18]
In this subsection, the proposed scheme is compared withthe CLWZC [18] scheme scheme by varying MN and noticingits effect on per user rate. The expression of rate R for [18]scheme is taken from Theorem 1 in [18]. The schemes arecompared for different values of r . The per user rate for the P e r U s e r R a t e ( R / K ) Access Degree (r) Normalized Cache size (M/N) r = 4 : Proposed Schemer = 3 : Proposed Schemer = 2 : CLWZC Schemer = 3 : CLWZC Schemer = 4 : CLWZC Schemer = 5 : CLWZC Schemer = 5 : Proposed Schemer = 6 : Proposed Schemer = 6 : CLWZC Schemer = 2 : Proposed Scheme (a) MN < r P e r U s e r R a t e ( R / K ) Access Degree (r)
Normalized Cache size (M/N) r = 4 : Proposed Schemer = 3 : Proposed Schemer = 2 : CLWZC Schemer = 3 : CLWZC Schemer = 4 : CLWZC Schemer = 5 : CLWZC Schemer = 5 : Proposed Schemer = 6 : Proposed Schemer = 6 : CLWZC Schemer = 2 : Proposed Scheme (b) MN ≥ r Figure 10: Comparison with the CLWZC scheme for num-ber of caches, C = 15 for different values of r i.e r ∈{ , , , , } .proposed scheme is found to be better than that of the CLWZCscheme for the range MN < r from Fig. 10a.In Fig. 11b, the two schemes are compared keeping MN constant i.e. MN = C . The two schemes have been comparedfor different values of r . In Fig. 11a, the two schemes arecompared by keeping the access degree same in both theschemes i.e. r = 2 . The two schemes have been comparedfor different values of t . From these plots it can be seenthat significantly low subpacketization levels can be attainedwith the proposed scheme when compared with the CLWZCscheme.In Table I the proposed scheme is compared with theCLWZC scheme for t = CMN = 1 . In can be observed that for t = 1 and the access degree r = C − the proposed schemeachieves lower subpacketization level ( F ) and supports largernumber of users ( K ) than the CLWZC scheme for the same Number of Users (K) S ubpa ck e t i z a t i on Le v e l ( F ) t = 2 : CLWZC Schemet = 3 : CLWZC Schemet = 4 : CLWZC Schemet = 5 : CLWZC Schemet = 2 : Proposed Schemet = 3 : Proposed Schemet = 4 : Proposed Schemet = 5 : Proposed Scheme (a) In the plot shown above, the two schemes are compared fordifferent values of t = CMN i.e t ∈ { , , , } , r = 2 . Number of Users (K) S ubpa ck e t i z a t i on Le v e l ( F ) r = 2 : CLWZC Schemer = 3 : CLWZC Schemer = 4 : CLWZC Schemer = 5 : CLWZC Schemer = 2 : Proposed Schemer = 3 : Proposed Schemer = 4 : Proposed Schemer = 5 : Proposed Scheme (b) In the plot shown above, the two schemes are compared fordifferent values of r i.e r ∈ { , , , } , t = CMN = 2 . Figure 11: Comparison with the CLWZC scheme in terms ofsubpacketization (F).fraction of each file at each cache MN , fraction of each fileeach user has access to M,N and rate ( R ) .IV. C ONCLUSION
We conclude that the proposed scheme is better than mostexisting schemes in literature since the number of users ( K )supported is very large at low subpacketization ( F ) levels.When MN < r , the rate per user achieved is lesser thanthat of existing schemes. The proposed scheme exists for anymulti-access network with N files, C caches equipped withmemories of size M file units each and access degree r . Thescheme can be designed for any integer CMN , and the ratepoints in between can be achieved through memory sharing,allowing convenience in designing.able I: Comparison between CLWZC and Proposed schemefor t = CMN = 1 . Parameters CLWZC Scheme Proposed Scheme
Number of Caches ( C ) C C
Number of Caches auser has access to ( r ) C − C − Number of Users ( K ) C ( C )( C − Fraction of each fileat each cache (cid:16) MN (cid:17) C C Fraction of each fileeach user has access to C − C C − C Subpacketization level C (cid:0) C − t ( r − t (cid:1) = 3 C (cid:0) Ct (cid:1) = C Rate (R) C − trt +1 = 1 (cid:16) Ct + r (cid:17)(cid:16) Ct (cid:17) = 1 A CKNOWLEDGMENT
This work was supported partly by the Science and Engi-neering Research Board (SERB) of Department of Science andTechnology (DST), Government of India, through J.C. BoseNational Fellowship to B. Sundar Rajan.R
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