Coded Caching with Multiple File Requests

Transcription

1 Coded Caching wih Muliple File Requess Yi-Peng Wei Sennur Ulukus Deparmen of Elecrical and Compuer Engineering Universiy of Maryland College Park, MD Absrac We sudy a wo-phase caching nework consising of one server wih N files conneced o users hrough an error-free shared link. Each user has a cache memory which can sore M files in he placemen phase. In he delivery phase, each user requess L files, and he server ransmis he messages accordingly. Using he message sen by he server combined wih he cache memory, each user reconsrucs he L files hey requesed. In his work, we focus on he case L > 1, i.e., he case of muliple file requess. We adop he symmeric bach caching scheme and propose a general delivery scheme. To prove he opimaliy of he proposed general delivery scheme, we apply wo converse echniques. The firs converse echnique is for general coding schemes and is obained hrough virual user consrucion. The second converse echnique is for vecor linear coding schemes and is obained using an inerference alignmen poin of view. Wih hese wo converse echniques, we characerize eiher he unconsrained opimal coding rae, or opimum linear coding rae, wih symmeric bach caching for cerain cases. I. INTRODUCTION Consider a wo-phase caching nework [1] consising of one server wih N files conneced o users hrough an error-free shared link. Each user has heir local cache memory which can sore M files. The wo phases are he placemen phase and he delivery phase. In he placemen phase, he nework raffic load is low. Each user can access he whole N files in he server and fill heir cache memory in advance. In he delivery phase, he nework raffic load is high. Each user requess L files from he server, and he server delivers messages hrough he error-free shared link o users. The reques of each user is unknown a priori in he placemen phase. Each user reconsrucs he L files hey requesed by he messages sen from he server and he side informaion sored in heir cache memory. The objecive is o minimize he raffic load in he delivery phase due o he high raffic load in his phase. The wo-phase caching nework is firs sudied in [1], wih he assumpion ha in he delivery phase, each user requess one file, i.e., L = 1. Reference [1] proposes symmeric bach caching for he placemen phase. Combined wih he coded mulicasing in he delivery phase, reference [1] shows ha global caching gain can be obained. Using a cu-se bound analysis, order opimaliy of rae-memory rade off is shown for he wors-case file requess in [1]. Independen and idenical caching is proposed in [2] o accoun for he decenralized naure of pracical caching neworks. Boh symmeric bach caching and independen and idenical caching are uncoded This work was suppored by NSF Grans CNS , CCF , and CNS cache placemen schemes [3], [4], i.e., each user sores a subse of he bis of he original files. For uncoded cache placemen, wih N, by using index coding converse bound [5], reference [3] shows he opimaliy of rae-memory rade off for he wors-case file requess. In addiion, reference [4] shows he opimaliy for arbirary N, and M no only for he wors-case file requess bu also for he average case. For coded placemen, order opimaliy resuls can be found in [6] and references herein. In he delivery phase, each user can reques more han one file, i.e., L > 1. This case is firs sudied in [7] which adops symmeric bach caching as in [1]. For he delivery phase, [7] reas each differen file reques as a differen index coding problem, and generalizes he achievabiliy scheme for muliple unicas index coding in [8] o group casing index coding. Reference [7] shows he order opimaliy for he wors-case file reques wih a muliplicaive consan 18 based on a cu-se bound analysis. Then, reference [9] shows he order opimaliy for he wors-case file reques wih muliplicaive consan 11 by improving he converse bound hrough Han s inequaliy. Reference [9] adops symmeric bach caching as in [1] and applies he delivery scheme in [1] L imes. In his work, we also adop symmeric bach caching as in [1], and focus on exac opimaliy as opposed o order opimaliy. We propose a general delivery scheme for muliple file requess. To show he opimaliy of he delivery scheme, we use wo converse echniques. The firs echnique is for general coding schemes and is inspired by he converse in [4]. The second echnique is for vecor linear coding schemes using an inerference alignmen poin of view and is inspired by [10], [11]. Wih hese wo converse echniques, we deermine eiher unconsrained opimal coding rae, or opimal linear coding rae wih symmeric bach caching for cerain cases. If L differen files are requesed, we characerize he opimal coding rae. For L = 2 and = 3, 4, when each subfile is cached only a one user (i.e., = 1, we characerize eiher he unconsrained opimal coding rae, or he opimal linear coding rae. II. SYSTEM MODEL AND PROBLEM SETTING We consider a caching nework (see Fig. 1 consising of one server and users. The server connecs o he users hrough an error-free shared link. The server has N files denoed by W 1, W 2,..., W N. Each file is of size F bis. Each user has a local cache memory Z k of size MF bis for some real number M [0, N]. There are wo phases in his nework, a placemen

2 server error-free shared link users F bis W 1. W N N files X(D; R(D; ZF bis caches Z 1 Z 2 Z M files requess d 1 d 2 d Fig. 1. Caching nework. phase and a delivery phase. In he placemen phase, user k can access all he N files and fill is cache memory Z k. Therefore, Z k = φ k (W 1, W 2,..., W N, where φ k : F NF 2 F MF 2. In he delivery phase, each user requess L files ou of he N files. Le us denoe each user s reques by an L 1 vecor d k = (d k,1,..., d k,l T, where d k,i is he index of he file, i.e., d k,i {1, 2,..., N}, 1 i L. A reques marix D of size L is formed accordingly. The server oupus X(D of size RF bis hrough he error-free shared link o he users, where R refers o he load of he nework in he delivery phase. A rae R is said o be achievable if each user k can decode he L files i requesed by uilizing X(D and Z k. Noe ha in he placemen phase, which L files will be requesed by each user is unknown in advance. Therefore, he cache memory Z k is deermined before knowing D. In addiion, we focus on he uncoded cache placemen as in [4], which means ha each user k chooses MF bis ou of NF bis o fill is cache memory Z k. An example of coded cache placemen is provided in [1, Appendix]. We denoe he minimum achievable rae in he delivery phase by R (D, Z, where Z = (Z 1,..., Z. An average rae for a given cache placemen Z is defined as R (Z = E D [R (D, Z], by assuming ha each user chooses he L files equally likely from he N files. The minimum rae is defined as R = min Z R (Z. If L = 1, reference [4] shows ha symmeric bach caching originally proposed in [1] aains he minimum rae R. We summarize he symmeric bach caching here. Given each user has cache memory size M = N, where {1,..., }, for each file W i, we pariion he file ino non-overlapping and equal-size subfiles, and denoe [] = {1, 2,..., } and T = {T : T [], T = }, where means he cardinaliy of a se. Noe T =. We label he subfiles of Wi as W i,t, where T T. Equivalenly, W i = T T W i,t and W i,t W i,t = if T T. In he placemen phase, user k places he subfile W i,t ino he cache memory Z k if k T. Thus, user k ges 1 1 subfiles of each file Wi. We denoe he symmeric bach caching wih parameer as Z sym. In his work, we adop he symmeric bach caching, Z sym, and sudy R (D, Z sym. Also, we denoe Rl (D, Z sym as he minimum achievable rae confined o vecor linear coding schemes. We propose a general delivery scheme. We characerize R (D, Z sym if L disinc files are requesed by he users. For L = 2, = 1, = 3, 4, we characerize R (D, Z sym for cerain reques marices D, while we characerize R l (D, Z sym for all oher reques marices. III. PROPOSED DELIVERY SCHEME To illusrae he delivery scheme, we inroduce he following vecor space represenaion. In he placemen phase, we pariion each file ino subfiles. We view each subfile as a vecor, and regard each subfile as he basis of he vecor space. Since he subfiles are linearly independen, wih all he subfiles, we have a N -dimensional vecor space. Noe ha in he delivery phase, we do no furher sub-packeize he subfiles. Therefore, our delivery scheme can also be viewed as scalar linear coding. We consider he vecor space over F 2. Le S = {S : S [], S = + 1}. For every S S and 1 l L, a candidae delivery message is as follows: Y S,l = s S W ds,l,s\{s}. (1 Here, d s,l idenifies he index of he lh file user s requess. Since S \ {s} =, i idenifies he subfile W ds,l,s\{s} owned by hese users. Noe ha among he + 1 erms on he righ hand side of (1 only one erm is unknown o each user s. By sending he candidae delivery message Y S,l, each user s in S can decode a subfile of W ds,l hey requesed. For L = 1, he delivery scheme proposed in [1] is o go over all he ses in S and send ou all he candidae delivery message given in (1. This resuls in +1 ransmissions. However, reference [4] shows ha going over all he ses in S is unnecessary. Insead, [4] goes over S = {S : S [], S U, S = + 1}, where U corresponds o he leaders defined in [4]. In [4], he number of ransmissions is reduced o ( +1 U +1. For L > 1, we know ha sending ou all he candidae delivery messages by going over all he ses in S and 1 l L is sufficien o saisfy all he requess. To reduce he raffic load, if he candidae delivery message can be obained hrough a linear combinaion of he sen messages, hen he server does no need o send his candidae delivery message. Since each candidae delivery message given in (1 has a vecor represenaion, he necessary delivery messages consis of he messages in he maximal linearly independen subse of he candidae delivery messages formed by going over all he ses in S and 1 l L. We use an example o illusrae he proposed delivery scheme. Le N = 4, = 4, M = 1, and L = 2. Then, = 1. To simplify he noaion, le A, B, C and D denoe he four files. By applying symmeric bach caching, Z 1 sym, we have Z k = (A k, B k, C k, D k, (2 where k = 1, 2, 3, 4. Suppose he reques marix is as follows A D =, (3 B C C C

3 which means user 1 requess files A and B, user 2 requess files A and C and so on. If he server sends ou all he candidae delivery messages as [9, Lemma 1], hen his resuls in R = 3 normalized raffic load, i.e., all he 12 candidae delivery messages are sen ou. If he server applies he delivery scheme in [4] wice, hen i resuls in R = 5 2 normalized raffic load, i.e., only 10 candidae delivery messages are sen ou. By applying he delivery scheme proposed here, only 9 candidae delivery messages are sen ou resuling in R = 9 4 normalized raffic load. Specifically, by reaing each subfile as he basis of he vecor space, we have A w e w, w = 1, 2, 3, 4 (4 B x e x+4, x = 1, 2, 3, 4 (5 C y e y+8, y = 1, 2, 3, 4 (6 D z e z+12, z = 1, 2, 3, 4 (7 where e i, i = 1, 2,..., 16 are he sandard bases of F Since file D does no appear in he reques marix given in (3, in he following we only consider he subspace spanned by {e i, i = 1, 2,..., 12}. We represen he candidae delivery messages as follows: Y {1,2},1 = A 2 A 1 (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (8 Y {1,3},1 = A 3 A 1 (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, (9 Y {1,4},1 = A 4 B 1 (0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, (10 Y {2,3},1 = A 3 A 2 (0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, (11 Y {2,4},1 = A 4 B 2 (0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, (12 Y {3,4},1 = A 4 B 3 (0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, (13 Y {1,2},2 = B 2 C 1 (0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, (14 Y {1,3},2 = B 3 C 1 (0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, (15 Y {1,4},2 = B 4 C 1 (0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, (16 Y {2,3},2 = C 3 C 2 (0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, (17 Y {2,4},2 = C 4 C 2 (0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, (18 Y {3,4},2 = C 4 C 3 (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1. (19 Here, (8-(13 are he candidae delivery messages for he requess of files (A, A, A, B, and (14-(19 are he candidae delivery messages for he requess of files (B, C, C, C. Reference [9] sends ou all he 12 candidae delivery messages in (8-(19 achieving a normalized raffic load R = 3. In [4], if we choose users {1, 4} as he leaders for he requess of files (A, A, A, B, and choose users {1, 2} as he leaders for he requess of files (B, C, C, C, hen we do no need o send ou candidae messages Y {2,3},1 and Y {3,4},2, since Y {2,3},1 = A 3 A 2 (20 = (A 3 A 1 (A 2 A 1 (21 = Y {1,3},1 Y {1,2},1, (22 Y {3,4},2 = C 4 C 3 (23 = (C 4 C 2 (C 3 C 2 (24 = Y {2,4},2 Y {2,3},2. (25 Tha is, if he server applies he delivery scheme in [4] wice, hen he normalized raffic load is R = 5 2. For our general delivery scheme, in addiion o Y {2,3},1 and Y {3,4},2, we also do no send ou Y {1,3},2, since Y {1,3},2 = B 3 C 1 (26 = (A 4 B 3 (B 2 C 1 (A 4 B 2 (27 = Y {3,4},1 Y {1,2},2 Y {2,4},1. (28 Therefore, sending ou he oher 9 candidae delivery messages in (8-(19 is sufficien. Our proposed delivery scheme achieves normalized raffic load of R = 9 4. By using our proposed delivery scheme, he following rae R(D, Z sym = rank (Y (29 is achievable in general, where Y is an L +1 n marix, wih n = N, where each row vecor of marix Y consiss of vecor represenaion of candidae delivery messages given in (1 by reaing each subfile as a sandard basis of F n 2. IV. CONVERSE We consider wo converse echniques in his work. The firs echnique is for general coding schemes and is inspired by [4]. The second echnique is for vecor linear coding schemes using an inerference alignmen poin of view and is inspired by [10], [11]. We use wo examples o illusrae hese wo converse echniques. A. Converse for General Coding Schemes This converse builds on he virual user consrucion, which is a generalizaion from [4, Appendix A]. The virual user consrucion is relaed o he reques marix D. I sars from he se consising of users, i.e., a he beginning le U = []. We remove user k U if user k requess he same se of files as an oher user k U. Now, we consruc he virual user according o U = {u 1, u 2,..., u n }. Iniially, he cache memory of he virual user, denoed by Z v, is empy. We fill Z v wih he cache memory of user u 1, i.e., Z u1. Then, we fill Z v wih he cache memory of user u 2, bu if he subfiles cached in Z u2 are he files requesed by previous user, i.e., d u1, hen we do no fill hese subfiles ino Z v. We coninue his process, i.e., filling Z v wih he cache memory of user u k, bu if he subfiles in Z uk are he files requesed by one of he previous users, i.e., d u1,..., d uk 1, hen no filling hese subfiles ino Z v. Noe ha he virual user does no have a memory size consrain. Noe also ha here are n! ways o build he virual user. Le us use an example o show he virual user consrucion and he converse bound i provides. Consider a caching nework wih N = 4, = 3, = 1, and he reques marix is D = ( B C D. (30 By applying symmeric bach caching, Z 1 sym, we have Z k = (A k, B k, C k, D k, (31

4 where k = 1, 2, 3. According o he reques marix D given in (30, we have U = {1, 2, 3}, since no wo users reques exacly he same se of files. We fill Z v according o he order of user 2, user 3 hen user 1. Due o user 2, Z v conains subfiles (A 2, B 2, C 2, D 2. For user 3, since subfiles (A 3, C 3 are requesed by previous user, i.e., d 2, we only fill subfiles (B 3, D 3 ino Z v. Finally, for user 1, since subfiles (A 1, B 1, C 1, D 1 are requesed by one of he previous users, we do no fill any new subfiles ino Z v. Afer he consrucion of he virual user, we consruc he converse bound as follows. Assume ha by receiving X( D each user can reconsruc he files hey requesed. Since he virual user has more side informaion han user 2, by receiving X( D he virual user can also reconsruc files A and C. This also shows ha X( D conains informaion of subfiles (A 1, A 3, C 1, C 3. Afer he virual user obains files A and C, ogeher wih he files in Z v, he virual user has more side informaion han user 3. Therefore, he virual user can also reconsruc files B and D. This also shows ha X( D conains informaion of subfiles (B 1, D 1. According o his virual user consrucion, we know ha X( D a leas needs o conain 6 F 3 = 2F bis, since informaion of subfiles (A 1, A 3, C 1, C 3, B 1, D 1 are from X( D, and each subfile is of size F 3 bis. By applying he delivery scheme proposed in Secion III, we noe ha normalized raffic load R = 2 is also achievable. Therefore, we have R ( D, Z 1 sym = 2. Noe ha if we consruc he virual user according o he order of user 1, user 2 hen user 3, hen we have subfiles (A 1, B 1, C 1, C 2, D 1, D 2, D 3 in Z v. From his consrucion, we can only show ha he raffic load mus be greaer han 5 3 F, since i only implies ha X( D conains informaion of subfiles (A 2, A 3, B 2, B 3, C 3. Therefore, differen virual user consrucions may resul in a differen converse bounds. We remark ha for L = 1, he virual user consrucion in [4, Appendix A] always resuls in a igh converse. However, for L > 1, he virual user consrucion may no always give a igh converse. To see his, consider a differen reques marix as follows: D =. (32 B C C In his case, no maer which ordering we use o consruc he virual user, we can only show ha he normalized raffic load R 5 3, while he proposed achievable scheme achieves R = 2, leaving a gap beween he wo. B. Converse for Vecor Linear Coding Schemes For he reques marix given in (32, if we are confined o vecor linear coding scheme, hen we have a igher converse bound. The reques marix given in (32 is equivalen o he following index coding problem [10, Sec. II], [11]. User 1 wih side informaion (A 1, B 1, C 1 requess subfiles (A 2, A 3, B 2, B 3. User 2 wih side informaion (A 2, B 2, C 2 requess subfiles (A 1, A 3, C 1, C 3. User 3 wih side informaion (A 3, B 3, C 3 requess subfiles (B 1, B 2, C 1, C 2. Since we apply he symmeric bach caching, Z sym, each subfile has he same size, here F 3 bis. Therefore, he upper bound of he symmeric capaciy of he index coding problem can be used o lower bound he raffic load of he caching nework [3]. Le RA IC 2 denoe he rae o ransmi message A 2 in he corresponding index coding problem. For he caching nework, A 2 refers o he subfile. For he corresponding index coding problem, A 2 represens he message. The superscrip IC refers o he rae for he index coding problem. For user 1, we have nr IC A 2 = H(A 2 (33 = I(A 2 ; S n, A 1, B 1, C 1 + H(A 2 S n, A 1, B 1, C 1 (34 I(A 2 ; S n, A 1, B 1, C 1 + o(n (35 = I(A 2 ; S n A 1, B 1, C 1 + o(n, (36 = H(S n A 1, B 1, C 1 H(S n A 1, B 1, C 1, A 2 + o(n (37 where S n in (34 refers o he received symbols in he index coding problem, (35 is due o Fano s inequaliy, and (36 is due o he independence beween A 2 and (A 1, B 1, C 1. Since we only use vecor linear coding schemes, (37 implies R IC A 2 + dim(v A3,B 2,B 3,C 2,C 3 dim(v A2,A 3,B 2,B 3,C 2,C 3, (38 where V A3,B 2,B 3,C 2,C 3 denoes he vecor space spanned by he messages (A 3, B 2, B 3, C 2, C 3. For he lef hand side of (38, we furher have R IC A 2 + dim(v A3,B 2,B 3,C 2,C 3 = R IC A 2 + dim(v A3 + dim(v B2,B 3,C 2,C 3 (39 = R IC A 2 + dim(v A3 + dim(v B2 + dim(v B3 + dim(v C2,C 3 (40 R IC A 2 + dim(v A3 + dim(v B2 + dim(v B3 + dim(v C2 (41 5R IC (42 To guaranee ha user 1 decodes correcly, we have dim(v A3 V B2,B 3,C 2,C 3 = 0, since user 1 requess A 3 and has no side informaion abou (B 2, B 3, C 2, C 3. Togeher wih he fac ha for vecor spaces A and B, dim (A + dim (B = dim(a B + dim(a B, (43 we have (39. Similarly, dim(v B2 V B3,C 2,C 3 = 0 and dim(v B3 V C2,C 3 = 0, due o he requess of user 1 and he lack of side informaion of user 1. Therefore, we have (40. By using (43 and he fac ha dimension is a nonnegaive quaniy, we have (41. To ge (42, by applying Fano s inequaliy we have nr IC A 3 I(A 3 ; S n A 1, A 2, B 1, B 2, B 3, C 1, C 2, C 3 + o(n (44 = H(S n A 1, A 2, B 1, B 2, B 3, C 1, C 2, C 3 + o(n. (45

5 Therefore, R IC A 3 dim(v A3 (46 Since we consider he symmeric capaciy upper bound, we remove he subscrip of R IC o simplify he noaion. For he righ hand side of (38, we have dim(v A2,A 3,B 2,B 3,C 2,C 3 + dim(v C1 = dim(v A2,A 3,B 2,B 2,C 1,C 2,C 3 (47 dim(v A1,A 2,A 3,B 1,B 2,B 3,C 1,C 2,C 3 = 1. (48 To guaranee ha user 2 decodes correcly, we have dim(v C1 V A3,B 3,C 3 = 0. Also, o guaranee ha user 3 decodes correcly, we have dim(v C1 V A2,B 2,C 2 = 0. Togeher wih (43 and he fac ha for vecor spaces A, B and C, dim((a B C = dim(a C + dim(b C dim((a B C, (49 we have (47. By using (43 and he fac ha dimension is nonnegaive, we have (48. Finally, combining (38, (42, and (48, we have 5R IC 1 dim(v C1. (50 Wih (46, we have R IC 1 6. Since he upper bound for he symmeric index coding capaciy is 1 6 and each subfile has he same size of F 3 bis, his resuls in he lower bound of 2F bis raffic load for he cache nework. By applying he delivery scheme proposed in Secion III, we know ha normalized raffic load R = 2 is achievable. Thus, Rl ( D, Z 1 sym = 2. A. Disinc File Requess V. TIGHT RESULTS If he users reques L disinc files in he delivery phase, hen +1 R (D, Z sym = L. (51 We sar wih he converse bound. Since L disinc files are requesed, his means ha each file is requesed by a mos one user. Therefore, we consruc he virual user according o he se U = [], and fill Z v according o he order of user 1, user 2 o user. Assume ha by receiving X(D, each user can reconsruc he file hey requesed. Since he virual user has more side informaion han user 1, by receiving X(D he virual user can also reconsruc he files user 1 requesed. This implies ha X(D should conain a leas L ( 1 F bis of informaion. For user 2, his implies ha X(D should conain anoher L ( 2 F bis of informaion, since differen L files are requesed. Coninuing his process up o user, he normalized raffic load is a leas L ( = L, (52 where he equaliy holds by he binomial heorem. For he delivery scheme, since every candidae delivery message occupies a leas one new dimension, by going over all he ses in S, and 1 l L, we observe ha all he candidae delivery message are linearly independen. Therefore, rank(y in (29 is L +1, and we have a igh resul for he case of disinc file requess. B. Reducion o One File Reques For L = 1, our general delivery scheme has he same resul as in [4], i.e., if L = 1, hen ( R (D, Z +1 n +1 sym =, (53 where n is he number of differen files in D. For he converse bound, since n differen files are requesed, we have U = n. By using similar argumens as in Secion V-A, we have ha he normalized raffic load is a leas 1 ( + 2 ( + + n = +1 ( n +1. (54 For he delivery scheme, by going over all he ses in S, we have +1 candidae deliveries. Reference [4, Lemma 1] shows ha some se of candidae delivery messages can be obained hrough linear combinaion of oher se of candidae delivery messages, which resuls in ( +1 n +1 messages delivered. Therefore, we have rank (Y ( +1 n +1. Combining wih (54, we have (53. C. Two File Requess In Secion V-A, we allow arbirary L bu require disinc file requess, while in Secion V-B, we allow arbirary file requess bu require L = 1. Here, we consider L = 2, and allow arbirary file requess. These resuls are obained hrough he delivery scheme presened in Secion III. To characerize R (D, Z sym, we use he converse echnique presened in Secion IV-A, while for Rl (D, Z sym, we use he converse echnique presened in Secion IV-B. We firs consider L = 2, = 1 and = 3. We lis all he possible cases of file requess here. Oher cases no lised here can be obained hrough permuaions and relabeling. A B C D 1 = D E F 1, Z 1 sym = 6 3. (55 D 2 = C D E 2, Z 1 sym = 6 3. (56 A A A D 3 = B C D 3, Z 1 sym = 5 3. (57 D 4 = B C D 4, Z 1 sym = 6 3. (58 A A C D 5 = 5, Z 1 sym = 6 3. (59 D 6 = B B D ( A A A B B C 6, Z 1 sym = 5 3. (60

6 D 7 =, R B C C l (D 7, Z 1 sym = 6 3. (61 A A A D 8 = B B B 8, Z 1 sym = 4 3. (62 Noe ha D 1 is a case of disinc file requess as shown in Secion V-A. For D 2, if we consruc he virual user according o he order of user 3, user 2, and user 1, hen we have a igh converse resul. However, if we consruc he virual user according o he order of user 1, user 2, and user 3, hen we can only show ha he normalized raffic load mus be greaer han 5 3. For D 7, no maer which ordering we use o consruc he virual user, we can only show ha he normalized raffic load R 5 3. Insead, by using he converse echnique in Secion IV-B, we characerize he opimal linear coding rae for D 7 as Rl (D 7, Z 1 sym = 6 3. We hen consider L = 2, = 1, and = 4. The following cases are all he represenaive cases for = 4. D A B C D 1 = E F G H 1, Z 1 sym = (63 2 = D E F G 2, Z 1 sym = (64 3 = C D E F 3, Z 1 sym = (65 D B 4 = C D E F 4, Z 1 sym = (66 5 = B D E F 5, Z 1 sym = (67 D A A C D 6 = B B E F 6, Z 1 sym = (68 7 = B C D E 7, Z 1 sym = 9 4. (69 8 = B C D E 8, Z 1 sym = (70 D A A A C 9 = B B D E 9, Z 1 sym = (71 10 = B B C D 10, Z 1 sym = 9 4. (72 D A A A C 11 = B B B D 11, Z 1 sym = (73 12 = B B C D 12, Z 1 sym = (74 D B 13 = C C D E 13, Z 1 sym = (75 14 = B C D E 14, Z 1 sym = (76 D D 15 =, R B C C E l (D 15, Z 1 sym = (77 16 = 16, Z 1 sym = (78 D 17 = B C C D ( A B C D C, R l (D 17, Z 1 sym = (79 D B 18 = C C D D 18, Z 1 sym = (80 19 = B C D D 19, Z 1 sym = (81 20 =, R B B C C l (D 20, Z 1 sym = 9 4. (82 21 = B B B C 21, Z 1 sym = 8 4. (83 22 = B B B B 22, Z 1 sym = 6 4. (84 Here, D 1 corresponds o disinc file requess as shown in Secion V-A. For D 19, if we consruc he virual user according o he order of user 2, user 3, user 1 and user 4, hen we have a igh converse resul. However, if we consruc he virual user according o he order of user 1, user 2, user 3, and user 4, we can only show ha he normalized raffic load mus be greaer han 9 3. For D 15, D 17 and D 20, we can only characerize he opimal linear coding rae. The converse bound derivaion is similar o ha in Secion IV-B, since D is a sub-marix of hese marices hrough some elemenary operaions. VI. CONCLUSION For he wo-phase caching nework wih muliple file requess, we adop symmeric bach caching and propose a general delivery scheme. The general delivery scheme is based on sending only hose candidae linear combinaions which provide new linearly independen equaions. The general delivery scheme is shown o be opimal for cerain cases hrough wo converse echniques. The firs echnique is based on virual user consrucion and he second converse echnique uses an inerference alignmen poin of view. REFERENCES [1] M. A. Maddah-Ali and U. Niesen. Fundamenal limis of caching. IEEE Trans. on Inf. Theory, 60(5: , May [2] M. A. Maddah-Ali and U. Niesen. Decenralized coded caching aains order-opimal memory-rae radeoff. IEEE/ACM Trans. on Neworking, 23(4: , Aug [3]. Wan, D. Tuninei, and P. Piananida. On he opimaliy of uncoded cache placemen. In IEEE ITW, Sep [4] Q. Yu, M. A. Maddah-Ali, and A. S. Avesimehr. The exac rae-memory radeoff for caching wih uncoded prefeching. In IEEE ISIT, Jun [5] F. Arbabjolfaei, B. Bandemer, Y.-H. im, E. Şaşoğlu, and L. Wang. On he capaciy region for index coding. In IEEE ISIT, Jul [6] Q. Yu, M. A. Maddah-Ali, and A. S. Avesimehr. Characerizing he rae-memory radeoff in cache neworks wihin a facor of 2. In IEEE ISIT, Jun [7] M. Ji, A. M. Tulino, J. Llorca, and G. Caire. Caching and coded mulicasing: Muliple groupcas index coding. In IEEE GlobalSIP, Dec [8]. Shanmugam, A. G. Dimakis, and M. Langberg. Local graph coloring and index coding. In IEEE ISIT, Jul [9] A. Sengupa and R. Tandon. Improved approximaion of sorage-rae radeoff for caching wih muliple demands. IEEE Trans. on Comm., 65(5: , Feb [10] H. Maleki, V. R. Cadambe, and S. A. Jafar. Index coding An inerference alignmen perspecive. IEEE Trans. on Inf. Theory, 60(9: , Sep [11] H. Sun and S. A. Jafar. Index coding capaciy: How far can one go wih only shannon inequaliies? IEEE Trans. on Inf. Theory, 61(6: , Jun