Coded Caching in a Multi-Server System with Random Topology

Transcription

1 Coded Caching in a Muli-Server Sysem wih Random Topology iish Mial, Deniz Gündüz and Cong Ling Deparmen of Elecrical and Elecronic Engineering, Imperial College London {n.mial, d.gunduz, c.ling}@imperial.ac.uk arxiv: v1 [cs.it] 2 Dec 2017 Absrac Cache-aided conen delivery is sudied in a muli-server sysem wih users, each equipped wih a local cache memory. In he delivery phase, each user connecs randomly o any servers ou of a oal of P servers. Thanks o he availabiliy of muliple servers, which model small base saions wih limied sorage capaciy, user demands can be saisfied wih reduced sorage capaciy a each server and reduced delivery rae per server; however, his also leads o reduced mulicasing opporuniies compared o a single server serving all he users simulaneously. A join sorage and proacive caching scheme is proposed, which explois coded sorage across he servers, uncoded cache placemen a he users, and coded delivery; and i is shown ha he achievable average sum delivery rae is comparable o ha achieved by a single server, while he gap beween he wo depends on, he available redundancy across servers. I is also shown ha he gap can be reduced by increasing he sorage capaciy a he SBSs. I. ITRODUCTIO The unprecedened growh in ransmied daa volumes across neworks necessiaes he design of more efficien delivery mehods ha can exploi he available memory space and processing power of individual nework nodes o increase he hroughpu, and simulaneously increase he reliabiliy of daa availabiliy. Coded caching and disribued sorage have received significan aenion in recen years as promising echniques o achieve he aforemenioned goals. Wih proacive caching, par of he daa can be pushed ino nodes local cache memories during offpeak hours, called he placemen phase, o reduce he burden on he nework, paricularly he wireless downlink, during peak-raffic periods when all he users place requess, called he delivery phase. Inelligen design of he cache conens creaes mulicasing opporuniies among he users, and hrough coded delivery muliple demands can be saisfied simulaneously. Coded caching is able o uilize he cumulaive cache capaciy in he nework o saisfy all he users a much lower raes [1] - [10]. A differen ype of coded caching is shown o improve he delivery performance in he so-called femocaching scenario [2]. In femocaching, files are replicaed or coded a muliple cache-equipped small base saions (SBSs so ha a user may reconsruc is reques from only a subse of he available SBSs. SBSs can ac as edge caches and provide conens o users direcly, reducing laency, backhaul load or he energy consumpion [2], [4]. Coding for disribued sorage sysems has been exensively sudied in he lieraure (see, for example, [11], [12], and in he femocaching scenario, ideal raeless maximum disance separable (MDS codes allow users o recover conens by collecing pariy bis from only a subse of SBSs hey connec o [2]. In his work, we (a = 2, q 1 = 2, q 2 = 4, q 3 = 2. (b = 2, q 1 = 4, q 2 = 4, q 3 = 0 (bes opology. (c = 2, q 1 = 3, q 2 = 3, q 3 = 2 (wors opology. Fig. 1: Examples of differen nework opologies for P = 3 and = 4 wih = 2. combine disribued sorage a SBSs, similar o he femocaching framework of [2], wih local cache sorage a he users, and coded delivery over errorfree shared broadcas links as in [1]. We consider a library of files sored across P SBSs each equipped wih is own limied-capaciy sorage space. Unlike he exising lieraure on coded caching, we consider a random conneciviy model; ha is, we assume ha,

2 during he delivery phase, each user connecs only o a random subse of SBSs, where P (see Fig. 1. This may be due o physical variaions in he channel, or resource consrains. Mos imporanly, hese connecions ha form he nework opology are no known in advance during he placemen phase; herefore, he cache placemen canno be designed for a paricular nework opology. Soring he files across muliple serving SBSs and allowing users o connec randomly o a subse of hese servers resuls in a loss in mulicasing opporuniies for he servers, indicaing a rade-off beween he coded caching gain and he flexibiliy provided by disribued sorage across he servers, which o he bes of our knowledge, has no been sudied in he exising lieraure. We propose in his paper a pracical coded cache placemen and delivery scheme ha explois MDS coding across servers simulaneously wih coded caching and delivery o users, and show ha he cos for he flexibiliy of disribued sorage is a scaling of he delivery rae by a consan. We also characerize he average wors-case delivery rae performance of he proposed scheme by assuming ha he users connec o a uniformly random subse of he servers; and show ha i is relaively close o he bes-case performance, which is he single-server cenralized delivery rae derived in [1], achieved when all users connec o he same se of servers. We observe ha, as he server sorage capaciies increase, he average delivery raeuser cache memory radeoff improves, approaching he single-server delivery rae. In a relaed work [7], he auhors analyze he performance of coded caching schemes presened in [1] and [6], when muliple servers aided by pariy servers are available. The auhors consider special scenarios wih one and wo pariy servers. They propose a scheme ha sripes he files ino blocks, and codes hem across he servers wih a sysemaic MDS code, and hey also propose a scheme for he scenario in which files are sored as whole unis in he servers, wihou sriping. Our work builds on he former scenario, in which files are coded across he servers, by proposing a scheme ha generalizes o he use of any ype of MDS code and any number of pariy servers. We sudy he impac of he opology on he sum delivery rae and he radeoff beween he server sorage space and he average sum delivery rae. In [8], he auhors consider muliple servers serving he users hrough an inermediae nework of relay nodes, each server having access o all he files in he library. The auhors propose schemes o reduce he delivery delay in he nework by uilizing he fac ha muliple servers may ransmi packes in parallel. These models are inherenly differen from he one considered in his paper. Anoher line of relaed work sudy caching in combinaion neworks [9], [10], which consider a single server serving cache-equipped users hrough muliple relay nodes. The server is conneced o hese relays hrough unicas links, which in urn serve a disinc subse of a fixed number of users hrough unicas links. A combinaion nework wih cache-enabled relay nodes is considered in [10]. However he symmery of a sandard combinaion nework which would be unrealisic in many pracical scenarios, he performance merics and he assumpion of a fixed and known nework opology during he placemen phase make he caching scheme and he analysis fundamenally differen from our paper. oaions. For wo inegers i < j, we denoe he se {i, i + 1,..., j} by [i : j], while he se [1 : j] is denoed by [j]. Ses are denoed wih he calligraphic fon, and A denoes he cardinaliy of se A. For A (a 1, a 2,..., a p, we define X A (X a1,..., X ap. II. SYSTEM MODEL We consider he sysem model illusraed in Fig. 1 wih P servers, denoed by S 1, S 2,..., S P, serving users, denoed by U 1, U 2,..., U. There is a library of files W 1, W 2,..., W, each of lengh F bis uniformly disribued over [2 F ]. Each user has access o a local cache memory of capaciy M U F bis, 0 M U, while each server has a sorage memory of capaciy M S F bis. The caching scheme consiss of wo phases: placemen phase and delivery phase. We consider a cenralized placemen scenario as in [1], which is carried ou cenrally wih he knowledge of he servers and he users paricipaing in he delivery phase. However neiher he user demands, nor he nework opology is known in advance during he placemen phase. In he delivery phase, we assume ha each user randomly connecs o servers ou of P, where P, and requess a single file from he library. For j [], le Z j denoe he se of servers U j connecs o, where Z j =, and d j [] denoes he index of he file i requess. For example, in Fig. 1a, Z 1 = {S 1, S 2 }, Z 2 = {S 1, S 2 }, Z 3 = {S 2, S 3 } and Z 4 = {S 2, S 3 }. Le he demand vecor be denoed by d (d 1, d 2,..., d. The opology of he nework, i.e., which users are conneced o which servers, and he demands of he users are revealed o he servers a he beginning of he delivery phase. The complee library of files mus be sored a he servers in a coded manner o provide redundancy, since each user connecs o a random subse of servers. is known while soring he library of files across he servers. The files may be sored in he servers using erasure codes. Since any user should be able o reconsruc any requesed file from is own cache memory and from he servers i is conneced o, he oal cache capaciy of a user and any servers mus be sufficien o sore he whole library of files; ha is,

3 we mus have M U + M S. Le p denoe he se of users served by S p, for p [P ], and define he random variable Q p p, which denoes he number of users served by S p. We shall denoe a paricular realizaion of Q p for a given opology as q p, where we have P q p =. For example, in Fig. 1a, 1 = {U 1, U 2 }, 2 = {U 1, U 2, U 3, U 4 }, 3 = {U 3, U 4 }. Each server S p ransmis a message X p of size R p F bis o all he users conneced o i, i.e., users in se p, over he corresponding shared link. The message X p is a funcion of he demand vecor d, he nework opology, he sorage memory conens of server S p, and he cache conens of he users in p. Each user U j receives he messages X Zj and reconsrucs is requesed file W dj using hese messages ogeher wih is cache conens. The objecive is o minimize he oal number of bis required o be sen over he nework during he delivery phase, P R p, such ha all he users can correcly decode heir requesed files, independen of he combinaion of files requesed by he users; ha is, we are ineresed in he wors-case delivery rae. Assuming ha (i.e., he number of files is larger han he number of users, we can assume ha all users requesing a differen file models he worscase scenario. III. CODED DISTRIBUTED STORAGE AD CACHIG SCHEME We firs noe ha our sysem model brings ogeher aspecs of disribued sorage and proacive caching/coded delivery. To see his, consider he sysem wihou any user caches, i.e., M U = 0. This is equivalen o a disribued sorage sysem wih unreliable servers. I is known ha MDS codes provide much higher reliabiliy and efficiency compared o replicaion in his scenario [11]. On he oher hand, when he servers are reliable, i.e., = P, our sysem is equivalen o he one in [1], and coded delivery provides significan reducions in he delivery rae. Accordingly, our proposed scheme brings ogeher benefis from coded sorage and coded delivery. To illusrae he main ingrediens of he proposed scheme we assume M S = in his secion. Exension o oher server sorage capaciies will be presened in Secion V. A. Server Sorage Placemen Wih reference o Fig. 2, we firs describe how he files are sored across he servers in order o guaranee ha each user reques can be saisfied from any servers he user may connec o. We define M U, and assume iniially ha i is an ineger, i.e., [0 : M U ]. The soluion for nonineger values of can be obained hrough memory- Fig. 2: Segmenaion, MDS coding and placemen of files ino P servers. sharing [1]. Each file is divided ino ( equal-size non-overlapping segmens. We enumerae hem according o disinc -elemen subses of [], where W j,a denoes he segmen of W j ha corresponds o subse A. We have W j = A []: A = W j,a. Each segmen is furher divided ino equal-size non-overlapping sub-segmens denoed by Wj,A l, l []. The sub-segmens of each segmen are coded ogeher using a (P, linear MDS code wih generaor marix G, giving as oupu P coded versions of he segmen W j,a, denoed by Cj,A l, l [P ]. Cl j,a is a linear combinaion of he subsegmens of he segmen corresponding o subse A, of he j h file, ha is sored in server S l. Since each sub-segmen is of lengh ( F, every linear combinaion Cj,A l is of he same lengh; and hence, server sorage capaciy consrain of F is me wih equaliy. Remark 1. We assume ha each user knows he generaor marix G of he MDS code employed for disribued sorage. This assumpion is necessary because in he delivery phase, he user mus be able o reconsruc any coded symbol Cj,A l from he uncoded segmen W j,a sored in is cache memory. B. User Cache Placemen For he user caches we use he placemen scheme proposed in [1]. Each segmen of a file, W j,a, is placed ino he caches of all he users U k for which k A. C. Delivery Phase We firs make he following observaion abou he above placemen scheme: in he wors-case demand scenario, consider any users. Any ou of hese users share in heir caches one segmen of he file requesed by he remaining user. Enumerae hese [ ( subses of users as H i, i +1 ]. Consider any server S p, and one of he q p users conneced o i, say U k. Then, for any subse H i, ha includes k, i.e., k H i, he segmen W dk,h i\{k} is needed by user U k, bu is no available in is cache because k / H i \{k}; while i is available in he caches

4 of he remaining users in p Hi. The MDS coded version of W dk,h i\{k} sored by S p is C p d k,h, and i\{k} since he users know he generaor marix G, each user which has W dk,h i\{k} in is cache can reconsruc C p d k,h as well. Then, for each H i\{k} i ha includes a leas one user from p, S p ransmis X p (H i = C p d k,h, (1 i\{k} k p Hi where { [ denoes he biwise XOR operaion. Then, ( i ] } ( +1 : k H i = 1 is he number of messages ransmied by server S p ha conain he coded version of a segmen requesed by U k, and is also equal o he number of segmens of W dk no presen in he cache of user U k. Overall, he message ransmied by S p is given by X p = X p (H i. (2 i [( +1]: p Hi φ From he ransmied message X p (H i in (1 for each se H i, user U k can decode he MDS coded version C p d k,h of is requesed segmen W i\{k} d k,h i\{k}. Wih he ransmissions from all he servers, U k receives coded versions of each missing segmen from he servers i is conneced o. Since each segmen is coded wih a (P, MDS code, he user is able o decode each missing segmen. oe ha each ransmied message X p (H i by a server is of lengh F / ( { [ bis. The number of ( ransmissions by S p is i +1 ] : p Hi φ} = ( { [ ( +1 i +1 ] } ( : p Hi = φ = +1 ( qp +1. In oher words, server Sp ransmis ( +1 ( qp ( +1 messages, each of lengh F / bis. Therefore, he sum delivery rae R is given by R = R p = 1 ( = P ( ( 1 ( [( ( ] qp ( qp. (3 In order o characerize he bes and wors nework opologies ha lead o he minimum and maximum delivery raes, we presen he following lemma wihou proof. Lemma 2. For n 1, n 2, r Z + saisfying r n 1 and n n 2, we have ( ( ( ( n1 n2 n1 + 1 n (4 r r r r The lemma above indicaes he convex naure of he binomial coefficiens in (3; ha is, he poins (r, ( ( r r, (r + 1, r+1 r,..., (n1 + n 2 r, ( n 1+n 2r r form a convex region. From Lemma 2, i can be deduced ha he second summaion erm in (3 akes is minimum when max p (q p min p (q p +1, p [P ], i.e., he values of q p are as close o each oher as possible. This corresponds o he class of opologies wih he highes sum delivery rae (see Fig. 1c for an example. The opology ha requires he minimum sum delivery rae of R = +1 is when q p is eiher 0 or for each server, or equivalenly, when all he users are conneced o he same servers (see Fig. 1b for an example. IV. AVERAGE SUM DELIVERY RATE In his secion we sudy he average wors-case sum delivery rae, where he average is aken over all he possible nework opologies, assuming a uniformly random user-server associaion; ha is, each user connecs o any ou of P servers wih uniform disribuion. As we have seen in he previous secion he delivery rae depends on he specific opology. We remind ha, for a given opology, he worscase delivery rae refers o he wors-case demand combinaion when each user requess a differen file from he library. Le T denoe he se of all possible opologies. The cardinaliy of T is ( P. Define (qp as he number of differen opologies, in which a paricular server S p serves q p [0 : ] users. Then (q p = ( ( P 1 qp ( P 1 qp. q p 1 The following heorem provides he average wors-case delivery rae. Theorem 3. The average wors-case delivery rae over all opologies under uniformly random user-server associaion is given by E[R] = P ( ( P ( ( qp Pr(Q p = q p, (5 q p=0 where Pr(Q p = q p = (qp is he probabiliy of ( P exacly q p users being served by a paricular server. Proof. Each opology τ T is represened by a paricular uple q τ = (q 1,..., q P. Each opology is disinc, bu no all uples are necessarily disinc. This is demonsraed in Fig. 1c, where wo disinc opologies have he same uple q τ = (3, 3, 2 associaed wih hem. The expecaion of he wors-case delivery rae over all possible opologies τ T can be wrien as [ P ( q τ,τ T (+1 1 P ( qp ] ( +1 E[R] = ( P

5 = P ( ( 1 ( P ( q τ,τ T = P ( P ( (q p q p=0 ( P ( = P ( ( P ( q p=0 ( qp ( qp ( (q p qp ( P. The average sum delivery rae vs. user cache capaciy is ploed in Fig. 4. V. REDUDACY I SERVER STORAGE CAPACITY In Secion III-A he server sorage capaciy is fixed as M S =. The minimum server sorage capaciy ha would allow he reconsrucion of any demand combinaion is given by M S = M U. In his case, we cache a M U fracion of he library in he user caches during he placemen phase, and ransmi he remaining fracion of he library from he servers. The wors-case delivery rae in his case is given by R = ( 1 M U. ex, we consider he case when here is redundancy in server memories; ha is, we have < M S. Assume ha M S = z for some ineger z [ 1]. For non-ineger values of z, he soluion can be obained by memory-sharing. In his case, a (P, z Fig. 3: An example 7 5 incidence marix (P = 7, = 5 wih = 4. MDS code is used for server sorage placemen, which allows each user o reconsruc any requesed file by connecing o z servers. The user cache placemen is done as in he previous secion. In he delivery phase, each user randomly connecs o servers. We now have a degree of freedom hanks o he addiional sorage space available a each server. Each user can ge a paricular segmen from only a z subse of he servers i is conneced o by receiving one copy from each of hose servers. The choice of he servers ha will deliver he coded subsegmens o each user is done such ha he mulicasing opporuniies across he nework are maximized. Consruc an incidence marix A of dimensions P such ha a ij = 1 if server i is conneced o user j, a ij = 0 oherwise. Consider he elemen subse H i, and he file segmens W dk,h i\{k} k H i. Consider he columns of A corresponding o he users in H i and he marix Q formed by hem. Define he minimum cover of H i as he smalles l for which a l +1 submarix of Q has a leas z non-zero values in each column. The servers corresponding o he l rows of his submarix have o ransmi one coded message each o saisfy compleely he requess for he missing segmens corresponding o H i. Therefore, he oal number of ransmissions required o deliver he segmens W dk,h i\{k}, k H i, is equal o he minimum cover of H i. As an example, consider he incidence marix as shown in Fig. 3 which corresponds o a sysem wih 7 servers and 5 users, where each user connecs o 4 servers. Assume ha he server sorage capaciy is M S = 2 and = 1. In his seing, coded subsegmens of requesed files can be delivered o +1 = 2 users hrough mulicasing, and i is sufficien for each user o receive coded segmens from 2 = 2 servers. Then, for he user se H i = {1, 2}, we consider he submarix corresponding o he columns 1 and 2 and rows 1 and 2 (marked by he blue dashed lines in Fig. 3, which is he smalles submarix saisfying he condiion ha each column has a leas z = 2 1s. Hence, he minimum cover for H i is equal o he number of rows of his submarix, ha is, 2. Similarly, for H i = {3, 4} (marked by he red dashed lines in Fig. 3, and he minimum cover is 3. Thus, from (1, for segmens W dk,{3,4}\{k}, k {3, 4}, S 3 ransmis he message X 3 ({3, 4} = k {3,4} C3 d k,{3,4}\{k}, S 4 ransmis X 4 ({3, 4} = Cd 4, and S 3,{4} 5 ransmis X 5 ({3, 4} = Cd 5 4,{3}. The oal number of ransmissions is 3. This algorihm can be applied o ransmi all he missing segmens of he requesed files. VI. DISCUSSIO OF THE RESULTS In Fig. 4 we illusrae he achievable rade-off beween he user cache capaciy and he sum delivery rae for he bes opology and he wors opology, and he average sum delivery rae over all opologies. The radeoff curves are ploed for differen server sorage capaciies. We observe ha he gap beween he wors and he bes opologies can be significan. From (3 and (5 we can deduce ha he wors opology sum delivery rae; and hence, he average sum delivery rae of he proposed scheme are boh wihin a muliplicaive facor of P of he bes opology sum delivery rae. In Fig. 5 he achievable average sum-delivery rae-server sorage capaciy radeoff is illusraed for he server sorage capaciies of M S [ M U, ]; he plo is obained by performing muliple simulaions wih random realizaions of he opology and averaging he achievable sum delivery rae over hem. We observe from Fig.

6 Sum delivery rae Bes opology Wors opology(m S = 5 4 Average over opologies(m S = 5 4 Average over opologies(m S = 5 3 Average over opologies(m S = 5 2 Average over opologies(m S = 5 Sum delivery rae = 3 = 4 = User cache capaciy (M U Fig. 4: Average sum delivery rae vs. user cache capaciy M U, for P = 7, = = 5, = 4. The bes and wors opologies are as illusraed in Fig. 1. The average delivery rae is ploed for server sorage capaciies M S = 5 4, 5 3, 5 2, 5. 4 ha he sum-delivery rae decreases significanly, paricularly for low M U values, as he redundancy in server sorage increases. We observe from Fig. 5 ha he average sum-delivery rae decreases rapidly for an iniial increase in he server sorage capaciy, and he decrease can become significanly fas for high values. This is because, hanks o he MDScoded caching a he servers, available mulicasing opporuniies increases wih available redundancy. VII. COCLUSIOS In his paper, we have exended he cenralized coded caching and delivery framework of Maddah- Ali and iesen o a random opology scenario, in which users randomly connec o a subse of muliple servers, each holding only a fracion of he whole library. While his allows each server, placed a a SBS, o have only a limied amoun of sorage capaciy, i requires coded sorage across servers o accoun for he random opology. We proposed a novel scheme ha joinly applies MDS-coded caching a he servers, and uncoded caching and coded delivery o users. The achievable delivery rae of his scheme for any opology is wihin a consan muliplicaive facor of he single server rae performance, wih he consan facor represening he redundancy of he erasure code used for disribued sorage. The average rae over all opologies is also derived. This analysis shows ha he price for robusness and reliabiliy using disribued sorage in erms of mulicasing gain is no much. Finally, we have shown ha he increase in he server sorage capaciies furher reduces he gap beween he wors and bes opologies Server sorage capaciy (M S Fig. 5: Average sum delivery rae vs. server sorage capaciy M S, for P = 7, = = 5, M U = 1. VIII. ACOWLEDGEMET This research is funded by he Scavenge H2020- Marie Sklodowska-Curie Acion(MSCA-Innovaive Training ework(it Gran Agreemen umber REFERECES [1] M. A. Maddah-Ali and U. iesen, Fundamenal limis of caching, IEEE Trans. Inf. Theory, vol. 60, no. 5,May [2]. Golrezaei, A. F. Molisch, A. G. Dimakis, and G. Caire, Femocaching and device-o-device collaboraion: A new archiecure for wireless video disribuion, IEEE Commun. Mag., vol. 51, no. 4, Apr [3] M. Mohammadi Amiri and D. Gunduz, Fundamenal limis of caching: Improved delivery rae-cache capaciy rade-off, IEEE Trans. on Communicaions, vol. 65, no. 2, Feb [4] M. Gregori, J. Gomez-Vilardebo, J. Maamoros, and D. Gunduz, Wireless conen caching for small cell and D2D neworks, IEEE Journal on Seleced Areas in Communicaions, vol. 34, no. 5, May [5] J. G. Vilardebo, Fundamenal limis of caching: Improved bounds wih coded prefeching, arxiv: , [6] C. Tian and J. Chen, Caching and Delivery via Inerference Eliminaion, IEEE Inernaional Symposium on Informaion Theory, July [7] T. Luo, V. Aggarwal, and B. Peleao, Coded caching wih Disribued Sorage, ArXiv: v1 [cs.it] ov [8] S. P. Shariapanahi, S. A. Moahari, and B. H. halaj, Muliserver coded caching, IEEE Trans. on Informaion Theory, Vol. 62, Issue 12, Dec [9] M. Ji, A.M. Tulino, J. Llorca, and G. Caire, Caching in combinaion neworks, Asilomar Conf. on Signals, Sysems and Compuers, Monerey, CA, ov [10] A.A. Zewail and A. Yener, Coded caching for combinaion neworks wih cache-aided relays, Inernaional Symposium on Informaion Theory 2017, pp , June [11] A. G. Dimakis, P. Godfrey, Y. Wu, M. J. Wainwrigh, and. Ramchandran, ework coding for disribued sorage sysems, IEEE Trans. Inf. Theory, vol. 56, no. 9, Sep [12] D.S. Papailiopoulos, J. Luo, A.G. Dimakis, C. Huang, and J. Li, Simple regeneraing codes: ework coding for cloud sorage, Proc. IEEE IFOCOM, March 2012.