An Efficient Delivery Scheme for Coded Caching

Transcription

1 201 27h Inernaional Teleraffic Congress An Efficien Delivery Scheme for Coded Caching Abinesh Ramakrishnan, Cedric Wesphal and Ahina Markopoulou Deparmen of Elecrical Engineering and Compuer Science, Universiy of California, Irvine {abinesh.r, ahina}@uci.edu Huawei Innovaion Cener, Sana Clara, CA Deparmen of Compuer Engineering, Universiy of California, Sana Clara cedric.wesphal@huawei.com, cedric@soe.ucsc.edu Absrac We consider a nework wih several users rying o access a daabase of files sored a a server hrough a shared link. Each user is equipped wih a cache, where files can be prefeched according o a caching policy which is mainly based on he populariies of he files. Coded caching ries o exploi coding opporuniies creaed by cooperaive caching and has been shown o significanly reduce he load on he shared link. Mos of he prior works focused on opimizing he caching policy so as o minimize his expeced load. Given he caching policy and he user demands, he problem of minimizing he load over he shared link is essenially an index coding problem. In his paper, we design a novel delivery scheme ha builds on a prior scheme for he uniform demand case, bu performs beer in he nonuniform demand case. We also evaluae his delivery scheme for differen caching policies. I. INTRODUCTION Today s Inerne raffic is dominaed by conen disribuion services like live-sreaming and video-on-demand. Demand for delivery of video conen o Inerne capable smarphones and oher mobile devices has been one of he mos driving force behind he explosive growh of raffic in he recen imes. Popular services like Neflix, YouTube, ec, exhibi wo imporan feaures: (i he user demands are predicable based on heir saisical hisory [] and (ii hey exhibi srong emporal variabiliy, resuling in highly congesed peak hours and underuilized off-peak hours. A common approach is o ake advanage of he memories disribued across he nework (a end users and/or inside he nework o sore some of he popular conens. This process, ermed caching, can be done during off-peak hours, so ha during peak hours user requess can be served from hese locally available memories wihou having o burden he nework. Design and analysis of caching echniques for various kinds of neworks have been researched exensively in he pas [1], [4], [11], [14], [18], [19] along wih he impac of user demand saisics (file populariies on he performance of caching [], [7], [23]. This body of prior work only considered uncoded caching. I is easy o see ha filling he cache wih he mos popular files (LU is indeed he opimal sraegy for a sysem wih only one user [13], bu as we move on o sysems wih requess from muliple users, new challenges and opporuniies arise. A recenly suggesed mehod combines caching of files on he user devices wih a common mulicas ransmission of nework-coded daa [20], [22]. A coded caching sraegy was proposed by Maddah-Ali e al. in [17] for a sysem of uniform user demands and cenralized conen placemen scheme and was exended o decenralized approach [16] This work was parially suppored by NS Awards (CAREER and A. Ramakrishnan and A. Markopoulou were affiliaed wih CPCC a UCI. Par of he work was conduced during A. Ramakrishnan s inernship a Huawei. and non-uniform Zipf based user demands in [21]. In [1], Llorca e al. presened a formulaion for he general conen disribuion problem, where nodes in an arbirary nework are allowed o cache, forward, replicae, and code messages in order o deliver arbirary user demands wih minimum overall nework cos. In a recen work [], Ji e al. showed order opimaliy for a caching policy ha is uniform or uniform over a subse of he files based on he Zipf parameer for a Zipf disribued file populariy disribuion. Anoher recenly suggesed approach is emocaching [6], which involves deploying a large number of dedicaed helper nodes ha cache popular file and serve he users demands locally. In his approach, he caching is done a he helper nodes raher han a he end users. A hird recenly suggesed approach combines caching of files on he user devices wih shor range device-o-device communicaions [8], [9]. In his approach, he caches of muliple devices form a common virual cache ha can sore a large number of video files, even if he cache on each separae device is no necessarily very large. In our work we sudy he coded caching approach, we consider a seup very similar o he one in [17]. The coded caching problem ypically consiss of wo phases, he placemen phase where he conens are placed in local memory based on he saisics of user demands (file populariies and he delivery phase where he remaining conen, which is no available locally, is delivered afer he demands of he users have been revealed. In his paper, we focus mainly on he delivery phase. Our main conribuion is he design and evaluaion of he Heerogenous Coded Delivery (HCD scheme ha improves upon he curren sae-of-he-ar in coded caching and significanly reduces he load on he server and he shared link during he delivery phase. The srucure of he res of he paper is as follows. In Secion II, we formulae he problem. In Secion III, we discuss he inuiion and background of coded caching. In Secion IV, we presen he proposed heerogenous coded delivery scheme and explain is working in deail. In Secion V, we presen an evaluaion of he proposed scheme. Secion VI concludes he paper. II. MODEL AND ASSUMPTIONS In his paper, we consider a se of users accessing he conen server hrough a shared link as shown in igure 1. By designing an efficien delivery scheme we can reduce he load on he shared link. A. Problem Seing We consider a sysem consising of a server conneced hrough a shared, error-free link o users as illusraed in ig /1 $ IEEE DOI.19/ITC

2 N files User 1 Z1 Z2 message X (d1,...,d k of size R (d1,...,d k bis over he shared link for some fixed real number R (d1,...,d k. R (d1,...,d k is referred o as he load in his paper and is a measure of he lengh of he message. Using is cache conen Z k and he message X (d1,...,d k received over he shared link, each user k aims o reconsruc is requesed file W dk. ig. 1. Server shared link Problem Seup for Coded Caching. User 2 User Zk size M caches 1. The server has access o a daabase of N files W 1,...,W N each of size bis. Wihou loss of generaliy, we assume all files o be of he same size 1. Each user k has an isolaed cache memory Z k of size M bis (MB packes for some real number M [0,N]. The populariy of a file W n is he probabiliy ha his file is requesed by a user. The file populariy disribuion in he server is p =[p n ] N n=1, where N n=1 p n =1. W.l.o.g. we can assume p 1 p 2 p N. The sysem operaes in wo phases: a placemen phase and a delivery phase. In he placemen phase, he users are given access o he enire daabase W 1,...,W N of files. Each user k is hen able o fill he conen of is cache Z k using he daabase. The users follow a caching policy q =[q n ] N n=1, where q n denoes he fracion of he cache space in each user ha will be allocaed o he file W n and N n=1 q n =1. The caching can eiher be done in a cenralized manner, where he server besides deciding he caching policy q also decides wha pars of each file is sored in each user s cache; or in a decenralized/disribued approach. In he decenralized approach, he server has no conrol over which pars of he file goes ino each user s cache, i can only conrol wha fracion of each file is cached (i.e. he caching policy q. The users randomly cache some porion (he size of his porion alone is dicaed by he server of each file in heir corresponding cache. The server is assumed o have complee knowledge of Z k, he conen of he cache of user k. By knowing he conen of cache Z k, he server knows which bis/packes of each file are sored in he cache of user k. In he delivery phase, only he server has access o he daabase. Each user k requess one of he files W dk in he daabase, where d k represens he index of he file requesed by user k. The vecor (d 1,...,d k is a vecor of indices of he files requesed simulaneously by all users ordered accordingly. The file requess are independenly and idenically disribued across all he users and follow he populariy disribuion p. The probabiliy ha a user requess file n is p n. The server is informed of hese requess and proceeds by ransmiing a 1 iles of differen sizes can be spli ino smaller files of he same size. B. Problem Saemen Problem Saemen. Given he cache conen Z k for each user k and he exac demand vecor (d 1,d 2,...,d, wha is he opimal lengh R(d of message X 1,...,d k (d 1,...,d k ha he server should ransmi o saisfy he given demand? In general, his is an opimizaion of he delivery scheme and assumes he conen of he caches o be given. As shown in he nex subsecion, his opimizaion is in general NP-hard. Here we design a pracical delivery scheme for he given cache sae ha helps o reduce he lengh of he message X (d1,...,d k compared o he curren sae of he ar scheme in [21]. The expeced load is defined as he expecaion of he loads over all possible demand vecors, i.e. R(p = R (d1,...,d p d1 p d2 p d and will be used o evaluae how he delivery schemes perform over all (d 1,...,d demands. C. ormulaion In general, he process of various users caching differen pars of a given file resuls in spliing a file ino several nonoverlaping subfiles, such ha each subfile is presen in he caches of a disinc subse of users. Consider a file in he server, say A, here are users in he sysem and during he placemen phase, based on he caching policy, differen pars of his file A migh ge placed a he cache of each user. Grouping he bis of his file based on he se of users hey are cached a, he file A can be spli ino several subfiles A S, where each subfile A S denoes he group of bis ha are sored only in he cache of he specific se of users given by he corresponding S {1,...,}. or example, le us say here are =2 users, hen he file A can be possibly spli ino A {},A {1}.A {2} and A {1,2}. Here A {} represens all he bis ha are cached a neiher of he wo users, A {1} (A {2} represens he bis ha are cached only a user 1(2 and A {1,2} represens he bis ha are cached a boh users 1 and 2. Noe ha hese subfiles are non-overlapping, i.e. if a bi of he file is presen in one subfile i canno be presen in any oher subfile. Also noe ha no all subfiles have o be populaed, in he example above if no even a single bi of file A is cached a boh user 1 and 2 hen A {1,2} is unpopulaed and hence can be discarded. The cache Z k of he user k can be hough of as a collecion of such subfiles ha are cached a his user k. The subfiles can in urn be classified ino ypes based on he number of users hey are cached a. We say a subfile is of ype if exacly users have cached he bis of his subfile. I is easy o see ha here are +1 ypes of subfiles and here are ( subfiles for each ype. In a cenralized approach, he server has complee conrol over which bi of each file ges placed in he cache of each user. This basically means ha he server can deermine how a file is spli in he subfiles defined above. In he decenralized 47

3 approach, since he users randomly choose he bis hey will cache, he spliing is also randomized. Now consider a demand vecor (d 1,d 2,...,d, which indicaes ha user 1 demands file W d1, user 2 requess file W d2, and so on. To simplify noaion, le V k denoe he file requesed by user k, i.e. V k = W dk and V k,s denoe he subfiles corresponding o he file W dk ha are requesed by user k and are only available in he cache of users in S. The noaion S is used o refer o some ordered se of users, S {1, 2,...,}. or a given demand, based on he cache conen across all users, he server creaes subfiles of he form V k,s, for each user k, which will be used for ransmission. The user k already knows some subfiles of he file V k, as hese subfiles are already presen in is cache Z k ; so he server only needs o ransmi he subfiles ha are no presen in is cache Z k o saisfy user k s demand, i.e. he server needs o send all populaed subfiles of he form V k,s\{k} S, for ha paricular user k. Noe ha even if wo users, i and j, reques he same subfile, say A S, we will use separae noaions V i,s and V j,s o denoe he subfile requesed by he respecive users. or a given caching and demand vecor, finding he delivery scheme ha minimizes he code lengh of he message is equivalen o solving an index coding problem [2], [3] whose side informaion graph is deermined by he caching configuraion. The side informaion graph G = (V, E, where V denoes he se of verices and E he se of (direced edges, can be consruced as follows Each bi requesed by each user is represened as a verex. More specificially, every verex v V corresponds o a bi in a subfile of he form V k,s\{k} ha is requesed by user k and is cached only a all he users in he se S\{k}. If wo users reques he same bi of a file, hen ha bi is represened by wo disinc verices. Noe ha he noaion V k,s\{k}, when used in conex of he side informaion graph, denoes he group of verices ha represen he bis in he subfile V k,s\{k}. There exiss an edge (u, v Eiff he cache of he user requesing he bi represened by u conains he bi represened by v. I is well-known ha he index coding problem is, in general, NP-hard [12]. One can use he chromaic number based approach o ge a sub-opimal soluion [] by consrucing an undireced graph G a = (V, E a similar o G. The verices of G a are he same as he verices of G and here exiss an undireced edge (u, v E a beween wo verices u and v iff he cache of he user requesing he bi represened by u conains he bi represened by v and he cache of he user requesing he bi represened by v also conains he bi represened by u. The chromaic number soluion for he complemen graph of G a, which is equivalen o he minimum clique cover of he graph G a, will give a subopimal approach for he delivery scheme in our problem, bu finding he chromaic number is also known o be NP-hard, in general. III. BACGROUND ON CODED CACHING In his secion, we presen background on coded caching and a raher deailed overview of prior work, including observaions and insighs ha inspired our new scheme. A. Uniform Demands We sar from he inuiion of he caching policy and he delivery scheme for he uniform demands scenario, considered in [17]. The N files in he daabase of he server are assumed o have equal populariy here, i.e. he probabiliy ha a given file is requesed by a user is 1/N. Due o he uniform populariy, i only makes sense ha an equal porion of each file be cached a he user caches and since each user has a cache of size M files, some M/N porion of each file should be cached a every user. In oher words, we choose a caching policy where q n = M/N for all n. Also noe ha, [17] considers a cenralized placemen scheme, where he server can deermine which bis of each file ges cached a each user. Le us consider a file A. The caching policy dicaes ha each user caches M/N bis of his file. In order o deermine which bis ge placed a which user s cache, he server splis his file A ino ( subfiles of ype each labeled as AS, for some S. The subse S comprises of users and i is easy o see ha here are ( such subses for a sysem wih users. Since he conen of he subfiles are nonoverlapping, each subfile will conain / ( bis of he file A. A user k mus cache all he subfiles A S, where k S. There are ( 1 1 such subfiles for each user and he oal bis in hem should add up o M/N bis for each user. ( 1 = M N 1 ( = M N Thus he server will spli each file ino subfiles of ype = M/N in order o help he users deermine which bis hey will be soring in heir respecive caches. Designing an opimal delivery scheme for a given demand vecor is equivalen o solving an index coding problem. Due o he NP-hardness of his problem, pracical soluions consider subopimal soluions. The cenralized placemen scheme, described above, coupled wih he uniform caching policy inroduces a symmery in he sysem ha makes i easier o find he minimum clique cover (chromaic number soluion for any given demand vecor. The undireced graph G a =(V, E a can be consruced for a given demand vecor as explained in secion II-C. or every user, here are ( 1, subfiles each wih / ( bis ha are no cached a ha paricular user. So he server needs o ransmi a oal of ( ( 1 / bis o saisfy he demand, i.e. V = ( ( 1 / = (1 M/N. An uncoded delivery scheme would require he server o send all hose bis one by one (in he wors case scenario, when each user demands a differen file o saisfy he demand. The minimum clique cover provides a coding mechanism which would significanly reduce he number of bis ha he server needs o ransmi. Consider a subse S of +1 users, for each k Sconsider a node in G a ha represens a bi in he subfile V k,s\{k}. These +1 nodes can form a clique of size +1. By coding (XORing ogeher all he +1 bis represened by hese nodes and ransmiing hem o all users in S, each user in S will be able o decode is desired bis using he informaion sored in is own cache. Noe ha / ( cliques are required o cover all he bis in a subfile and each clique covers a bi in +1 subfiles. Each clique here is a 48

4 maximal clique and subsequenly i is no hard o see ha he cover indeed uses a minimum number of cliques. This whole process of clique cover based coding can be hough of biwise coding (XOR he subfiles of he form V k,s\{k} k S and repeaing his for all S of size +1. In he end here will be / ( ( cliques for each subse S of size +1 and +1 such subses, ( so he server only needs o send a message of size ( +1 / bis o saisfy all he demands. The ransmied message lengh R (d1,...,d k = ( ( +1 / remains he same across all demand vecors because of he symmery and he uniformiy in he placemen and delivery schemes. or easier noaion, le us denoe his load R (d1,...,d k as jus R. In [17], R is shown o be informaion heoreically opimal. R = ( +1 1 ( = +1 = (1 M/N 1+M/N In he decenralized approach for conen placemen, he server has no conrol over which bis are cached a each user, so i is no possible o look a he files as a collecion of subfiles of a single ype. The users randomly selec M/N bis o sore in heir cache and based on his random placemen of bis, each file can be seen as a collecion of subfiles of various ypes. An algorihm was provided in [16] for he delivery in he decenralized seup, which was sill based on he clique cover approach, bu he clique cover soluion yielded by his algorihm is no necessarily he minimum. Insigh. More imporanly, he algorihm in [16] was resriced o form cliques only beween nodes (i.e., code he corresponding bis ogeher in he subfiles of he same ype, and hus missed coding opporuniies We build upon his observaion and design a new delivery scheme, which considers more coding opporuniies by forming cliques beween nodes no only of he same, bu also of differen ype. B. Non-Uniform Demands Alhough uniform demands faciliae analysis, file populariies are far from uniform in pracice; in fac, hey could vary several orders of magniude. In he uniform case, each file has he same probabiliy of being requesed by a user, hus i is naural o allocae equal cache o each file a every user. In he non-uniform case, he leas popular file almos never ge requesed by users. Therefore, cache should no be allocaed equally o highly popular files and leas popular files. An inuiive way o share he cache is o make he caching policy q follow he populariy disribuion p. The work in [21] considers Zipf-disribued file populariies and hey propose a caching policy ha groups files ogeher. The files are divided ino groups based on he closeness in heir populariies and he caching policy q is designed such ha all he files in he same group will have he same amoun of cache space, q n, which is deermined by dividing he cumulaive populariy of all he files in he corresponding group by he oal number of files in ha group. However, files in differen groups will have differen q n (cache space allocaion. They also explicily sae ha he grouping can be opimized o minimize he expeced load. The work in [] considers Zipf file populariies, bu proposes a simpler and differen caching policy han he one in (1 [21]. [] divides he files ino jus wo groups. The files in one group are no allocaed any cache space a all, i.e. q n =0for all he files in his group, whereas he files in he oher group ge o divide he enire cache space equally among hemselves (q n = M/group size, for all files in his group. Boh [21] and [] assume a decenralized placemen scheme, because a cenralized approach would require a lo of work from he server, which is no as pracical as he decenralized approach. Unlike he uniform case, he load R (d1,...,d k varies across all demand vecors, hus i is more meaningful o consider he expeced load R(p insead. The delivery scheme used in [21] is essenially he same as he one in [16], as discussed briefly in secion III-A. The key idea behind he scheme in [21] is o code (biwise XOR ogeher all he subfiles of he form V k,s\{k} k S and repeaing his coding procedure for all valid S. If he size of hese subfiles is no he same, which is usually he case when employing non-uniform caching policy and/or decenralized placemen scheme, he scheme jus pads hem wih zeros o make all heir sizes he same. Considering a graph G a consruced based on he cache conen of he users, for each S, he algorihm ries o cover all he nodes in he groups of he form V k,s\{k} k S,by rying o form cliques of maximum size S \ {k} beween he nodes across he groups V k,s\{k} k S. If i canno find any uncovered node wihin he groups V k,s\{k} k S o form a clique of size S \ {k}, hen he algorihm simply chooses o form a clique of lower size wih he available nodes. Insigh. In he siuaion jus described above, he algorihm unnecessarily resrics iself from considering all coding opporuniies: i could form a bigger clique wih nodes from a differen group of he form V k,s \{k}, where S S. This is he key poin we will exploi in he improved algorihm ha we presen in he nex secion. The delivery scheme in [] makes use of he minimum clique cover soluion, bu finding he minimum clique cover is NP-hard and [] does no provide any pracical algorihm o do ha efficienly. Boh [21] and [], prove ha heir respecive placemen and delivery scheme combinaions are indeed order opimal. In paricular, [21], chooses a specific grouping, where files wih populariy differing by a mos a facor of wo are grouped ogeher, o prove he order opimaliy of heir approach. IV. HETEROGENOUS CODED DELIVERY In his secion, we propose he Heerogenous Coded Delivery scheme (HCD. irs, we define he algorihm and we discuss he core ideas and inuiion on how his scheme achieves beer performance han he sae-of-he-ar in [16] and [21]. Then, we walk hrough he deails of his new scheme hrough an illusraive example. A. The Scheme The pseudocode for he HCD scheme is presened in Algorihm 1. HCD can be used wih boh cenralized and decenralized placemen approaches. The new scheme, similar o he ones in [16], [21], is sill based on finding a clique cover for he graph G a. Bu HCD explois he possibiliies of forming cliques wih nodes from subfiles of higher ypes, which could poenially reduce he number of cliques required for he cover. The users have already populaed he cache 49

5 based on some caching policy and placemen scheme. Once he server is informed of he user requess, i.e. he demand vecor (d 1,...,d, i would be able o forma he subfiles V k ha are required for ransmission. Algorihm 1 Heerogenous Coded Delivery (HCD 1: procedure DELIVERY(d 1,...,d 2: for =1, 2,..., 1, do 3: 4: for S [] : S = do binsize =max k S V k,s\{k} : 6: for k S do if V k,s\{k} <binsizehen 7: Move (binsize V k,s\{k} bis from non-empy bins V k,s \{k} : S S o emp 8: Creae new subfile V k,s\{k} 9: V k,s\{k} k,s\{k} + emp : coded coded V k,s\{k} 11: else 12: coded coded V k,s\{k} 13: end if 14: end for 1: server ransmis subfile coded 16: end for 17: end for 18: end procedure The procedure DELIVERY is called for he given demand vecor o deermine he message X (d1,...,d k ha needs o be ransmied o saisfy he demands. The scheme requires o ierae over all possible subses of users, S, and he wo ouermos loops in he algorihm help o do ha. Noe ha in Algorihm 1, we use he noaion [] o refer o he se {1, 2,...,}, he operaor + refers o concaenaion and refers o biwise XOR operaion. The innermos loop helps o ierae hrough each user wihin a given subse S. The seps wihin he inner mos loop are he core of he algorihm and he difference beween our algorihm and hose presened in [16], [21]. Recall ha he algorihms in [16], [21] code ogeher all he subfiles of he form V k,s\{k} k S, afer appending zeros o make hem all of he same size (equal number of bis. In our algorihm, insead of appending zeros righ away, we firs ry o borrow bis from he immediae higher ype subfiles of he form V k,s \{k}, where S S. Afer exhausing all he opions for borrowing, we append zeros. In he algorihm, his borrowing process is accompanied by a sep involving he creaion of new subfile V k,s\{k}. We do his o avoid conflic wih he definiion of V k,s \{k}. Noe ha here are bis in he new subfile ha, alhough presen in he caches of users in S\{k}, are no longer cached only a hese user in S\{k}. Since S can be any superse of S, all he bis in he new subfile V k,s\{k} are cached a all he users in S\{k} and so hey would sill be able o decode heir respecive bis using he informaion presen in heir cache. Also noe ha he borrowing process acually involves moving bis from he original subfile, no jus copying hem. I is imporan o firs code and ransmi he subfiles of lower ype before moving on o higher ypes, because a subfile can only borrow bis from he corresponding subfiles of higher ypes. In graph heoreic erms, we consider a graph G a consruced based on he cache conen of he users, as in [16], [21]. or each S he algorihm ries o cover all he nodes in W1 p1 =0.3 W2 p2 =0.2 W3 p3 =0.2 W4 p4 =0.2 W p =0.1 ig. 2. ile Server wih N =and populariy disribuion p. he groups of he form V k,s\{k} k S, by rying o form cliques of maximum size S \ {k} beween he nodes across he groups V k,s\{k} k S. Our algorihm differs in he fac ha, if i canno find any uncovered node wihin he groups V k,s\{k} k So form a clique of size S \ {k}, insead of jus seling wih forming a clique of lower size, our algorihm ries o cover he nodes from he corresponding higher ype groups V k,s \{k}, where S S. We would like o poin ou ha, since our algorihm ries o cover nodes from higher ype groups whenever possible, even in he wors-case scenario he performance of our algorihm will be a leas as good as he ones in [16], [21]. Observe ha we do no ry o opimize he borrowing sep, i.e. we do no ry o deermine he subfiles o borrow bis from such ha final clique is minimized. The opimizaion of his sep is he core of he minimum clique cover and so could be NP-hard. Insead, we choose o go wih a simpler and more pracical approach wherein a subfile V k,s\{k},if required, will firs borrow from a valid immediae higher ype subfile of he form V k,s \{k}, where S S. or example, consider an ieraion where we are rying o code ogeher he subfiles V 1,{2}, which has bis, and V 2,{1}, which only has 1 bi, and he oher subfiles relaed o user 2 presen in he sysem are V 2,{1,3,4},V 2,{1,4,} and V 2,{1,3,4,} wih 2 bis each. The algorihm will firs ry o borrow from he immediae higher ype subfiles, which in his case are V 2,{1,3,4} and V 2,{1,4,}. The nex higher ype subfile V 2,{1,3,4,} will only be considered if V 2,{1,3,4} and V 2,{1,4,} do no have enough bis o borrow from. B. Example We now presen a deailed walkhrough of our algorihm hrough an illusraive example. This will help highligh he subleies in he proposed scheme. Example 1. Consider a sysem wih a server consising of N =files, each of size bis, =users, each wih a cache of size M =2files and populariy disribuion p as shown in fig. 2. The file populariy disribuion akes hree disinc values: p 1 =0.3, p 2 = p 3 = p 4 =0.2 and p =0.1. or his example, we will consider a caching policy ha follows exacly he populariy disribuion, i.e. q = p, and a cenralized placemen scheme. We choose he cenralized placemen scheme as i helps o highligh he key difference beween our algorihm and he one in [16], [21]. The cenralized scheme for he non-uniform caching policy is quie similar o he one described in secion III-A. The server splis a file W i ino ( i subfiles of ype i = p i M, each geing / ( i bis as shown in fig. 3. The server splis he file W1 ino =subfiles of ype 1 = p 1 M =3, namely W 1,{1,2,3}, ( 3 0

6 TABLE I. User i Subfiles requesed bu no cached a user i Size of each bin 1 V 1,{2,3,4},V 1,{2,3,},V 1,{2,4,},V 1,{3,4,} / 2 V 2,{1,3},V 2,{1,4},V 2,{1,},V 2,{3,4},V 2,{3,},V 2,{4,} / 3 V 3,{1,2},V 3,{1,4},V 3,{1,},V 3,{2,4},V 3,{2,},V 3,{4,} / 4 V 4,{1,2},V 4,{1,3},V 4,{1,},V 4,{2,3},V 4,{2,},V 4,{3,} / V,{1},V,{2},V,{3},V,{4} / W 1,{1,2,4}, W 1,{1,2,}, W 1,{1,3,4}, W 1,{1,3,}, W 1,{1,4,}, W 1,{2,3,4}, W 1,{2,3,}, W 1,{2,4,} and W 1,{3,4,}, each wih / bis. Similarly, each of he files (W 2,W 3,W 4 wih populariy 0.2, will be spli ino ( 2 =subfiles of ype 2 ( 2 = 3 = 4 =0.2M =2, each wih / bis. or insance, he file W 2 will be spli ino he following subfiles: W 2,{1,2}, W 2,{1,3}, W 2,{1,4}, W 2,{1,}, W 2,{2,3}, W 2,{2,4}, W 2,{2,}, W 2,{3,4}, W 2,{3,} and W 2,{4,}. inally, he file W will be spli ino ( 1 =subfiles of ype =0.1M =1, each wih / bis, namely W,{1},W,{2},W,{3},W,{4} and W,{}. A user i will sore in is cache all he subfiles of he form W k,{s:i S} k, i.e. i will sore all he subfiles wih a label of he form W k,s, where i S. Observe ha now each user will have cached p i M bis of he file W i and since pi =1, he users end up wih heir caches fully occupied. We will firs consider he demand vecor (W 1,W 2,W 3,W 4,W o see he improvemens his new scheme offers. Afer he server is informed abou he demand vecor, i would be able o creae and populae he subfiles V k,s using he corresponding W dk,s. User 1 demands he file W 1. I already has he subfiles V 1,{1,2,3}, V 1,{1,2,4}, V 1,{1,2,}, V 1,{1,3,4}, V 1,{1,3,} and V 1,{1,4,} in is cache, so he server only needs o send he remaining subfiles V 1,{2,3,4}, V 1,{2,3,}, V 1,{2,4,} and W 1,{3,4,}. Table I shows he subfiles, along wih he number of bis in each, ha he server has o send for each user. The algorihm goes hrough all he subfiles in Table I, saring wih he lower ype ones. In his example he lowes ype subfiles are V,{1},V,{2},V,{3} and V,{4}, each conaining / bis of he file V i ha is no cached a user. If here is a subfile V 1,{} conaining bis demanded by user 1 and cached only a user, we will be able o code (biwise XOR V,{1} wih V 1,{}. Bu such a subfile is no presen, so we creae a new subfile V 1,{} and populae i wih bis borrowed from V 1,{2,3,}, V 1,{2,4,} and V 1,{3,4,}, which we refer o as he donor subfiles. V 1,{} has /, so we borrow /1 bis from each of he hree donor files reducing heir size o / bis. Noe ha he bis in hese donor subfiles are all demanded by user 1 and are cached a user (and also a few oher users, so by ransmiing he / bis obained by coding V,{1} wih V 1,{} user 1 would sill be able o recover is requesed bis and a he same ime user would also be able o receive and recover some of is requesed bis. This process is repeaed for all he remaining subfiles ype 1 as shown in able II and fig. 4. Moving on o he ype 2 subfiles, we can see ha here are 18 subfiles under his ype. Originally, all of hese subfiles had / bis each, bu he borrowing seps shown in able II has reduced he size of some of hem o / bis. Consider he subfiles V 2,{3,4}, V 3,{2,4} and V 4,{2,3}, none of hem were used in able II and so hese subfiles can be coded ogeher and ransmied as a single subfile of size / bis. TABLE II. Subfiles Size Donor subfiles : Bis aken Bis lef Bis lef (before (afer V 1,{2,3,} : 1 1,{} V 1,{2,4,} : 1 V 1,{3,4,} : 1 V 2,{1,} : 1 2,{} 3,{} 4,{} V 2,{3,} : V 2,{4,} : V 3,{1,} : V 3,{2,} : V 3,{4,} : V 4,{1,} : V 4,{2,} : V 4,{3,} : V,{1} 1,{} V,{2} 2,{} V,{3} ig. 4. Cliques of size 2 ig.. 3,{} V V 4,{},{4} V 2,{3,4} V 3,{2,4} V 4,{2,3} Cliques of size 3, no bis borrowed. TABLE III. Subfiles Size Donor subfiles : Bis aken Bis lef Bis lef (before (afer 1,{2,} V 1,{2,3,} : V 1,{2,4,} : 1,{3,4} V 1,{2,3,} : 0 V 1,{3,4,} : V 1,{2,4,} : 1,{4,} 0 V 1,{3,4,} : 0 Nex, consider he subfile V 2,{1,}, i could be poenially coded ogeher wih bins V 1,{2,} and V,{1,2} if hey were presen. We can creae new subfile V 1,{2,} and fill i up wih / bis from donors V 1,{2,3,} and V 1,{2,4,}, bu here is no poin in creaing a subfile V,{1,2} as i does no have any donor subfiles of higher ypes o borrow from. V 3,{1,} and V 4,{1,} encouner a similar siuaion and he enire process for hose subfiles can be seen in able III and fig. 6. Consider subfile V 2,{3,}, i could be poenially coded wih V 3,{2,} and V,{2,3}, one of which is already presen. As in he 1

7 W 1 p 1 =0.3 W 2 p 2 =0.2 W p = W 1,{1,2,3} W 1,{1,2,4} W 1,{1,2,} W 1,{3,4,} W 2,{1,2} W 2,{1,3} W 2,{1,4} W 2,{4,} W,{1} W,{2} W,{3} W,{4} W,{} Type 3 subfiles Type 2 subfiles Type 1 subfiles ig. 3. Placemen Phase,{1,2},{1,3},{1,4} 1,{2,} V 2,{1,} ig. 6. V 1,{3,} V V 1,{4,} 3,{1,} V 4,{1,} Cliques of size 3 ha are reduced o cliques of size 2, bis borrowed.,{2,3},{2,4},{3,4} TABLE IV. Subfiles Size Donor subfiles : Bis aken Bis lef Bis lef (before from each bin (afer 1,{2,3} 1,{2,4} 1,{3,4} 0 V 1,{2,3,4} : V 1,{2,3,} : V 1,{2,3,4} : V 1,{2,4,} : V 1,{2,3,4} : V 1,{3,4,} : ,{2,3} 1,{2,4} 1,{3,4} V 2,{3,} V 3,{2,} V 2,{4,} V 4,{2,} V 3,{4,} V 4,{3,} ig. 7. Cliques of size 3 ha are reduced o cliques of size 2, no bis borrowed. previous case, i does no make sense o creae a new subfile V,{2,3} as i does no have any higher ype donor subfiles o borrow from. Boh V 2,{3,} and V 3,{2,} have / bis as seen in able II each, so we could code hese wo ogeher and ransmi a single subfile of size / bis. This, along wih a similar process for V 2,{4,}, V 4,{2,}, V 3,{4,} and V 4,{3,}, are shown in fig. 7. There are sill a few more subfiles of ype 2 ha have no been ransmied ye, namely V 2,{1,3}, V 3,{1,2}, V 2,{1,4}, V 4,{1,2}, V 3,{1,4} and V 4,{1,3}. None of hem were used as a donor so far and so each of hem sill has / bis. V 2,{1,3} and V 3,{1,2} can be coded ogeher wih V 1,{2,3}, which is no presen. So we creae a new subfile V 1,{2,3} and ry o fill i up wih bis from donor subfile. We can see from Table IV ha here is only one non-empy donor subfile ha we could borrow from and we will borrow all / bis from i. or he oher subfiles, we are lef wih no donor subfiles o borrow from and hence i does no make sense o creae new subfiles. The process is illusraed in able IV and fig. 8. In oal, we have ransmied 4 coded subfiles of size / bis (fig. 4, 4 coded subfiles of size / bis (figs. and 8 and 6 coded subfiles of size / bis. Therefore, he oal lengh of he message ha was ransmied o saisfy he demands vecor (W 1,W 2,W 3,W 4,W is V 2,{1,3} V 3,{1,2} V 2,{1,4} V 4,{1,2} V 3,{1,4} V 4,{1,3} ig. 8. Cliques of size 3, some are reduced o size 2. 4 =1.4 bis Baseline. The scheme in [16], [21] does no borrow bis from higher ype subfiles during he coding process, so heir scheme will end up ransmiing 4 subfiles of size / bis for user s reques, coded subfiles of size / bis for user 2, 3 and 4 s requess and 4 more subfiles of size / for user 1 s reques. This sums up o a oal of 2.2 bis, which is considerably more han he lengh of our scheme. V. EVALUATION We presen an evaluaion of he Heerogeneous Coded Delivery (HCD scheme using simulaions for boh he cenralized and decenralized approaches. Our focus is mainly on he evaluaion of he delivery scheme, no of he caching policy; he laer is fixed for each evaluaion. The load R (d1,d 2,...,d k of he coded message depends on he demand vecor (d 1,d 2,...,d,. Therefore, o evaluae he overall performance, we will use he expeced load meric R(p = R (d1,...,d p d1 p d2 p d hroughou his (d 1,...,d secion. We will mainly be comparing he expeced load of our scheme o he sae-of-he-ar delivery scheme provided in [16], [21]. 2

8 A noe on he decenralized scenario. We would like o highligh some poins before diving ino he evaluaion resuls. In he decenralized approach, he users randomly cache q n M bis of file W n in heir respecive caches. Because of he randomness here, he bis of a file are disribued across subfiles of various ypes, unlike he cenralized approach where hey are conained in subfiles of a single ype. I is easy o see ha a bi in file W n has a probabiliy q n M of being cached a any given user. Noe q n M 1, as i does no make sense o allo more ha he 1 bis for caching he file W n. The probabiliy ha a bi ges cached a users can be easily deduced as: ( Pr(a bi is cached a users = (q n M (1 q n M Thus he number of bis in ype subfiles can be modeled as a binomial disribuion B(, q n M and he bis wihin a ype are uniformly disribued across all he subfiles of ype. We will use his modeling o simulae he decenralized caching. A. Uniform Caching Under a uniform caching policy, all N files will be allocaed an equal amoun of cache space a each user, i.e. q n =1/N n. If such a caching policy is used, irrespecive of he file populariy disribuion, boh our scheme and he scheme in [16] would yield similar load values. Especially, a cenralized placemen scheme following a uniform caching policy will have a srong symmery in he cache conens, which would resul in our scheme essenially reducing o he scheme in [16]. In he cenralized approach, he bis of a file are spli wihin subfiles of he same ype, and due o he uniform naure of he caching policy, a similar phenomenon can be observed across all he files. Here he sep creaing new subfiles ha borrow bis from subfiles of higher ype becomes unnecessary. In he decenralized approach, he subfile ypes are no longer limied o a single value. The bis of a file ge disribued across several subfiles ypes, which can be modeled as a binomial disribuion. This disribuion mainains is parameers across all files because of he uniformiy of q n. All he subfiles which would be coded ogeher in he delivery phase will have almos he same number of bis, hereby minimizing he need o borrow bis from higher ype subfiles in our algorihm. Thus he improvemen we ge from our algorihm, compared o he sae of he ar, would be negligible. This was indeed observed in our simulaions. B. Non-Uniform Caching We define non-uniform caching as a caching policy ha does no allocae equal cache space for all files irrespecive of heir populariies. This definiion includes he mulilevel grouping in [21] and he wo level policy used in []. The caching policy considered in [] divides he files ino wo groups, one group wih files of low populariy which will no be cached a any user and he remaining files of higher populariy which will divide he cache space equally among hemselves. The bis of hese files canno be coded ogeher wih anoher bi, as he users do no have he side informaion necessary o decode. The bis of he files in he Expeced Load Expeced Load Caching policy q ( Zipf parameer Caching policy q ( Zipf parameer Expeced Load Expeceed Load Caching policy q ( Zipf parameer Caching policy q (4 HCD Baseline Zipf parameer ig. 9. Plo of expeced load for N =, =4and M =and cenralized caching. q (1, q (2, q (3 : arbirary mulilevel caching polices - around % improvemen can be noed. q (4 : wo level caching scheme wih levels 0 and no improvemen. laer groups are uniformly divided in subfiles of single ype in a cenralized approach. Therefore, he HCD scheme will no have performance gain compared o he sae-of-he-ar. The same argumens from decenralized uniform caching can be used o explain he negligible difference in load observed in he decenralized approach for his caching policy. If he considered caching policy has muliple levels like he one in [21] or example 1, hen we can see a significan difference in he expeced load. Example 1 uses a caching policy where q = p, we were able o calculae he expeced load for HCD scheme as 1.16 and he expeced load of sae of he ar scheme as This brings an improvemen of nearly % for his example. We also evaluaed our scheme for a sysem wih N =files, =users and M =4 sized cache, where he user demands are Zipf disribued. The resuls of he evaluaion for a cenralized caching approach is presened in fig. 9. The figure shows a plo of he expeced load vs Zipf parameer for four differen caching policies. The firs hree are arbirary policies wih muliple level groups, we are able o see a significan improvemen for hese; he fourh one is similar o he policy in [], where an uniform caching policy is followed for he firs four popular files (level 1/4and he remaining files are no cached (level 0. HCD scheme does no provide any improvemen in his case because of he uniform naure of he caching policy. The performance evaluaion of HCD for a decenralized caching approach is shown in fig., again for a sysem wih N =, =and M =4. The plos in he op row of fig. are for a caching policy ha is also Zipf based, namely Zipf(0.,N, Zipf(0.7,N and Zipf(1,N. We can see ha for he firs wo plos in he op row, here is a performance benefi of around %. The hird plo in he op row shows a huge performance difference due o he inefficiency of he caching policy; his caching policy resuls in allocaing more han bis in he cache of each user for some of he mos 3

9 called he Heerogenous Coded Delivery (HCD scheme, ha performs significanly beer han he curren sae-of-he- ar scheme in coded caching for all mulilevel (more han wo caching policies. An open quesion for fuure work is wheher muliple levels are necessary for opimal caching policies, or wo levels are sufficien. ig.. Plo of expeced load for N =, = 4 and M = and decenralized caching. Top row: Zipf based caching policy; Boom row: Zipf based caching policy wih facor 2 grouping applied. Big gap in performance in he op righ plo is an anomaly due o q i M>1. Improvemens of around % are seen in almos all cases. popular files which is obviously inefficien. C. Comparison o Sae-of-he-Ar. Performance. HDC performs always a leas as well as he baseline scheme, by design: HCD considers all he coding opporuniies ha he baseline does, and hen some more. In he uniform case, however, here is no subsanial benefi: due o he symmery of he problem, our scheme essenially degeneraes o he baseline. The significan benefis come in he non-uniform case, for which our scheme was specifically designed o exend beyond he baseline approach. Complexiy. HDC has a polynomial complexiy in he number of subfiles V k,s (nodes of he graph G d, he same as he coded caching schemes in [16] and [21]. The main difference beween HCD and he one in [21] is he added sep of borrowing bis from higher ype subfiles o fill up he lower ype ones. Consider he wors case scenario, where a subfile of very low ype needs o borrow some bis, he algorihm will firs access he nex immediae higher ype and will keep moving up o higher ype unil i is full. The algorihm will move o higher ypes only afer exhausing all bis from he immediae higher ype. So even if i ends up borrowing bis from he highes ype subfiles, in he process i has covered all he bis for ha user s demand. VI. CONCLUSION The coded caching problem consiss of wo phases: one involves opimizaion of he caching policy and he oher is he design of he delivery scheme ha minimizes he load on he shared link for any given demand. The caching policy opimizaion is a very ineresing problem, wih recen works from [], [21] showing some order opimaliy resuls, bu no he focus of his paper. Given he cache conen and he demands of he users, he delivery phase opimizaion is basically an insance of he index coding problem, which is known o be NP-hard. We provide a pracical algorihm, REERENCES [1] I. Baev, R. Rajaraman, and C. Swamy. Approximaion algorihms for daa placemen problems. SIAM Journal on Compuing, 38(4: , [2] Z. Bar-Yossef, Y. Birk, T. Jayram, and T. ol. Index coding wih side informaion. Informaion Theory, IEEE Transacions on, 7(3: , [3] Y. Birk and T. ol. Coding on demand by an informed source (ISCOD for efficien broadcas of differen supplemenal daa o caching cliens. 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