Coded Caching with Heterogenous Cache Sizes

Transcription

1 Coded Cachng wth Heterogenous Cache Szes Snong Wang Department of Electronc Engneerng, Shangha Jao Tong Unversty arxv:50403v [csit] 3 Jul 05 Abstract We nvestgate the coded cachng scheme under heterogenous cache szes I ITRODUCTIO Coded cachng s a novel mechansm to releve wreless congeston durng peak-traffc tmes for content dstrbuton [] [3], where the temporal varablty of wreless traffcs s utlzed Wth coded cachng, popular contents are partally prefetched at users local cache durng the placement phase, e, off-peak traffc tmes, and the rest of the contents are delvered usng coded multcastng durng delvery phase, e, peak traffc tmes upon request Compared wth the tradtonal cachng scheme that adopts the orthogonal uncastng transmsson and the cachng gan s straghtforwardly dependent on the users cache sze, coded cachng provdes codng opportuntes among dfferent requests durng the delvery phase, whch further explots cache resources by jontly optmzng the placement and delvery phases Efforts have been made to reveal the fundamental lmts of coded cachng n an nformaton-theoretcal perspectve that the coded cachng scheme n the bottleneck network can acheve a global cache gan to reduce the delvery-phase traffc volume and guarantee the order optmalty compared to the nformaton-theoretcal lower bound The semnal works attract much attentons n the communty and encourages further nvestgatons on the coded cachng scheme [4]- [8]; however, exstng works have the shared assumpton, that s, users have the dentcal cache sze, whch s extremely dffcult to satsfy n practce In ths paper, we present a comprehensve study on coded cachng wth heterogeneous cache sze at the user end Several non-trval challenges need to be addressed before we get the nsght of coded cachng scheme wth heterogeneous cache sze Frst, what s the fundamental bound n an nformaton theoretcal perspectve under heterogeneous cache sze? Second, how to mplement ths lower bound by desgnng the placement and delvery algorthms? In partcular, could the algorthms desgned for the settng of homogeneous cache sze be appled to the heterogeneous cache sze scenaro? If not, what s the root cause and how could we desgn algorthms dedcated for the heterogeneous cache sze scenaro? Thrd, how the heterogeneous cache sze nfluences the wreless traffc volume durng the delvery phase? Is t possble to obtan a smple form trade-off between cache sze and traffc rate as n the homogeneous settng [] [3]? Ths paper tres to shed lght on how to resolve these challenges, where some nterestng results have been derved T ra ffc V o lu m e /F b ts H e te ro g e n o u s B o u n d H o m o g e n o u s B o u n d U n c o d e d A v e ra g e C a c h e S z e /F b ts Fg Comparson of traffc volume versus average memory sze produced by coded cachng scheme between heterogenous and homogenous cache sze The number of users K = 5, the number of contents = 00 The aggregate cache sze s dentcal under above two scenaros We focus on a cachng system conssted of K users connectng to a server through a shared, error-free lnk The server has a database of contents Each user has accessed to a cache memory bg enough to store M of the contents In the tradtonal coded cachng scheme [3], t s assumed that each user has an dentcal memory space and the random cachng procedure n the placement phase wll produce the content segments of approxmate equal sze, and form a maxmal clque of dfferent segments that can be fully utlzed to create the codng opportuntes n the delvery phase However, heterogeneous cache sze ncurs that the sze of content segments n the delvery phase s also heterogeneous, whch causes problems for codng A straghtforward soluton s to perform paddng to smaller-szed segments so that all segments can be algned for codng Apparently, such an approach can cause larger-szed segments to mss codng opportuntes n the delvery phase thus ncrease the traffc volume In ths settng, the delvery-phase traffc volume of conventonal uncoded cachng scheme s M K = K M, whch has a local cache gan that s dependent on aggregate sze of cache K M, In contrast, under the assumpton that M M M K, our smple modfed coded cachng scheme proposed n

2 ths paper attans a traffc volume of M M j j= K Thus, n addton to the local cache gan of M, coded cachng also acheves an extra global cache gan of j= Mj for the user, whch s a nonlnear term of users cache szes that are smaller than hm Then, through dervng the nformaton-theoretcal lower bound [6] n ths case An nterestng fndng from our nvestgaton s revealed that: although ntroducng potental loss of codng opportuntes, coded cachng scheme adoptng paddng under heterogeneous cache sze stll presents a constant gap to the optmal scheme, and the constant gap s less than 0 The man reason s that the mss of codng opportuntes s an nherent lmtaton of bottleneck network under the heterogenous cache szes, whch appears not only n the coded cachng scheme, but also n the nformatontheoretcal lower bound As llustrated n Fg, the lower bound under the heterogenous cache szes s also larger than that under homogeneous case To further nvestgate the fundamental lmts of heterogenous cache sze, we ntroduce the concept of probablstc cache set and characterze such memory-traffc volume tradeoff and order optmalty va the numercal statstcs of the cache sze dstrbuton We analytcally show that both gaps of traffc volume produced by coded and uncoded cachng scheme to the lower bound wll decrease when the devaton of all users cache szes ncreases, and the uncoded cachng scheme shows a faster decreasng speed Ths result mples: frst, the coded cachng scheme wll gradually degenerate to the uncoded verson as the dfference among users cache szes ncreases; second, the mss of codng opportuntes s a ntrnsc characterstc n the bottleneck cachng network under heterogenous cache szes, snce the traffc volume of uncoded cache s rrelevant to devaton of cache szes, e, only dependent on the aggregate cache szes of all users, the decreasng gap manly comes from the ncreasng of nformaton-theoretcal bound Besdes that, we pont out such mss of codng opportuntes occurs not only n the heterogenous cache szes but also the group coded delvery GCD, where users are dvded nto groups based on ther cache szes and coded cachng are performed on each group separately An nterestng and counter-ntutve fndng s that: f we adopt GCD n the heterogeneous cache sze scenaro, such mss of codng opportuntes caused by GCD and heterogenous cache sze neutralze each other, nstead of degradng the system performance n a collaboratve manner Moreover, we fnd that as devaton of users cache szes ncreases, the GCD can be approxmately order optmal to the lower bound Ths fndng ndcates that GCD could be mplemented n real systems, because GCD can gan sgnfcant decrease n computatonal complexty at the cost of very lmted performance Fnally, we wll prove that the mss of codng opportuntes s nherent from random cachng procedure and t s mpossble to overcome t The remander of the paper s organzed as follows We descrbe the servce model and problem settng n Secton II and provde some prelmnares and motvatons n Secton III Secton IV presents our man results about coded cache under heterogenous cache szes, ncludng our modfed coded cachng scheme, traffc volume-memory trade-off and order optmalty analyss Secton V further nvestgates ths problem under the probablstc cache set and group coded delvery umercal analyss are presented n Secton VI Secton VI concludes ths work and exhbts some nterestng extensons The proof of our man results and detals are provded n the Appendx II MODEL AD ASSUMPTIO In ths paper, we consder a set of users connectng to a content server through a shared wreless lnk that s smlar as the setup n [3] A Problem Settng We consder a network conssted of a content server connected to K users through a shared, error-free lnk, as llustrated n Fg The error-free lnk can be acheved wth error correcton scheme or relable transmsson scheme n the upper layer The user set s denoted by K = {,, K} The content server has accessed to a database of unform dstrbuted contents W, W,, W wth each of sze F bts The same-sze-content assumpton s for the theoretcal convenence, whch however does not hnder the practcablty of the coded cachng operatons n the real world, because the man body of content objects can be talored as the same sze for coded cachng based dstrbuton and the rest n a small quantty can be dstrbuted n the tradtonal way The content ndex set s denoted by = {,, } Each user k has an solated cache space Z k of sze M k F bts for some real number M k [0, ] All users cache szes consttutes a cache set M = {M,, M K } Wthout loss of generalty, we assume the cache set M s a ordered set, e, M M M K The system operates n two phases: a placement phase and a delvery phase In the placement phase, the content server push contents W, W,, W to the shared lnk and the cachng of each user can be done n multple ways For example, n the centralzed manner, the server wll decde whch parts of each content s stored n each user s local cache, and there s no feedback of cache state In the decentralzed manner, the sever has no control over what parts of content goes nto each user s local cache The users dvde ther cache space nto dentcal parts and randomly choose whch bts to cache usng a random number generator By uploadng the seed value of each user s The sze can also be packet-based For smplcty, we use the bts as the metrc n the followng Here we use the cache set to denote the set conssted of users cache szes nstead of each user s local cache Z k

3 server shared lnk user cache Fg set database V V V etwork archtecture of coded cachng under heterogenous cache random number generator, the server can reconstruct the cache contents of each user In the delvery phase, each user frst sends ts request d k d k s the ndex of content W dk, d k, k K These requests are ndependently and dentcally dstrbuted d across the contents and users Then the server collects all users requests d, d,, d K and proceeds by transmttng a sgnal X d,d,,d K of sze R d,d,,d K F F bts over the shared lnk Ths metrc are referred to as the load or the traffc volume of the shared lnk under scheme F Usng the cache contents and sgnal receved over the shared lnk, each user can reconstruct ts requested contents B Problem Statement Then based on above settng, we present the basc defntons n our problem A memory-traffc volume par M, R d,,d K F s achevable for scheme F under requests d,, d K f every user k s able to reconstruct ts requested content W dk wth error probablty P e 0 and produce the traffc volume R F d,, d K F bts under the cache set M A memorytraffc volume par M, R F s achevable for scheme F f ths par s achevable for every possble requests d,, d K n the delvery phase Defned by Defnton : Achevable scheme R F 3 K M M M3 MK max R d,,d K d,,d K K F 3 the worst case normalzed traffc volume for scheme F We use the R to represent the smallest traffc volume such that M, R s achevable Defned by Defnton : Optmal scheme M, R nf {M, R F, M, F} 4 the nfmum of all achevable M, R F Clearly, R F s functon of cache set M and number of users K and number of contents To emphasze ths dependency, we rewrte above traffc volume as R F M,, K and R M,, K The am of ths paper s to fnd a scheme F such that R F M,, K guarantees the order optmalty, defned as Defnton 3: Order optmalty The scheme F s order optmal f only f R F M,, K R C, 5 M,, K C s a constant ndependent of the system parameters M, and K We can see that the order optmalty can be guaranteed only f the traffc volume produced by scheme F has the constant gap to the optmum For the smplcty of the followng llustraton, we refer M I as the homogeneous cache set for the heterogenous cache set M, both of whch have the dentcal aggregate cache sze Defned by Defnton 4: Homogeneous cache set The average set of M s defned as M I = {M, M,, M}, where M = E k [M k ] the average sze of all users local caches III PRELIMIARY AD MOTIVATIO In ths secton, we revew the basc coded cachng scheme about more detals and present the man motvaton of our work A Related Works Coded cachng s a novel technque to mtgate the wreless traffc volume durng the peak-traffc tme Ths result has been extended to nonunform demands n [4] [5], onlne cachng system n [6], herarchcal cachng system n [7] and heterogenous network wth mult-level cache access n [8] Snce our work s manly based on ths semnal work [3], we now brefly revew ths scheme wth an example so that the dscussons n the followng sectons can be understood Remark that, we use the notaton F to represent ths scheme Example Decentralzed Coded Cachng Suppose the system s dstrbutng contents A and B to users, each wth the cache sze MF bts The sze of each content s also F bts In the placement phase, each user randomly caches MF/ bts of content A and B ndependently Let us focus on content A The operatons of placement phase partton content A nto four subcontents, A = A Ø, A, A, A,, where U {, }, and A U denotes the segments of content A that are prefetched n the memores of users n U For example, A represents the segments of A only avalable n the memory of user We use to denote the expectaton sze of each segment, and A Ø = M/ F bts, A = A = M/ M/F bts and A, = M/ F bts The same analyss holds for content B In the delvery phase, we assume the worst case that user and user request content A and B, respectvely User has cached subcontent A and A, n the placement phase and lacks A Ø and A Smlarly, user has already cached B and B,, and lacks B Ø and B Wth tradtonal uncoded cachng scheme, the server s requred to uncast A Ø and A to user and uncast B Ø and B to user The total traffc volume s M M F + M F = MF bts

4 Wth the coded cachng scheme, the server can satsfy the requests by transmttng A Ø, B Ø and A B over the shared lnk, where denotes the bt-wse XOR operaton The traffc volume over the shared lnk s M M F + M F = M 4 < MF bts MF We can see that, the coded cachng scheme has a coded gan of M 4 n contrast to the uncoded cachng scheme The traffc volume for general case that has contents, K users and cache sze M s K M [ KM M ] K 6 The term K M can be seen as the traffc volume produced by uncoded cachng scheme, and only contans the local cache gan that are lnear to the cache sze Whle the coded [ cachng scheme has another multplcatve term KM M K ], named as the global cache gan, whch s nverse proportonal to the cache sze B Motvaton In the regme of heterogenous cache set, an ntuton s whether the decentralzed coded cachng scheme F can be appled straghtforward In fact, makng the followng smple modfcatons of scheme F, we can adopt t n our regme In the placement phase, user k randomly prefetches M k F/ bts of content n Based on ths cachng strategy, the same content stored n dfferent users local caches wll occupy the dfferent sze of memory space Thus, n the delvery phase, for each multcast user group U, the sze of V k,u/{k} of each user k wll be dfferent Snce these segments should partcpate bt-wse XOR, all the segments n one group should be btspadded to the longest one A possble paddng method s zerobts-paddng and we refer ths scheme as F o The followng example llustrate the performance of scheme F o Example Coded Cachng wth Heterogenous Cache Set Suppose that there are three smlar system dstrbutng contents to K = users The frst system adopts our modfed coded cachng scheme F o and the users have the cache set M = { αm, + αm}f bts wth 0 < α < The second system dvdes users nto two groups and adopts scheme F o n each group The users take the same cache set M The thrd system adopts tradtonal coded cachng scheme F, shown n Algorthm, and have the unform cache sze M I = {M, M}F bts Remark that above three systems have the same aggregate cache sze of 4MF bts Our objectve s to compare the traffc volume of above three systems Assume that user and user request content A and content B Based on the coded delvery technque, the server transmtted sgnal A B, A and B, and the correspondng sgnal sze va frst system s αm +αm R Fo M,, = max αm +αm αm + αm + = αm + αm The correspondng sgnal sze va second system s αm R U M,, = + = 4M + αm 7 8 Based on the traffc volume formula 6, the correspondng sgnal sze va thrd system s R F M I,, = M M 9 Then we consder two knds of scenaros: the cache set of small devaton α = 0 and the cache set of large devaton α = 8 The frst scenaro refers to the stuaton that the sze of each user s local cache s smlar, e, M = {8M, M} The second scenaros refers to another stuaton that the sze of each user s local cache s extremely dfferent, e, M = {0M, 38M} The number of contents = 4M Base on the traffc volume formula 7-9, we get the followng results, shown n TABLE I It can be seen that, when the sze of each user s local cache s smlar, the traffc volume produced by system approxmates to the system 3 and much lower than system ; when the sze of each user s local cache s large, t approxmates to the system and much larger than the system 3 Ths result comes from a bts waste phenomenon that when the user cache sze vares wdely, the sze of requested segments A, B are largely dfferent, and the scheme F o wll pad lots of useless zero bts to the smaller segment B, whch wll dmnsh part of codng opportuntes for the longer segment A and thus ncrease the traffc volume TABLE I THE CODED DATA AD TRAFFIC VOLUME ICURRED Scenaro/Traffcbts system system system 3 α = F F 075F α = F F 075F Example shows that when the cache set s approxmately unform, the coded cachng scheme F o stll has a global cache gan, whle when the cache set s skewng, ths global gan wll dmnsh and the groupng of users system wll not reduce the performance of whole system Ths phenomenon and the reason behnd t nspres us n three dmensons Frst, whether the scheme F o s stll order optmal or whether can we develop some new technques to overcome the mss of codng opportuntes when the users cache szes are extremely dfferent Second, as can be seen n the traffc formula

5 7, there exsts amount of maxmzaton operaton n traffc volume-memory trade-off and the number of such operaton wll ncrease exponentally wth number of users K ncreasng The key s whether we can get a close-form expresson of such trade-off Thrd, what condton the cache set should satsfy such that GCD can stll guarantee the order optmalty The followng work answers these questons IV FUDAMETAL BOUD AD IMPLEMETATIO SCHEME In ths secton, we frst utlze the Fano s nequalty and cut set-bound argument [6] to get the nformaton-theoretcal lower bound of the traffc volume, namely, the bound of M, R Then we derve a close-form expresson of traffc volume R Fo M,, K produced by our modfed scheme F o, based on whch we get a lot of nterestng nsghts Fnally, we show the order optmalty of scheme F o A Informaton-theoretcal lower bound The nformaton-theoretcal lower bound s ndependent of any specfc schemes, nstead, only dependent on the system parameters ncludng M, and K The followng theorem gves ths lower bound on the optmal achevable traffc volume R M,, K It s a standard cut-set bound [6]: for a feasble M, R par, the total nformaton contaned n the memory of any subset of caches and the server transmtted sgnals must be at least the sze of contents that users accessng these caches can reconstruct Theorem : Cut-set Bound For cachng problem wth contents, KK users, and ordered cache set M = {M, M,, M K }, we have R M,, K R c M,, K max s K max U [K], U =s s /s M U 0 Proof: The proof can be seen n Appendx-A In fact, prevous work [] ponts out the cut-set bound s sometmes loose, and tghter bounds on R M,, K can be derved va stronger arguments than the cut-set bound There are some works nvestgatng how to mprove ths lower bound [] [9] For example, n the Appendx of [], they show a possble method to get a sharper bound for K =, = Usng ths method, we can get the followng bound n our regme { 3 R M,, max M + M, M } that s much sharper than cut set bound 0 However, the cut-set bound along s suffcent for llustraton of the man dea of our work and we wll use ths bound n the followng dscusson Insght on the lower bound: In the tradtonal homogeneous case, the lower bound acheves the maxmum R c M,, K = 4M when s M, whch mples that the optmal scheme attans the delveryphase traffc volume that s nverse proportonal to the cache sze The tradtonal coded cachng scheme also attans such nverse proportonal trade-off and only exhbts constant gap less than Whle n the regme of heterogenous cache set, above analyss wll become more complcated Consder the assumpton that M M M K, the nner maxmzaton operaton n the lower bound s equal to s s M / /s Then the cut set bound R c M,, K can be regarded as the functon of s and t can be proved that R c M,, K acheves the maxmum when s satsfes s M + s M s s+ M + sm s+ > It s obvously that ths condton has a dscrete form and t s mpossble to derve such a smple form as n the homogenous case But we can use ths condton to smplfy such lower bound and make the order analyss n the sequel As can be seen n 0, when the users has the dentcal cache szes, above lower bound wll embody the bound derved n [] and we have the followng result Corollary : Relaton between two knds of cut-set bound For cachng problem wth contents, KK users wth ordered cache set M = {M, M,, M K }, we have R c M,, K R c M I,, K, equal f only f M = M I Proof: The proof can be seen n Appendx-B Corollary shows that when the aggregate cache sze s fxed, f we regard the lower bound as the functon of the cache set M, ths functon acheves the condtonal mnmum when cache set s homogenous amely, the cut-set bound wll ncrease when the aggregate cache szes s dstrbuted n a nonunform manner Ths result can be regarded as a prelmnary llustraton for the degenerated performance under heterogenous cache set The reason wll be dscussed n the last subsecton and the further quanttatve dscusson wll be presented n the Secton V B Coded Cachng Scheme For clarty, we frst present the man procedure of our modfed scheme F o mentoned before The pseudocode s shown n Algorthm Then we utlze the nherent characterstc of scheme F o and a seres of mathematcal dervaton to determne the close-form expresson of ts traffc volumememory trade-off Operator refers to btwse XOR operaton In the placement phase, the random cachng procedure wll dvde content nto K segments: W,U, U K In the delvery phase, the element V k,u/{k} n sgnal X U represents the segment of user k s requested content that beng cached n users of set U/{k} Snce we take the zero-bts-paddng for each segment before delvery, the sze of sgnal X U s determned by the longest

6 Algorthm : Decentralzed coded cachng scheme wth nonunform cache sze M Placement Phase for k = 0; k < K; k + + do for n = 0; n < ; n + + do user k randomly prefetches M k F/ bts of content n ; Delvery Phase for k = K, k > 0; k do for choose k users from K users to form a subset U do Maxsze max k U V k,u/{k} ; for l U do f V k,u/{k} < Maxsze then temp Maxsze V k,u/{k} bts of all zero; V k,u/{k} V k,u/{k} + temp; X U X U V k,u/{k} ; Multcast the coded data X U to users n U segment V k,u/{k}, k U The followng lemma shows an mportant characterstc of ths longest segment Lemma : Ordered segment sze Based on Algorthm, the sze of each transmsson sgnal X U s, X U = max k U V k,u/{k} = V nf U,U/{nf U} Proof: The proof can be seen n Appendx-C Lemma shows that the longest segment of sgnal X U s user nf U s requested content V nf U,U/ nf U, who has the mnmum cache sze of user set U Ths result s ntutve, snce the segment V k,u/k represents the logcal unt of content W dk that only stored n the caches of users n U/k and related to the the cache sze of these users, f user k has the mnmum cache sze of all users n U, the users n U/k wll has the largest memory space to store ths logcal unt and yeld the largest sze of t Based on Lemma, we can get the formula of traffc volume produced by scheme F o Lemma : Traffc-memory trade-off For cachng problem wth contents, KK users each wth ordered cache set M = {M, M,, M K }, the scheme F o produced the traffc, s R Fo M,, K = P Q S + 3 where P = Q = K j=+ s= s m, < K;, = K; 4 m j, K; 5 j= S = j=+ S + s j 3=j + m j, K; 6 j =+ m j 3 m j K j =j + j s =j s + m j m j s 7 and m = M / the normalzed cache sze by the number of contents Proof: The proof can be seen n Appendx-D Lemma presents a possble formula to calculate the traffc volume under ordered cache set M We use the followng toy example to llustrate the elte of Lemma and Lemma Example 3 K = 3, M = {M, M, M 3 } Consder the cachng problem wth contents and K = 3 users wth cache set {M, M, M 3 } Assume the users request content A, B and C n the delvery phase The transmsson sgnal and correspondng sgnal sze are, W,3 W,3 W 3, a = W,3 = M M M3 ; W, W, a = W, = M M M 3 ; W 3, W,3 a = W 3, = M M M3 ; W,3 W 3, a = W,3 = M M M3 ; W,Ø, W,Ø, W 3,Ø = 3 M M M 3 ; The operaton a s based on the Lemma After smplfcaton, the total traffc volume s P Q + P Q S + P Q + P Q S S + P Q S + P 3 Q 3 In fact, the trade-off formula n Lemma s dffcult to calculate due to the combnatoral term 7 snce t has C s K terms And we cannot get any useful nsght, e, how the user cache szes nfluence the traffc volume, from such tradeoff However, ths formula shows an useful teratve structure n aspect of number of users K that we can utlze to get a close-form expresson of the traffc volume under scheme F o For example, P K = m K P K, where P K represents the operaton P n 4 when there are K users Theorem : Close-form expresson of the traffc-memory trade-off For cachng problem wth contents, KK < users, wth ordered cache set M = {M, M,, M K }, the scheme F o produces the traffc volume of, R Fo M,, K = M j 8 j= Proof: The proof can be seen n Appendx-E Both Lemma and Theorem present the formula to calculates the traffc volume of scheme F o Lemma calculates the traffc volume based on the transmssons, whch seems

7 lke knd of a top bottom portrat, whle the Theorem calculates the traffc volume based on the users, whch seems lke knd of a left rght portrat Besdes, the proof of Theorem and the traffc volume formula 8 provde us wth some observatons: n the aspects of sth transmsson, the traffc volume due to the ntroducton of user K can be seen as the average combnaton of s th and sth transmssons when there are K users; n the aspects of the total traffc formula, the ncrement s denoted by followng term K M F 9 The man reason s that the random cachng n the placement phase and coded multcastng procedure n the delvery phase connects the users local caches together, such that the requested sent from a new user can be not only satsfed by ts own cache but also by other users caches We use the followng equvalent network model, cachng and delvery scheme to present a more nterestng llustraton Sever broadcast lnk Cache Sever Uncast lnk Cache M M M3 MK M M M3 MK Fg 3 Uncast lnk Equvalent network dagram of coded cachng As shown n the rght part of Fg 3, the equvalent network dagram conssts of a content server connectng to K users through an error-free uncastng lnk Each user can utlze the contents that beng stored n the users havng smaller cache sze, and communcate wth them by an uncastng lnk The cachng procedure s same as scheme F o and the delvery procedure s lsted n the Algorthm 3 Algorthm : Equvalent scheme F u of scheme F o Delvery Phase for k = K, k > 0; k do for = k, > 0; do user sends W dk Z to user k; Sever sends the rest part of W dk to user k; The operaton W dk Z n Algorthm 3 represents the part of content W dk that stored n user s local cache The delvery phase of scheme F u uses the herarchcal uncastng transmsson When user k sends ts request d k, t frst uses ts local cache to reconstruct part of ts requested content W dk, then t successvely uses user k to s local cache to reconstruct W dk, fnally the server uncasts the part of content W dk that are not stored n user to k s local cache to user k Thus, the traffc volume ncurred n the uncast lnk between server and user k s 9 Summmng the traffc volume produced by all users, we can get the traffc formula that s dentcal to 8 From ths equvalent network dagram and scheme F u, we can see more clearly how the coded multcast plays a role n connectng user s local cache For the tradtonal uncoded cache, each user only uses ts local cache, whle for the scheme F o, each user can access other user s cache that has a smaller cache sze Based on above close-form expresson of the traffc volumememory trade-off and the cut-set bound, we have Theorem 3: Order optmalty of scheme F o For cachng problem wth contents, KK users wth heterogenous cache set M = {M, M,, M K }, we have, R Fo M,, K R 0 0 M,, K Proof: The proof can be seen n Appendx It can be seen that the scheme F o exhbts the constant gap 0 to the cut-set bound, even less than the unform case As analyzed before, snce the mss of codng opportuntes n ths regme wll lead to the ncrease of the traffc volume, the only reason for the decreasng constant gap s that the ncrease of cut-set bound, e, a more ncreasng speed due to the heterogenous cache set Here we dscuss the man reason behnd ths trend The cut-set bound argument presents an essence that the broadcastng sgnal of cachng network plays a role n combnng users caches to jont reconstruct ther requested contents, and for a user group U of s users, the mnmum sze of broadcastng sgnal wll always be bounded by those s mnmum caches If the cache szes are dstrbuted n a nonunform manner, the gan comng from such combnng nformaton retreval wll be weaken, namely, the broadcastng sgnal s lmted n ths regme Thus, although codng scheme F o wll mss the codng opportuntes, t stll guarantee the order optmalty Moreover, ths constant gap s related to the devaton of the cache set Consder two extreme cases: n the frst case, the cache set has the smallest devaton that the all users cache sze s dentcal, then the gap has been proved to 0; n the second case, the cache set has the largest devaton that the cache sze of half users s 0 whle anther half s, then the traffc volume and the correspondng nformaton-theoretcal lower bound s all K/, the constant gap reduced to Ths result mples that the lower bound mght have a more ncreasng speed than the coded cachng scheme wth the devaton of the cache set ncreasng and the coded multcastng effcency n such network wll gradually degenerated to the uncoded manner In the next Secton, we wll use the concept of probablstc cache set to quanttatvely nvestgate the relatonshp between ths constant gap and the characterstcs of the cache set C Dscusson Before we further analyze the degenerated performance of such network, we frst use the above close-form expresson of traffc-memory tradeoff to derve some meanngful results

8 Corollary : Traffc volume under unform cache set When the sze of each user s local cache all equals to M, and the correspondng cache set s M u = {M,, M} The number of users s KK <, we have, R Fo M u,, K = K and M KM [ R Fo M u,, K R M u,, K M ] K, Proof: The proof can be seen n Appendx-F It can be seen that, the traffc formula s same as 6, whch has the same order of the nformaton theoretcal lower bound Corollary 3: Traffc volume under sngularty cache sze When the the cache set s M s = {M, M,, +αm}, α > 0, and the number of users s KK <, we have, R Fo M s,, K = RM u,, K α M M K Proof: The proof can be seen n Appendx-G Corollary shows that, the ncrease of aggregate cache sze KM wll reduce the traffc volume n an nverse proportonal manner, as show n the term /KM However, Corollary 3 shows that the ncrease of aggregate cache sze K + αm, e, the ncrease of parameter α, wll reduce the traffc n a lnear manner Thus, the effect of the heterogeneous cache set on the traffc volume has a dfferent form compared wth the homogeneous cases Besdes, another observaton s, although there exsts the mss of codng phenomenon, the ncrease of a sngle cache sze wll also decrease the traffc volume Corollary 4: Traffc volume under two groups of cache sze When the number of users s KK <, the cache set s M d = {M, M }, where M = {M,, M}, M = {αm αm}, α > 0, and M = M = K Assume K s an even number, we have, R Fo M d,, K =R Fo M,, K + Q K RM,, K, where Q K = M K Proof: The proof can be seen n Appendx-H In Example, when the cache set has large devaton, the GCD only ncreases a lttle traffc volume Here Corollary 4 makes a analytcal explanaton for ths result Snce αm, we have Q K K α Then the cache set M d has a large devaton corresponds to the large value of parameter α and that Q K approxmates to Thus, the traffc formula R Fo M d,, K wll approxmate to the traffc produced by two groups The followng corollary shows the relatonshp between traffc volume under heterogenous cache set and homogenous cache set, whch s same as Corollary about the relatonshp of the cut-set bound Corollary 5: Relaton between two knds of traffcvolume For cachng problem wth contents, KK users wth ordered cache set M = {M, M,, M K }, we have, R Fo M,, K R Fo M I,, K, equal f only f M k = E k [M k ], k K Proof: The proof can be seen n Appendx-I Corollary 5 states a fact that, when the system aggregaton cache sze s fxed, the unform dstrbuton of user cache sze wll accomplsh the mnmal traffc volume under the scheme F o Ths s easy to understand for the scheme F o, snce the there s no bt waste phenomenon when the sze of each user s local cache s dentcal, the traffc volume wll be mnmzed V THEORETICAL AALYSIS In ths secton, we quanttatvely nvestgate the degenerated performance of such network We assume the cache sze of each user s a random varable and the correspondng cache set s referred to probablstc cache set We frst rewrte the basc defntons of ths problem under probablstc cache set Then, we derve the expected gap between the traffc volume produced by scheme F o and the cut-set bound under the specfc cache sze dstrbuton Besdes that, we prelmnarly nvestgate the mss of codng phenomenon n both heterogeneous cache set and GCD A Order Optmalty under Probablstc Cache Set Let M K = {M,K, M,K,, M K,K } denote the cache set, where M K s a set of order statstcs that comes from a common parental dstrbuton F Here, the order statstcs represent the a seres of ndependent random varables such that M,K M,K M K,K Further detal can be seen n [7] A memory-traffc volume par M K, E[R d,,d K F ] s achevable for scheme F under requests d,, d K f every user k s able to reconstruct ts requested content W dk wth error probablty P e 0 and produce the expected traffc volume E[R F d,, d K ]F bts ote that expectaton s on the cache sze A memory-traffc volume par M K, E[R F ] s achevable for scheme F f ths par s achevable for every possble requests d,, d K n the delvery phase Defned by Defnton 5: Expected achevable scheme R F max E d,,d K K [ R d,,d K F ] the expected worst case normalzed traffc volume for scheme F We use the R to represent the smallest traffc volume such that M K, R s achevable Defned by Defnton 6: Expected optmal scheme M K, R nf {M K, E[R F ], F, F} 3 the nfmum of all achevable M K, R F Correspondngly, the order optmalty s defned by

9 Defnton 7: Expected order optmal The scheme F s order optmal f only f [ ] RF M K,, K E R CF 4 M K,, K We can see that ths constant s functon of the parental dstrbuton F In the followng dscusson, we wll show that ths constant s a functon of th moment and th moment of F when K = The followng theorems use the smple case K = to llustrate the both constant gaps of coded cachng scheme and uncoded verson are related to the varance of the cache set Theorem 4: Order optmalty for normal dstrbuted cache sze For the cachng problem, contents, K = users each wth cache sze M = {M,, M, }, and comes from a normal dstrbuton µ, σ, then E [ RFo M,, R c M,, ] πµ + σ π 5 Proof: The proof can be seen n Appendx-J Theorem 5: Gap of the uncoded cachng scheme For the cachng problem, contents, K = users each wth cache sze M = {M,, M, }, and comes from a normal dstrbuton µ, σ, then [ ] M, + M, / E R c M,, σ π 6 Proof: The proof can be seen n Appendx-K From Theorem 4 and 5, we can clearly see how the constant gap related to the varance of the cache set, where the varance of the cache set plays a lnear negatve role n such constant And ths gap has a faster speed n the uncoded cache, seen σ rather than the σ of coded verson Consder R Fu = E [ M, + M, ] = µ, the traffc volume produce by the uncoded cachng scheme F u s rrelevant to the varance of cache set Thus the descendng gap of uncoded verson manly come from the ncrement of the cut-set bound, whch further mples the devaton of cache set wll strengthen the nherent degenerated performance of wreless coded multcastng Remark that the varance σ cannot be nfnty snce the range of M K s [0, ] B Mss of Codng Opportuntes In the delay-senstve regme, such as vdeo-on-demand streamng, each user wll have the delay constrant and cannot wat for the arrval of all users requests Thus, we should dvde the users nto groups based on ther delay tolerant threshold, e, users wth hgh threshold are n the large group, whle the users wth low threshold are n the small group Then operate the scheme F o n each group separately Ths strategy wll also gve rse to the mss of codng opportuntes among the requests of dfferent groups We use the followng toy example to llustrate Example 4 Group coded delvery Consder two smple cachng scenaros, the frst scenaro s same as Example 3, the second scenaro dvdes users nto two groups {M, M } and {M 3 } In the frst system, the content A, B and C are dvded nto 8 logcal segments, separately Let us focus on content A A = {A, A, A 3, A, A 3, A 3, A 3, A ø } whle n the second system, each content s dvded nto 4 segments n the frst group and segments n the second group A = {A, A, A, A ø}, A = {A 3, A ø} Consderng the meanng of above logcal segments, we have the followng one-to-multple correspondence A = {A, A 3 }, A = {A, A 3 }; A = {A, A 3 }, A ø = {A 3, A ø }; A 3 = {A 3, A 3, A 3, A 3 }, A ø = {A, A, A, A ø } Then assume the users request content A, B and C, separately In the frst scenaro, server wll send the followng sgnals A 3 B 3 C, A B, A 3 C, B 3 C, A ø, B ø, C ø The second wll send A B, A ø, B ø, C ø {A, A 3 } {B, B 3 } {A 3, A ø }, {B 3, B ø }, {C, C, C, C ø } Comparng these two sequences of sgnals, we can fnd that the segments of the boldface n the second system mss the codng opportuntes Thus the traffc volume n the second system wll be larger than the frst system However, when we consder the mss of codng opportuntes due to the heterogenous cache set, we can fnd the zero-bt-paddng s appeared n the segments B 3, C, B, C, C of frst scenaro, whle just appeared n the segments B, B 3 of second scenaro, whch means that GCD can dmnsh the effect of the heterogenous cache set As for that heterogenous cache set dmnsh the effect of heterogenous cache set, we adopt the followng method We use the concept of probablstc cache set to develop an upper bound of the ncrement of traffc volume due to the GCD and show that ths upper bound s related to the devaton of the cache set Theorem 6: Expected upper bound of GCD For cachng problem wth contents, KK < users wth probablstc cache set M K, we dvde all users nto L groups K = {K,, K L }, M K = {M K,, M KL }, then we have, [ L ] E R F o M,, K H Fo,LM K,, K, L R Fo M,, K 7 It can be seen the left-sde n 7 s the expected rato between the traffc volume wth GCD and wthout GCD, the rght-sde s the upper bound, whch s dependent on the structure of parental dstrbuton F, the number of groups L, the number of contents and the number of users K Due to analytcally ntractable of the close-form expresson of

10 In c re m e n t n u m e rc a l v a lu e u p p e r b o u n d u m b e r o f G ro u p s : L a σ/µ= n u m e rc a l v a lu e u p p e r b o u n d u m b e r o f G ro u p s : L b σ/µ= n u m e rc a l v a lu e u p p e r b o u n d u m b e r o f G ro u p s : L c σ/µ= when K > Then, we nvestgate the mpact of the system parameters on the delvery-phase traffc volume Moreover, we systematcally nvestgate the performance of GCD A The Effect of Cache Set We have proven that when K =, the constant gap s related to the varance of the cache set For large K, we use the numercal analyss to show how the constant gap scales as the devaton of the cache set To guarantee the devaton of the cache set, the average cache sze cannot be too large In fact, ths constrant s reasonable, snce the number of contents s mostly much larger than the cache sze Thus, we set the maxmum average cache sze µ = 03, and the cache set comes from a normal parental dstrbuton 4 S m a ll d e v a to n L a rg e d e v a to n Z e ro d e v a to n 5 4 S m a ll d e v a to n L a rg e d e v a to n Z e ro d e v a to n 3 3 Fg 4 The comparson between approxmate upper bound 8 and real value of ncrement of traffc volume under four knds of devaton of cache set The system parameters = 500, K = 300 and µ = 00 H Fo,LM K,, K, L, we use the numercal methods to study the lkely form of t under probablstc cache set Denote by H Fo,LM K,, K, L + e Kµ L µ µ+σ, L > 8 It can be seen that ths bound s dependent on the expectaton and varance of dstrbuton F and shows a trend that the ncrement of the traffc volume wll be reduced when the varance ncreases Ths result means that the heterogenous cache set wll dmnsh the effect of the GCD The reason that we get ths knd of form of ths upper bound can be seen n the full verson of ths paper [0] The Fg 4 partly shows the tghtness and effectveness of ths bound As show n Fg 4, when the varance s extremely small, our approxmate bound s tght, especally when the number of groups s small, whle when the varance s large, ths approxmate bound wll be a lttle loose Besdes that, another mportant observaton s that he ncrement of traffc volume s factually convergent to a constant In partcular, when the varance σ = µ, ths ncrement s bounded by only a constant factor! Moreover, Theorem 6 provdes us wth an nsght that the groupng method under heterogenous cache set wll guarantee the order optmalty under specfc condton, seen n Fg 4d that σ = µ If H Fo,LM K,, K, L =constant ndependent of any system parameters, the GCD under L groups s stll order optmal Further, f L = ΘK, we can get a lnear complexty coded cachng scheme that only need to transmt ΘK sgnals Ths s further dscussed n next two sectons VI UMERICAL RESULTS In the prevous secton, all analyss s n the scenaro that K In ths secton, we present the numercal results A v e ra g e C a c h e S z e / F b ts a = 0 0, K = A v e ra g e C a c h e S z e / F b ts b = 5 0, K = 0 0 Fg 5 The mpact of average cache sze on the gap between R Fo M,, K and R c M,, K ote that the scale n the a s a sem-logarthmc coordnate system The gap n Fg 5 and Fg 6 refers to the constant gap between traffc volume produced by scheme F o and cut-set bound Fg 5 plots the gap versus the average cache sze µ under fxed varance of cache set The varance under small and large devaton are denoted by σ = 0µ and σ = 07µ, separately The zero devaton refers to the tradtonal case that the cache sze s unform and we regard t as a baselne In the Fg 5a, the number of contents = 00 s larger than the number of users K = 50 The gap under large devaton s strctly less than that under small devaton, and less than the baselne Whle n the Fg 5b that = 50, K = 00, ths relaton shows pecewse characterstcs: when the average cache sze s small µ < 05, t has the same manner as > K, when the average cache sze s large µ > 05, the gap under homogenous cache set decreases dramatcally and less than the other two cases 0 0 U n c o d e d c a c h e C o d e d c a h e T h e v a ra n c e o f c a c h e s e t σ/00µ a = 0 0,K = U n c o d e d c a c h e C o d e d c a h e T h e v a ra n c e o f c a c h e s e t σ/00µ b = 5 0,K = 0 0 Fg 6 The mpact of devaton of cache set on the gap Fg 6 plots the gap versus the varance of the cache set under fxed average cache sze µ = 30 It can be seen that, the

11 σ=0 σ=05µ σ=05µ σ=µ σ=0 σ=05µ σ=05µ σ=µ L = L =0 L =30 L = L = L =0 L =30 L = T h e n u m b e r o f g ro u p s L a µ= T h e n u m b e r o f g ro u p s L b µ= T h e v a ra n c e o f c a c h e s e t σ/00µ c µ= T h e v a ra n c e o f c a c h e s e t σ/00µ d µ= 5 0 Fg 7 The mpact of number of groups L, the varance σ and the average cache sze µ on the performance of GCD System parameters: = 300, K = 00 performance gan of coded cache compared wth the uncoded cache s 5 under small devaton σ = 005µ, whle only approxmately 3 under large devaton σ = 06µ For the case that > K, seen n Fg 6a, both gap-varance curves of coded and uncoded cache have a declnng trend and uncoded verson performs faster Whle for the case that K <, seen n Fg 6b, the gap-varance curve of coded cache shows an unque sngle valley manner: under small cache szes, t frst ncreases, then t decreases lnearly when the varance s large B The Impact of number of contents and users: and K Then we present how the gap scales as the system parameters such as and K Assume there are two relatonshps between K and : = ΘK and = ωk The frst case refers to the number of users and the number of contents have the same order The second case refers to the number of content s extremely larger that the number of users The characterstc of cache set s µ = 30, σ = 03µ H e te ro g e n o u s B a s e ln e T h e n u m b e r o f c o n te n ts a = θk 4 3 H e te ro g e n o u s B a s e ln e T h e n u m b e r o f c o n te n ts b = ωk Fg 8 The mpact of system parameters and K on the gap between R Fo M,, K and R c M,, K The average cache sze s µ = 30 In Fg 8a, we can see that the gap gradually ncreases to a constant when s suffcently large, whle n Fg 8b, the gap gradually decreases to Based on the asymptotc analyss, we can get easly get the reason for = ωk Snce, each term n 8 wll approxmate to and R Fo M,, K wll approxmate to K In the smlar manner, the cut-set bound R c M,, K wll also approxmate to K Thus the gap wll approxmate to The reason for = ΘK s based on an extremely complcated seres analyss and can be seen n our full verson paper [0] C The Performance of GCD Here we nvestgate the performance of GCD compared wth the cut-set bound R c when = 300, K = 00 For the scenaro that < K, the results are smlar and we omt ths part Eght knds of cache set tested The average cache sze µ evaluates from {30, 50} to denote the small and large cache sze, separately The varance of cache set σ evaluates from {0, 05µ, 05µ, µ}, to denote the ncreasng trend of devaton of the cache set The results are shown n Fg 7 Here the gap means the constant gap between the total traffc volume produced by GCD and the cut-set bound of the orgnal system, and the partton of users s completely random In the Fg 7a and b, we plot the gap versus the number of groups under four knds of cache sets wth ncreasng varance, t can be seen that the gap except the homogeneous cache set wll be approxmate to a constant when the number of groups ncreases, and ths constant under small cache szed has a faster convergent rate than that under large cache sze Moreover, from Fg 7c and d, we can see that the gap decreases sharply when the varance of the cache set ncreases and larger average cache sze wll produce a faster decreasng rate In the practcal scenaro, the average cache sze µ = 0 and the varance s σ = 05µ If we dvde the 300 users nto 50 groups and each group has 6 users, the gap shown n the Fg 7a s just 9 that s much less than our estmated bound 4 50 /3 50 VII COCLUSIO AD EXTESIO In ths paper, we have nvestgated the fundamental lmts of coded cache under the heterogenous cache set Through dervng the cut-set bound n ths regme, we have ponted out that even tradtonal coded cachng scheme wth zero-btspaddng can guarantee the order optmalty Moreover, usng the concept of probablstc cache set, we have proven such gap s closely related to the devaton of cache set, e, the gap wll decrease when the dfference among users cache szes ncreases Besdes that, we also have studed the group coded delvery scheme n ths regme and presented that, although both heterogenous cache set and group coded delvery wll lead to mss of codng opportuntes, the combnaton of them wll weaken ths effect, even the group coded delvery stll shows the constant gap the optmum when the cache set has the small devaton A Heterogenous Coded Delvery In ths subsecton, we dscuss whether there exsts an effcent scheme can counteract ths bt waste phenomenon

12 and produce better performance As analyzed n the prevous secton, ths phenomenon s manly caused by the dfferent length of the segment n V k,u/k In fact, ths phenomenon s also emerged n the regme of nonunform populated contents In ths case, each content s allocated the dfferent sze of memory space and the segment V k,u/k wll have the dfferent length across dfferent k, further detals can be seen n [5] To counteract ths phenomenon, recent work [5] proposes a novel technque called heterogenous coded delveryhcd, whch s based on the followng observaton: the tradtonal decentralzed coded cachng scheme only consder the codng opportuntes restrcted to form clques only between the segments of the same type and thus mssed codng opportuntes They buld upon ths observaton and desgn a new delvery scheme that consders more codng opportuntes by formng clques between segments not only of the same type, but also of dfferent types We ncorporate ths technque nto our scenaro and prove ths technque cannot produce extra performance gan We refer ths scheme as F h The pseudocode s lsted n Algorthm 4 Algorthm 3: Delvery phase n HCD scheme F h for k =, k K; k + + do for choose k users from K users to form a subset U do Maxsze max k U V k,u/{k} ; for l U do f V l,u/{l} < Maxsze then temp Maxsze V l,u/{l} bts of all segments V l,u /{l} : U U ; V l,u/{l} V l,u/{l} + temp; X U X U V l,u/{l} ; Multcast the coded data X U to users n U It can be seen that the core of ths scheme s to pad the bts comng from the hgher type segments nstead of useless zero bts to the lower type segments that has the shorter length Theorem 7: The HCD scheme F h produced the same traffc volume as scheme F o, denote by R Fh M,, K = R Fo M,, K 9 Proof: To prove the equvalence of the traffc volume produced by these two schemes, we only need to prove the hgher-bts-paddng operaton n step K and user group U, U = wll not reduce the sgnal sze X U for U, U U Based on the Lemma that segment sze s ordered, we have X U = max k U V k,u/{k} = V nf U,U/{nf U} For user m U, m > nf U, the sze of ts segment V m,u/{m} s less than X U and should be hgher-bts-padded by all nonempty segments V m,u /{m}, U U Snce U U, the k > nf U > nf U, thus we have X U = V nf U,U /{nf U } = max k U V k,u/{k} > V m,u /{m}, whch means that although the operaton n step for user group U reduce the sze of segment V m,u /{m}, t cannot reduce the sze of the sgnal X U The above theorem utlzes the property of ordered sgnal sze to prove that the HCD scheme produced the same traffc volume From the procedure of the proof, we can see that the although paddng the bts comng from the hgher type segments can reduce the sze of them, ths knd of reducng effect never happens n the longest segment n each user group, thus t cannot reduce the traffc volume B Coded Cache under Heterogenous Cache Set and Random User Demands As we dscussed before, the mss of codng opportuntes not only occurs n the coded cache under heterogenous cache set, but also under the nonunform populated contents or random user demands Smlarly, an open queston s, f we consder random user demands under the heterogenous cache set, whether such mss of codng phenomenon wll be strengthened Ths lne of work s further dscussed n [3] C Centralzed Coded Cache Under Heterogenous Cache Set In ths paper, we manly focus on the decentralzed coded cachng scheme n [3] Ths scheme does not requre a centralzed control n the placement phase and the random cachng procedure n the placement phase guarantees that the dvson of each content s automatcal and the sze of each segment s determned by the sze of cache that storng them Whle n the centralzed coded cache, the dvson of each content s pre-defned For the homogeneous cache set M I, K each content s dvded nto equal segments, where t t = KM/ However, ths knd of dvson cannot be appled straghtforwardly to the heterogenous cache set Snce the cache sze of each user s dfferent, the number of segments of each content beng dvded wll be bound by the smallest cache sze KM / In fact, we can dvde each content unequally We use the followng example to llustrate our man dea Example 3 Centralzed coded cache under heterogenous cache set Consder = 3, K = 3 so that there are three contents W, W, W 3 Assume the ordered cache set M = {M, M, M 3 } satsfyng M = In the placement phase, we splt each content nto 3 segments, namely W = W,, W,3, W,3, and the cachng strategy s Z = W,, W,3, W,, W,3, W 3,, W 3,3 ; Z = W,, W,3, W,, W,3, W 3,, W 3,3 ; Z 3 = W,3, W,3, W,3, W,3, W 3,3, W 3,3 Assume the content W,, W,3 and W,3 occupes the α, β and γ percent of each content Then we have the followng