Coded Caching for Files with Distinct File Sizes

Transcription

1 Coded Caching for Fies with Distinct Fie Sizes Jinbei Zhang iaojun Lin Chih-Chun Wang inbing Wang Department of Eectronic Engineering Shanghai Jiao ong University China Schoo of Eectrica and Computer Engineering Purdue University USA Emai: {abechina Abstract Coded caching can expoit new muticast opportunities even when mutipe users request different pieces of content and thus can significanty reduce the backhau requirement for serving high-voume content However existing studies of coded caching have been imited to the scenarios where a fies of interest are of a common size his work studies the performance imits of coded caching when the fie sizes are different We derive a new ower bound and an achievabe upper bound for the worstcase transmission rate under coded caching and show that these two bounds differ by at most a Θ(og K) factor where K is the number of users in the system here are two key noveties in our anaysis First our ower bound is derived by considering a new cut-set bound where arger fies are requested more times he anaysis of this new cut-set bound requires carefu concatenation of severa entropy inequaities Compared to a ower bound using standard cut-set arguments our ower bound is improved by a Θ(og K) factor Second our achievabe scheme uses a caching probabiity that increases proportionay with the fie size Compared to schemes that use a common caching probabiity K the achievabe rate of our scheme is reduced by a Θ( ) og K factor I INRODUCION A new form of caching schemes caed coded caching have received significant attention atey in reducing the backhau requirement for serving a arge voume of content to mutipe users [] he significant performance improvement of coded caching arises from its capabiity to expoit potentia muticast opportunities even when each user requests a different piece of content Consider the setting of [] where one server serves K users via a broadcast channe Each user has a storage/cache of size bits he server has N fies (N > K) with equa size F bits (F > N ) and each user can request any one of the N fies Note that in the worst case each user may request a distinct fie In conventiona uncoded caching schemes the server woud be unabe to expoit the broadcast capabiity of the channe in such a worst case and thus had to transmit to each user a difference piece of content that is not stored in the user s cache It is then easy to see that the tota transmission rate needed from the server in the worst case is KF ( NF ) since each user can ony cache NF fraction of each fie In contrast in a coded caching scheme [] the worst-case transmission rate from the server /NF is reduced to KF +K/NF he additiona reduction factor of /[ + K/(NF )] is significant when the goba storage capabiity of the system is arge and thus is caed the goba caching gain in [] he key idea behind this performance improvement is to expoit muticast opportunities even when different users request different fies For exampe suppose that there are two users A B and each user requests a different fie If user A has cached some content requested by user B and user B has cached some content requested by user A the server can broadcast the OR of these two parts which aows both users to decode their requested content Aong this ine [] [7] have studied the fundamenta imits of coded caching under the assumption that a fies are of the same size A common theme of these studies is to first find an information-theoretic ower bound of the transmission rate hen characterize the performance of an achievabe scheme with a common caching probabiity ie which caches a common fraction of every content in each user s cache Finay prove that the performance of the achievabe scheme is a constant factor away from the ower bound Such approaches are ater generaized to decentraized coded caching schemes [] hierarchica networks [3] and muti-eve coded caching [4] average rate [5] [6] and onine caching [7] respectivey However one imitation of these previous studies is that they assume a fies to be of the same size In practice it is common that different types of content are of significanty different sizes In this paper we carry out the first study of the performance imits of coded caching when the fies are of different sizes We derive a ower bound and an achievabe upper bound for the worst-case transmission rate that differ by at most a Θ(og K) factor he contributions of our resuts are two-fod First our ower bound (Proposition ) considers a new cut-set bound that is different from those used in prior work As a resut we tighten the ower bound by a Θ(og K) factor (see the comparison with Lemma ) In contrast to the cut-set bounds in [] [7] where each fie is requested ony once in our new cut-set bound each fie coud be requested mutipe times in proportion to its fie size Anayzing this cut-set bound aso requires carefu concatenation of severa entropy inequaities [8] and coud be of independent interest Second our achievabe upper bound (see Sec IV) uses a caching probabiity that is proportiona to the fie size In other words the amount of cached content for a fie is quadraticay proportiona to its fie size In contrast to a scheme that uses a common caching probabiity for a fies the transmission rate of our achievabe scheme is reduced by a Θ(K/ og K) factor As iustrated in abe I for a system with 8 users and types of fie sizes (the detaied setting provided in Section V) our new ower bound (LB Proposition ) is higher than the conventiona ower bound (LB Lemma ) by about % and our achievabe scheme (UB) attains a transmission rate ower

2 than that using a common caching probabiity (UB) by about 6% For a system with 8 users and 3 types of fie sizes our ower bound is 30% higher and our achievabe upper bound is about 3% ower he trend when the number of users and number of fie types further increase is iustrated in Figure hese resuts thus provide significant new insights in the performance imits of coded caching in the more practica s- cenarios of distinct fie sizes he rest of the paper is organized as foows In Section II we present the network mode We provide the ower bounds of the worst-case transmission rate in Section III and derive the achievabe rate in Section IV Numerica comparison is presented in Section V Finay we concude ABLE I: Comparisons Rate LB/LB Gain UB/UB Gain types 4/36 % 53757/353 6% 3 types 5/ % 593/696 3% og(rate) LB LB UB UB Number of types K/og K Fig : An iustration on the trend when the number of types increases II NEWORK ODEL We assume there are N fies from the set F = {F F F N } he size of the i-th fie F i is denoted by F i = Fi Without oss of generaity we assume that the fie size is non-increasing ie F i F j if i j A the fies are stored in a server which serves K users through a broadcast channe Each user is equipped with a cache of a common size he content of P the data cached N in user k is denoted by k We assume that i= F i and N K which is true in most situations where cache size is imited During off-peak hours the server can pace some of the contents (possiby coded) in each user s cache k with the hope of reducing the rate needed to satisfy users requests during peak hours his process is caed the caching phase We emphasize that the caching phase must be competed before any fie requests are made During peak hours the k-th user wi request the content of the w k -th fie namey F wk Denote the requests of a K users as a K-dimensiona vector w Since there are N fies to choose from there are totay N K request patterns w and we denote the coection of a patterns by W If part of the fie F wk has been cached in k it wi be retrieved ocay For those uncached content the server wi broadcast some additiona ogk ogk ogk content denoted by R to a K users simutaneousy he goa is that every user k must be abe to reconstruct the fie F wk based on the content R received during peak hours and the cached data k Obviousy what content R to transmit depends on the request pattern w and on the set of fies F As a resut we often denote it by R w (F) From the above discussion given a set of fies F a coded caching scheme needs to decide (a) what is the content to be cached in each k k = K; and (b) for each pattern w W what is the additiona data to send ie {R w (F) : w W} he objective is to minimize the worst-case transmission rate max w W R w (F) denoted by R(F) We said that the rate R(F) is achievabe if for every ε > 0 and every arge enough fie size there exists a caching and transmission scheme such that regardess of the request pattern w with probabiity of error ess than ε every user can reconstruct the requested fie Let R (F) denote the infimum over a achievabe R(F) A wo Approximate Systems In the previous formuation the minima rate R (F) depends on the sizes of the N fies and we did not impose any constraints on the fie sizes We now consider two quantized versions from F constructed as F UB = {Fi UB F UB i = F og F F i i N} F LB = {Fi LB Fi LB = F og F F i i N} It is easy to see that Fi LB thus naturay impies R (F LB ) R (F) R (F UB ) () F i Fi UB for any i N his P N For the practica scenarios in which i= F i and K N one can prove that the two rates R (F LB ) and R (F UB ) differ by at most a constant term since F LB i = F UB i / for a i Hence for the purpose of proving constant-factor performance gaps it is sufficient to focus on those fie sets F such that the fie sizes differ by power-of- factors hus in the seque we wi ony consider those F with the above property Assume that there are at most distinct fie sizes in F We now ca the fies of the same size as of the same type hus we introduce the foowing equivaent notations he -th type = has fie size F = F where F is the argest fie size in the system Suppose that the -th type has N different fies hen the coection of a N fies of type is denoted by F and we re-index each fie of the -th type by F = {F j j N } III LOWER BOUNDS OF HE WORS-CASE RANSISSION RAE We now present two ower bounds on R (F) he first ower bound (Lemma ) is derived using standard cut-set bounds in the iterature which is then compared with the second and tighter ower bound (Prop ) We then highight the difference in the anaysis Lemma : he worst-case transmission rate is ower bounded by R (F) max Φ P N Φ /L Φ N F L! ()

3 where L = P Φ N F / ž and the maximization is taken P over any subset Φ of { } that satisfies Φ N F K If the expressions inside the foor and cei functions in () are integers the ower bound in () can be represented as R (F) max Φ P ( Φ P N F ) 4 Φ N (3) he ower bound in Lemma can be easiy obtained using standard cut-set bounds in the iterature which we provide a high-eve sketch beow For a subset Φ of fie types we choose h (h K) users to request fies from F Φ As in prior studies [] [7] we can construct P Φ N h request patterns for these h users such that every fie in F Φ is requested once Since a the fies requested can be retrieved from the transmissions and the oca storages we have the foowing cut-set ie P Φ N h R (F)+h P Φ N F Choosing h = L we obtain () We next present the second and tighter ower bound For ease of exposition we first focus on the case when the foowing two assumptions hod Assumption : F and N F are both integers for a types such that P min( og K) min(og Assumption : K) N F = K With Assumption we wi not need to be concerned with the use of foor and ceiing functions which simpifies the discussions beow On the other hand Assumption is simiar to the constraint on the set Φ in Lemma Whie Assumptions - simpify the anaysis of Proposition they sti aow us to expose the key new insights of the proof We wi then reax Assumptions - in Proposition Proposition : Under Assumptions - the worst-case transmission rate can be ower bounded by R (F) min(og K) = N F 4 (4) o compare (3) and (4) we et = og K and N + = 4N Reca that F + = F hen the fie set Φ chosen in Lemma equas to { og K} he RHS of (3) is smaer than NF On the other hand the RHS of (4) equas to og K NF 4 which is a Θ(og K) factor higher than (3) he key novety in the proof of Proposition is to use a new cut-set bound that is different from that in () Unike the derivation of () where we consider a set of request patterns so that every fie is requested once in the new cut-set bound we consider a new set of request patterns so that arger fies are requested more times he study of this new scenario aso requires more carefu concatenation of severa entropy inequaities [8] some of which have been used in [] and [4] For ease of exposition next we focus on the simper case with ony two types = which however sti iustrates the novety of our constructions Let s = N F and s = N F By Assumption both s and s are integers and min(s s ) We then choose two user sets U and U which satisfy U = s U = s U U = By Assumption we can aways find U and U among the K different users Divide fie set F into two non-overapping subsets F and F with equa size N / We then choose two disjoint sets of request S pattern D and D which satisfy D = D = N S S and w D k U F wk = F w D Sk U F wk = F S s S S w D Sk U F wk = F w D k U F wk = F We first expain the construction of D In the very first request pattern w in D we et the s users in U request the first s fies in F and et the s users in U request the first s fies in F hen in the second request w in D we et the s users in U requests the second s fies in F and et the s users in U request the second s fies in F Continue this construction unti D = N /s ie each of the N fies in F has a been requested once At the same time totay N s s fies of F have been requested by users in U Since s = NF s = N F and F = F we have N s s = N / hat is a fies in the first haf F have been requested he construction of D is simiar he difference is that we aow the fies in F to be requested by users in U the second time during D whie the users in U wi now request the second haf F instead See Figure for iustration Fig : An iustration on the requesting process for fie systems with two types However even with this new set of request patterns new anaysis is needed in order to produce a better ower bound on the transmission rate Specificay if we directy appy the idea that the overa rate from a the transmissions pus a the cache sizes must be greater than the tota size of a fies combined (as in the derivation of ()) we woud obtain N R (F) + NF + NF N F + N F (5) s On the other hand a stricty better bound than (5) can be obtained as stated in the foowing emma Lemma : For the request patterns constructed as in Figure the worst-case transmission rate shoud satisfy N R (F) N F + N F (6) s hus the ower bound on N s R (F) is increased by another term N F his increase is the key step towards the tighter ower bound in Proposition Indeed substituting s = NF and noting that F = F the = case of Proposition foows immediatey from Lemma ie R (F) NF +NF 4 3

4 4 Proof of Lemma : For ease of presentation we denote R D S w D R w R D S w D R w U S k U k S and U k U k Summing over a rates for each transmission we have N s R (F) H(R D ) + H(R D ) = H(R D F ) + I(R D ; F ) + H(R D F ) + I(R D ; F ) H(R D R D F ) + I(R D ; F ) + I(R D ; F ) = I(R D R D ; F F ) + I(R D ; F ) + I(R D ; F ) Here the first second and third ines foow from the definition and basic properties of entropy he fourth ine is due to H(R D R D F ) = H(R D R D F F ) + I(R D R D ; F F ) and the fact that H(R D R D F F ) = 0 since R w is generated by fie-sets F and F We then bound each of the mutua-information terms in (7) Noting that a fies in F can be reconstructed from: (i) the transmissions for the request patterns in D and (ii) the oca storages of users in U we have H(F ) = I(F ; R D U ) (8) I(F ; R D ) + H( U ) where the second equaity is due to the chain rue Since the tota cache size S k U k is s = N F we have I(F ; R D ) H(F ) H( U ) NF (9) Simiary we have I(F ; R D ) NF Finay using a simiar ogic the entropy of type- fies F can be written as H(F ) = I(F ; R D R D U F ) I(F ; R D R D F ) + H( U ) (7) (0) Here the first equaity is due to the independence of F and F aong with the fact that F can be decoded from S w D D R w and S k U k S Since the size of k U k is s = NF we have I(F ; R D R D F ) NF () he resut of Lemma then foows We can then easiy generaize the above proof for the case of 3 by choosing groups of users U to U each having s = N F users his is aways possibe when Assumptions - hod he rest of the derivation for Proposition foows by appying the above techniques iterativey A Reaxing Assumptions & o reax Assumptions - the anaysis is more compicated as we need to consider many corner cases In the foowing we provide the genera ower bound without these assumptions and provide a sketch of the proof Proposition : he worst-case transmission rate is ower bounded by R (F) where R (F) is the infimum of a vaues of R that satisfy N F R > max + N F 4! 3 + N 4 F 4 3 = + = 3 N F P where = = N (F R) and the parameters to 4 N and N 4 are of integer vaues and can be uniquey computed (for any given R) in the foowing way (i) is the argest index such that F > R If no such exists choose = 0; (ii) is the argest P index satisfying (a) > (b) F > and (c) = + N < K If no such exists then choose = P and N = 0 Otherwise choose N = min(n K = + N ) (iii) 3 is the smaest index such that F If no such exists then 3 = + ; (iv) P 4 is the argest index satisfying (a) 4 3 and (b) 4 = 3 N F < K If no such 4 exists then choose 4 = 3 and P N 4 = 0 Otherwise choose N 4 = min(n 4 (K 4 = 3 N F )/F 4 ) he main intuition of this genera ower bound is as foows We divide the fie types into three groups Group : types to Group : ( + ) to and Group 3: 3 to 4 Suppose that there exists a scheme that can achieve a worstcase transmission rate R It impies that the transmission rate R has to cover a possibe request patterns chosen from a three groups of fies hen we quantify the impact of the requests for each fie group and derive the condition on R First note that each fie of type in Group is arger than R o satisfy a requests for a given fie from Group each node needs to store at east (F R) amount of its contents herefore the remaining cache size that can be used to satisfy fie requests for Groups and 3 is upper bounded by (his argument can be made precise using conditiona entropy) As a resut when considering Groups and 3 we can treat it equivaenty as if the effective cache size has been reduced to We now consider Group We first notice that for any = + to we must P have < F a request pattern in which P = + N + N We then consider = + N + N users request distinct fies By a simpe cut-set bound P argument we can obtain R > N F = + + N F For Group 3 we use the same bounding techniques as in Prop ie we choose arger fies mutipe times in the set of request patterns We can then show that in order to satisfy the requests for fies in Group 3 it requires a minimum rate that is P arger than 4 N F = N F Proposition then foows Note that the rate for fies in Group 3 can be compared to (4) It is ooser now since we have reaxed Assumptions - IV A NEW ACHIEVABLE SCHEE In this section we compare two achievabe schemes One assumes a uniform caching probabiity he other adopts a proportiona caching probabiity It wi be shown that the second scheme achieves a ower rate than the first one Consider an achievabe scheme where the fractions of every fie are cached with an equa probabiity q as in [] [7] Due P to the memory constraint we have Φ qn F = and q = Using the resuts in [] we can show that its P Φ N F achievabe rate R uni is ower bounded by R uni KF ( q) + Kq P F = N F () 4

5 5 By choosing = og K and N + = 4N we can show that the ower bound (4) is og K N F /(4) and the achievabe bound () is arger than KN F /(8) he gap between those bounds can be as arge as Θ(K/ og K) Next we present the second scheme Again we first impose Assumptions - Let = min( og K) For every fie in F ( ) denote its caching probabiity as q Namey each user k wi cache q F of every P fie in F he overa cache-size constraint thus impies that = q N F = he key difference from the first scheme is that we choose q to be ineary proportiona to F In other words the amount of cached content for a fie of type is quadraticay proportiona to F he motivation for this choice of q is as foows Note that if we consider a request patterns where a K users request ony the fies with size F the rate needed can be approximated by F q hus in order to minimize the F worst-case vaue ie max q we shoud choose q to be proportiona to the fie size P F ie q = QF where the constant Q equas to /( = N F ) his choice is aso consistent with the choice of request patterns D in the proof of Lemma and Proposition Since the arger fies in F is requested twice as frequenty as the shorter fies in F it suggests that more cache space may be aocated to F We now describe the transmission design Initiaization: each user k reconstructs the portion of the requested fie F wk that is stored in its oca cache k ransmission: for any non-empty subset U { K} we do the foowing For each k U we assembe the portion of the requested fie F wk that is cached by a users h U\k but not by user k as a continuous bit-string and denote the assembed L bit-string by B k hen we send the bit-wise ORed string k U B k Note that some bit string B k may be shorter than the other B k We simpy zero-padded the shorter bit strings during OR In the end the tota amount of bits sent for a given U is max k U B k After finishing transmission for a U it is guaranteed that a users can recover the desired packets Note that in our construction those fies of type > wi never be cached As a resut if any user k requests such a fie the entire fie wi be treated as a singe bit-string and transmitted separatey By anayzing the bit-ength of each transmission U we can upper bound the transmission rate by R prop = max B k ( + ) N F / (3) k U U Comparing eqs (3) and (4) the gap is at most 4(og K +) which is a dramatic improvement from () hus using a caching probabiity q proportiona to F is critica Note that whie our scheme aows coding across different fie-types the inequaity in (3) does not expoit this gain which may be the reason for the Θ(og K)-factor gap between our upper and ower bounds For future work it woud be interesting to see whether this gap can be removed his approximation can be observed from the first inequaity in () If users request the same fie then we ony OR the string once he reason is that OR the same string twice wi give a zero-string = When reaxing Assumptions - the anaysis becomes more compicated due to many corner cases We can have the foowing resut Proposition 3: We can construct a modified scheme that achieves R prop (3 og K + )R (F) where R (F) is specified in Proposition Namey the gap to the ower bound is at most Θ(og K) V NUERICAL COPARISON We compare the two ower bounds (3) and (4) and the two upper bounds () and (3) in the foowing two numeric exampes here are K = 8 users each with a cache size = 8 System has fie types with F = 8 N = 6 F = 4 and N = 64 System has 3 fie types with F = 8 N = 6 F = 4 N = 64 F 3 = and N 3 = 8 For the achievabe schemes instead of deriving the bounds we ist the exact R prop (UB in abe I) P and R uni (UB in abe I) vaues by numericay computing U max k U B k hese numerica resuts verify our findings ie not ony the proposed ower bound (4) is greater than the resut in (3) that uses the traditiona cut-set bounds but aso R prop is much arger than R uni hat is proportiona caching probabiity significanty outperforms uniform caching probabiity VI CONCLUSION In this paper we study coded caching for systems where fies of interest are of different sizes We provide tighter owerbound and achievabe bound for the worst-case transmission rate which differ by at most a Θ(og K) factor he key novety is a new cut-set (ower) bound that considers request patterns where arger fies are requested more times ACKNOWLEDGEN his work was partiay supported by NSF grants: CCF ECCS and CCF-4997 a grant from the Army Research Office W9NF and two grants from NSF China (No ) REFERENCES [] A addah-ai and U Niesen Fundamenta Limits of Caching in IEEE rans Inform heory vo 60 no 5 pp ay 04 [] A addah-ai and U Niesen Decentraized Coded Caching Attains Order-Optima emory-rate radeoff to appear in IEEE/AC rans Netw 04 [3] N Karamchandani U Niesen A addah-ai and S Diggavi Hierarchica Coded Caching ariv: v [csi] Jun 04 [4] J Hachem N Karamchandani and S Diggavi uti-eve Coded Caching ariv: [csi] Apr 04 [5] U Niesen and A addah-ai Coded Caching with Nonuniform Demands ariv:308078v [csi] ar 04 [6] Ji A uino J Lorca and G Caire On the Average Performance of Caching and Coded uticasting with Random Demands ariv:404576v [csi] Ju 04 [7] R Pedarsani A addah-ai and U Niesen Onine Coded Caching ariv:33646 [csi] Nov 03 [8] RW Yeung A Framework for Linear Information Inequaities in IEEE rans Inform heory vo 43 no 6 pp Nov 997

6 6 Appendix : Suppementa aterias A Proof of Proposition Proposition : he worst-case transmission rate is ower bounded by R (F) where R (F) is the infimum of a vaues of R that satisfy N F R > max + N F 4! 3 + N 4 F 4 3 = + = 3 N F P where = = N (F R) and the parameters to 4 N and N 4 are of integer vaues and can be uniquey computed (for any given R) in the foowing way (i) is the argest index such that F > R If no such exists choose = 0; (ii) is the maximum P vaue satisfying (a) > (b) F > and (c) = + N < K If not such exists then choose = P and N = 0 Otherwise choose N = min(n K = + N ) (iii) 3 is the smaest index such that F If no such exists then 3 = + ; (iv) P 4 is the maximum vaue satisfying (a) 4 3 and (b) 4 = N 3+ F < K If no such 4 exists then choose 4 = 3 and P N 4 = 0 Otherwise choose = min(n 4 (K 4 = N 3+ F )/F 4 ) N 4 Proof ogic: Suppose that a rate R is achievabe to satisfy users worst-case request We wi first show the achievabe rate shoud satisfy (4) hen since we do not know the exact vaue of R before-hand we take the minimum over a vaues of R that satisfy (4) which then becomes a ower bound for R (F) he resut of Proposition then foows owards that end we divide the fies into 4 groups he division is cosey reated to the parameter As we wi see shorty can be interpreted as the effective cache size in each user s storage that can be used to retrieve fies outside Group Group : First consider the fies in Group which contains P a the fies with size F > R Now consider Φ N request patterns as foows A fixed user say k wi request one distinct fie in Group for each of request patterns hen since user k must be abe to retrieve each requested fie we must have H( [ [ j N j F j [ [ S S j N j R w k ) = 0 (4) Let A = j N j F j and B = j N R j w hen Eq (4) can be simpified to H(A B k ) = 0 According to the property of entropy we have S S 0 = H(A B k ) = H(A k ) I(A; B k ) Further we have H(A k ) H(B) H( k A) = H( k ) I(A; k ) = H( k ) H(A) + H(A k ) H( k ) H(A) + H(B) (5) (6) P where in the ast step we P have used (5) Note that H(A) = = N F H(B) = N R and H( k ) = herefore P for any user k we must have H( k A) = (F R) = Reca that A is exacty a the information in Group herefore when user k request a fie outside Group user k must be abe to retrieve the fie with k A and its received rate R Intuitivey this means that the amount of cached information in k that can be used for recovering fies outside Group is no arger than herefore when we quantify the impact of fies in Group and Group 3 we wi take as iteray the cache size Group P : Reca that the tota number of fies in Group is L = = + N + N Note that L K according to our construction Consider one request pattern where L users request the L fies in Group hen we must have H(R) + L L k= H(F wk ) (7) For every fie F wk in Group requested by user k H(F wk ) > Reca that H( k ) herefore we have R > = + N F + N F (8) Group 3: For fies in Group 3 we want to use the resuts in Proposition herefore we need two conditions ie F 3 N 4 F and N F For fies in Group 3 we have with 3 4 are a integers F We now construct a system where user k is equipped with a cache of size which is chosen as the smaest vaue satisfying F is an integer and herefore < Ceary if there exists a scheme that can satisfy a the requests using the storage size there must exist another scheme that can satisfy a the requests using the arger storage size Let H = N for 3 4 and H 4 = N 4 he fies in Group 3 can then be divided into two subsets ie Φ 4 = { H F < } and Φ 5 = { H F } For fies in Φ 4 we have H F H F 4 < (since H F < for a ) Φ 4 Φ 4 < F Φ 4 (since F + = F / for a ) < max F Φ 4 (since F R for a ) 4R (9) For fies in Φ 5 we further choose G H to be the argest vaue satisfying that G H and G F is an integer G aways exists since F is an integer and H N Hence we have H < G H Note that Assumption now hods with G F and Further Assumption hods because K herefore for G F and we can use P 4 = 3 H F

7 7 Proposition and have G F R 4 Φ 5 H F > 6 Φ 5 Combining Equations (9)(0) we have 4 (0) H F R > 3 () = 3 Combing Equations (8) and () we concude that R must satisfy (4) he resut of the proposition then foows B Proof of Proposition 3 Proposition 3: We can construct a modified scheme that achieves R prop (3 og K + )R (F) where R (F) is specified in Proposition Proof: Caching: For each fie F in Group reca that we have F R (F) We divide F into two parts One part is of size F R (F) which is cached in every user s storage he other part is of size R (F) For the second part each user caches of it herefore at each user P the amount of storage used to cache fies in Group is = (F R (F)) hen the amount of storage that can be used for other fies is exacty as defined in Prop Let 5 be the argest vaue satisfying (i) F 5 > R (F) K and (ii) 5 4 For fies F j 3 5 j N proportiona caching as in Sec IV is used For a other fies they are not cached ransmission: For users requesting fies in Group the rate can be upper bounded by R (F) o see this consider another system where each fie is of size R (F) and the caching probabiity is / which woud requires a rate at most K R (F)( q)/( + Kq) < R (F) Let H = N for + and H = N For fies outside P Group we have two cases P Case : If = + N > K we have = + H = K For fies outside P Group the tota rate needed must be no arger than = H + F which is smaer than R (F) according to P Prop Case : If = + N K there wi be no fie with type + 3 For users requesting fies in Group transmitting P the fies directy woud require a rate ess than = + H F R (F) For fies of type with 3 5 the caching probabiity for F is F and thus arger than Q = F P 5 = 3 H F P 4 = 3 H F herefore the rate for serving fies of type can be upper bounded by F Q 3R (F) Note that 5 3 og K since F 3 R (F) and F 5 > R (F) K herefore the tota rate needed for fies of type with 3 5 is no arger than 3 og K R (F) For users requesting other fies with type arger than 5 the rate needed is ess than max(kf 4 KF 5 +) Note that P 4 = 3 H F > K herefore R (F) > K and KF 4 < 8R (F) And KF 5 + R (F) Combing a these rates together we concude 6 F 4