Centralized Coded Caching of Correlated Contents

Transcription

1 Centraized Coded Caching of Correated Contents Qianqian Yang and Deniz Gündüz Information Processing and Communications Lab Department of Eectrica and Eectronic Engineering Imperia Coege London arxiv: v1 [cs.it] 10 Nov 2017 Abstract Coded caching and deivery is studied taking into account the correations among the contents in the ibrary. Correations are modeed as common parts shared by mutipe contents; that is, each fie in the database is composed of a group of subfies, where each subfie is shared by a different subset of fies. The number of fies that incude a certain subfie is defined as the eve of commonness of this subfie. First, a correation-aware uncoded caching scheme is proposed, and it is shown that the optima pacement for this scheme gives priority to the subfies with the highest eves of commonness. Then a correation-aware coded caching scheme is presented, and the cache capacity aocated to subfies with different eves of commonness is optimized in order to minimize the deivery rate. The proposed correation-aware coded caching scheme is shown to remarkaby outperform state-of-the-art correation-ignorant soutions, indicating the benefits of expoiting content correations in coded caching and deivery in networks. I. INTRODUCTION In proactive caching, popuar contents are stored in user devices during off-peak traffic periods even before they are requested by the users [1] [3]. Proactive caching is considered as a promising soution for the recent exposive growth of wireess data traffic, and can aeviate both the network congestion and the atency during peak traffic periods (see [1] [4], and references therein. Proactive caching typicay takes pace in two phases: the first phase takes pace during off-peak traffic periods, when users caches are fied as a function of the whoe ibrary of fies, referred to as the pacement phase; whie the second, deivery phase, takes pace during the peak traffic period when the users demands are reveaed and satisfied simutaneousy. In contrast to traditiona uncoded caching schemes, which simpy empoy orthogona unicast transmissions during the deivery phase, recenty proposed coded caching [4] creates coded muticasting opportunities, significanty reducing the amount of data that needs to be deivered to the users to satisfy their demands, even when these demands are distinct. Coded caching benefits from the aggregate cache capacity across the network, rather than oca cache capacities as in conventiona uncoded caching [4]. This significant improvement has motivated intense research interest on coded caching in recent years [5] [12]. Some works foow the simpified mode proposed in [4], and aim to improve the fundamenta imits of caching [6], [7], [13]; others consider more reaistic settings, such as decentraized caching [5], nonuniform popuarities across fies [9], [10], audience retention rate aware caching [14], or heterogeneous quaity of service requirements [12]. An important feature of video contents, which is the main source of the recent exposive traffic growth, is that, there may be significant overaps among different fies, e.g., the recordings of the same event from different anges and different cameras, or even the frames of the same scene in the same video. In [15], Hassanzadeh et a. propose a correationaware caching scheme, which groups the contents in the ibrary into two sets according to their correations as we as popuarity, where each fie in second set is compressed with respect to a fie in the first set. This scheme is shown to outperform correation-ignorant caching schemes. A more information theoretic formuation for caching of correated sources is considered in [16], focusing on a specia scenario with two receivers and one cache. A simiar informationtheoretic anaysis is carried out in [17] for two fies and two receivers, each with its own cache. In this paper, we consider a server with a ibrary of N correated fies, serving users equipped with oca caches. Different from [15], which ony expoits a fixed eve of common information among the fies, we consider a more genera mode in order to fuy expoit the potentia correations among different subsets of fies. We mode each fie in the server to be composed of a group of subfies, such that each subfie is shared by a different subset of fies. The number of fies to which each subfie beongs is defined as its eve of commonness. Equivaenty, in our mode, any subset of fies S share a common part that is independent of the rest of the ibrary, and shared excusivey by the fies in S. We first propose a correation-aware uncoded caching scheme, and show that the optima pacement for this scheme is achieved by giving priority to the subfies with the highest eves of commonness in the pacement phase. We then propose a correation-aware coded caching scheme, and derive a cosed-form expression of the achievabe deivery rate, based on which the cache capacity is optimay aocated to the subfies according to their eve of commonness. Then, we compare the performance of the proposed correation-aware schemes with those that ignore the correation, and the cutset bound; and show that, expoiting fie correations in coded caching can significanty reduce the deivery rate. Notations: The set of integers {i,..., j}, where i j, is denoted by [i : j], particuary, {1,..., j} is denoted by [j]. For sets A and B, we define A\B {x : x A, x / B}, and A denotes the cardinaity of A. ( j i represents the binomia coefficient if j i; otherwise, ( j i = 0. For event E, 1{E} = 1 if E is true; and 1{E} = 0, otherwise.

2 II. SYSTEM MODEL We consider a server with a database of N correated fies, W 1,..., W N, where each fie consists of 2 independent subfies, e.g., W i = W S, i [N]. Here, W S denotes S [N] i S the subfie shared excusivey by the subset of fies {W i : i S}. For simpicity, we assume that for S [N], W S = F, if S =, i.e., the common subfies shared excusivey by fies are of the same size of F bits. Let F (F 1,..., F N. As a resut, each fie in the ibrary is aso of the same size of F bits, given by F = N =1 ( N 1 F. (1 1 For S [N], S =, we say that the subfies W S have a commonness eve of. For exampe, W {1,2,3} and W {3,4,5} both have eve 3 commonness. For brevity, we refer to a the subfies with eve commonness as -subfies, = 1,..., N. We consider users connected to the server through a shared, error-free ink, each equipped with a cache of size MF bits. We consider centraized caching; that is, the server has the knowedge of the active users during the pacement phase, though not the knowedge of their demands. Centraized caching aows the server to fi the user caches in a coordinated manner. After the pacement phase, each user requests a singe fie from the ibrary, where d k [N] denotes user k s request, k []. A the requests are satisfied simutaneousy over the error-free shared ink. An (F, M, R caching code for this system consists of: caching functions f k, k [], f k : [2 F ] [2 F ] [2 MF ], (2 }{{} N fies such that the contents of user k s cache at the end of the pacement phase, denoted by Z k, is given by Z k = f k ({W i } N i=1 ; a deivery function g, g : [2 F ] [2 F ] D [2 RF ], (3 }{{} N fies where D (d 1,..., d, such that a singe message of RF bits, X D = g((w 1,..., W N, D, is sent by the server over the shared ink according to users demands; decoding functions h k, k [], h k : D [2 MF ] [2 RF ] [2 F ], (4 where Ŵd k = h k (D, Z k, X D, is the reconstruction of W dk at user k. Definition 1. A user cache capacity-deivery rate pair (M, R is achievabe for a system described above, if there exists a sequence of (F, M, R codes such that for any demand reaization D [N], im Pr } {Ŵdk W dk = 0. (5 F 1,...,F N k [] For a system with N fies and users, our goa is to characterize the minimum achievabe rate R as a function of the user cache capacity M, i.e., R (M inf{r : (M, R is achievabe}. III. CORRELATION-AWARE UNCODED CACHING AND DELIVERY (CAUC SCHEME We first present an uncoded caching and deivery scheme expoiting the correation among fies, referred to as CAUC. 1 Pacement phase: Each user caches the same p F bits from each -subfie, where 0 p 1, [N], such that N ( N MF = p F, (6 =1 which meets the imitation of the cache capacities. We refer to P (p 1,..., p N as the cache aocation vector, which wi be specified in the seque. 2 Deivery phase: The server deivers the remaining bits of each requested subfie that have not been cached by the users, i.e., W S for which 1{d k S} 1. k=1 In the worst case, when the demand combination is the most distinct, i.e., users request distinct fies for the case N, or each fie is requested by at east one user for the case N <, the deivery rate is given by N (( ( N min{n, 0} R CAUC (P = (1 p F. =1 (7 The optima P can be derived by soving the foowing optimization probem min R CAUC (P N ( N such that p F MF, =1 which, straightforwardy, eads to: p = 1, if C( MF ; p MF C(+1 =, if C( + 1 < MF < C(; and p ( N F = 0, otherwise; where we have defined C( N Fi, for [N]. We remark that the optima cache aocation gives priority to the subfies with the highest eve of commonness. i= IV. CORRELATION-AWARE CODED CACHING AND DELIVERY (CACC SCHEME In this section, we present a correation-aware coded caching and deivery scheme, referred to as CACC. Simiary to the CAUC scheme, we aocate different cache capacities to subfies of different eves of commonness, again specified by the cache aocation vector, P = (p 1,..., p N, which satisfies the constraint in (6, such that each user caches p F bits from each -subfie, [N]. In the foowing, we first present how coded caching and deivery of the subfies with the same eve of commonness is carried out, and then specify the aocation of cache capacity. ( N i (8

3 A. Coded Caching and Deivery of -subfies Here, for a given cache aocation vector P, we present the coded caching and deivery of -subfies, [N]. We define t p, 0 t. If t = 0, users do not cache the -subfies at a, whie if t =, each user stores a the - subfies in its cache. In the foowing, we focus on the cases where t [ 1]. 1 Pacement Phase: We empoy the prefetching scheme proposed by [4] for the subfies rather than the fies themseves: each -subfie is partitioned into ( t disjoint parts, each with approximatey the same size of F / ( t bits. We abe these ( t disjoint parts of each -subfie W S by W A S, where A = t, A []; that is, we have W S = A: A =t, A [] W A S. Each of these parts, W A S, is paced into the cache of user k if k A. Thus, each user caches a tota of ( 1 t 1 disjoint parts of each -subfie with a tota size of t F / bits, which sums up to p F bits. 2 Deivery Phase: We first focus on the case when N. There are a tota of ( N -subfies. We denote the set of these -subfies by W = { W S : S [N], S = }. Each user requires a tota of ( 1 -subfies, i.e., user k needs to recover subfies in { W S : S [N], S =, d k S }, k []. For each user, we can regard these ( 1 -subfies as ( 1 distinct demands. Our deivery scheme for the - subfies operates in ( 1 steps, and satisfies one demand of each user at each step. We define C j (c 1j,..., c Nj, where c ij {S : S [N], S =, i S}, i [N], j [ ( 1 ], which specifies which subfie shoud be deivered in the jth step of the deivery phase. C j is generated by Agorithm 1 by setting R = [N] and R =. Note that these vectors are generated independenty of the number of users or their demands. Exampe 1. Consider N = 5 and = 2. From Agorithm 1 we obtain: C 1 = ({1, 2}, {1, 2}, {3, 4}, {3, 4}, {1, 5}; C 2 = ({1, 5}, {2, 3}, {2, 3}, {4, 5}, {4, 5}; C 3 = ({1, 3}, {2, 5}, {1, 3}, {2, 4}, {2, 5}; C 4 = ({1, 4}, {2, 4}, {3, 5}, {1, 4}, {3, 5}. This means, for exampe, that, in the first step, subfies W 12, W 34, and W 15 wi be deivered (if there is a user requesting them. We denote by d j k c d k j the demand of user k to be satisfied in the jth step, i.e., user k recovers W d j after the jth step. k ( 1 We emphasize that W cij = {W S : S [N], S = j=1, i S}, i [N]; that is a the required -subfies wi be recovered by each user after step ( 1 for any demand combination. Exampe 2. Consider N = = 5 and = 2 as in Exampe 1. Consider distinct demands, i.e., D = {1, 2, 3, 4, 5}. Thus, Agorithm 1 Generate C j, j [ ( R 1 R 1 ] 1: procedure ASSIGNMENT 2: W t W, j 1, W ast, c ast 3: whie W t do 4: c t R 5: whie c t do 6: if c t R then 7: if c ast = then 8: Randomy seect one -subfie W S from W t such that S \ R c t, remove W S from W t, and c t c t \ S 9: for i S \ R do 10: c ij W ast 11: end for 12: ese 13: for i c ast \ R do 14: c ij S 15: end for 16: c t c t \ c ast, W ast 1, and c ast 17: end if 18: ese 19: Randomy seect one -subfie W S from W t such that c t S 20: for i c t do 21: c ij S 22: end for 23: for i R do 24: c ij 25: end for 26: W ast S, c ast S \ c t, c t, and j j : end if 28: end whie 29: end whie 30: end procedure based on C 1, we have d 1 1 = d 1 2 = {1, 2}, d 1 3 = d 1 4 = {3, 4}, and d 1 5 = {1, 5}; that is, at the end of the first step, users 1 and 2 shoud recover W 12, users 3 and 4 shoud recover W 34, whie user 5 shoud recover W 15. Exampe 3. With the same setting as in Exampe 2, consider now a non-distinct demand combination D = {1, 1, 1, 3, 4}. Based on C 1, we have d 1 1 = d 1 2 = d 1 3 = {1, 2}, and d 1 4 = d 1 5 = {3, 4}; that is, at the end of the first step users 1, 2 and 3 shoud recover W 12, whie users 4 and 5 shoud recover W 34. Based on the deivery scheme proposed in [13], we present our coded transmission scheme in Agorithm 2 according to C j, j [ ( 1 ], where R =, R = [N]. In Agorithm 2, we define A j as the number of distinct d j k for each j [( 1 ]. We note that, among the CODED DELIVERY and RANDOM

4 Agorithm 2 Coded transmission based on C j 1: procedure CODED DELIVERY 2: for k = 1,..., do 3: d j k c d k j 4: end for 5: U j Any subset of A j users with distinct d j k 6: for V [] : V = t + 1, 1{k V} 1 do k U j 7: Send k V W V\{k}. d j k 8: end for 9: end procedure 10: procedure RANDOM DELIVERY 11: for S R : S = R do 12: Server sends enough random inear combinations of the bits of -subfie W S R to enabe the users demanding it to decode it. 13: end for 14: end procedure DELIVERY procedures of Agorithm. 2, the one that requires a smaer deivery rate is performed. Exampe 2 - continued. In Exampe 2, assume that t = 1, i.e., each -subfie is divided into disjoint parts of equa size, and each disjoint part is cached exacty by one user. Based on D, we have A 1 = 3. Assume that U 1 = {1, 3, 5}. Then, the server sends {1,2} W {3} W {3} W W W W {3} W, W W {3}, W. By receiving these coded bits, users 1 and 2 can recover W {1,2} together with the contents of their own caches. Simiary, users 3 and 4 can recover W, whie user 5 recovers W. In the same manner, by coded transmission based on C 2, user 1 can recover W, users 2 and 3 recover W {2,3}, and users 4 and 5 recover W {4,5} in the second step. After four deivery steps based on C 1,..., C 4 each user decodes a the 2-subfies of their requests. The tota number of bits deivered in these four steps is 36F 2 /5. Exampe 3 - continued. Assume again that t = 1. Based on D, we have A 1 = 2, and et U 1 = {1, 4}. Then, the server sends W W {3} {1,2} W {1,2} W {3} W W, such that users 1, 2 and 3 can recover W {1,2}, whie users 4 and 5 can recover W. Based on C 2, the server sends {3} W W W {2,3} W {4,5}, {2,3}, W {3} {2,5}, W {3} {4,5}, {4,5}, W {2,3},, {4,5},, such that users 1, 2 and 3 can recover W, whie user 4 and user 5 can recover W {2,3} and W {4,5}, respectivey. Based on C 3, the server sends W {1,3} W {3} {1,3} {1,3}, {1,3}, W {1,3}, {1,3} {1,3}, W {3} {1,3}, {1,3}, W W {1,3}, based on which users 1,2,3 and 4 can recover W {1,3}, whie user 5 recovers W. Finay, based on C 4, the server sends {3} W W {1,4} W {1,4}, {1,4} {1,4}, W {3} {1,4} {1,4}, {1,4}, W {1,4}, {1,4}, {1,4}, such that users 1, 2, 3 and 4 are abe to recover W {1,4}, whie user 4 recovers W. Thus, a the users are abe to decode the 2-subfies they requested. The tota number of bits deivered in this case is 30F 2 /5. Next, we consider the case N >. We first seect a subset of fies R, R [N], such that R =, and d k R for k = 1,...,. For any subset R [N] \ R, R = s, s [max{, 0} : min{ 1, N }}], Agorithm 1 is appied to a subset of -subfies W = {W S R : S R, S = R } to derive C j, j [ ( R 1 R 1 ], based on which Agorithm 2 is appied to enabe each user to decode its demanded -subfies in W. Therefore, each user can decode a the -subfies it is demanding. B. Achievabe rate The foowing theorem presents the deivery rate achieved by the proposed coded caching and deivery scheme for any demand combination, for a given cache aocation vector P. Theorem 1. For the caching system described in Section II, given a cache aocation vector P, the foowing deivery rate is achievabe N R CACC (P = R (t, (9 =1 where t = p, and for t [0 : ], and α (t R (t = min{α (t, m (t }, (10 max{min{ 1,N },0} s=max{,0} ( ( N min{n, } 1 s s 1 [ ( ( max{ min{n,} ] s 1, 0} F t + 1 t + 1 F (, t (( ( (11 N min{n, 0} m (t (F t F //F. (12 For t / [0 : ], R (t is given by the ower convex enveop of the above achievabe points. Proof. We show that R (t given above is achievabe t [0 : ]. The ower convex enveop of these integer points can then be achieved by memory sharing. For the case N, reca that the requested -subfies are deivered in ( 1 steps based on C j, j = [ ( 1 ], derived by Agorithm 1. In each step, the server sends at most N/ + 1 -subfies. Therefore, for any demand combination, the number of distinct d j k based

5 on C j, i.e., A j N/ + 1, n = 1,..., ( 1. Simiar to the deivery scheme proposed in [13], by the CODED DELIVERY procedure of Agorithm 2, the server broadcasts binary sums that hep at east one user in U j based on C j. The tota number of such subsets of t users that contain at east one user in U j is given by ( ( t +1 max{ N/ 1,0} t +1. Hence, given t {1,..., 1}, the tota number of bits sent by CODED DELIVERY procedure of Agorithm 2 for the deivery of the -subfies is bounded by (normaized by F : ( N 1 [ ( ( max{ N/ 1, 0} ] F R (t 1 t + 1 t + 1 F ( t (13 The right hand side (RHS of 13 is equa to (11 for N. Since each user caches t F / bits of each -subfie, according to [5, Appendix A], the number of bits sent by the RANDOM DELIVERY procedure is bounded by ( N R (t (F t F //F. (14 The RHS equas to (12 for N. Hence, for t {1,..., 1}, we have proven R (t given in (10 is achievabe for N. We then focus on the case where N >. For any ( max{, 0} s min{ 1, N }, there are a tota of N s subsets R such that R [N] \ R, R = s. Foowing the simiar anaysis for the case where N, given R containing a the demanded fies such that R =, and any R such that R [N] \ R, R = s, requested -subfies in W = {W S R : S R, S = R } are sent in ( 1 s 1 step. At each step, there are at most N s + 1 -subfies to be sent. Therefore, with simiar arguments, the tota number of bits sent by CODED DELIVERY procedure of Agorithm 2 in each step is bounded by ( ( ( 1 ( s 1 t +1 max{ N s 1,0} F t +1 F( t, whie the number of bits send by the RANDOM DELIVERY procedure is bounded by ( s (F t F //F. By summing over a ( N s subsets R for each s [max{, 0} : min{ 1, N }], we have and, R (t min{ 1,N } ( N R (t s s=max{,0} ( ( ( max{ N 1, 0} s t + 1 t + 1 min{ 1,N } s=max{,0} ( N s ( 1 s 1 F F ( t, (15 ( (F t F //F, (16 s by which, we have proven the correctness of (10 for the case N >. Remark 1. At each step of sending -subfies in W = {W S R : S R, S = R }, there are sometimes N N s + 1 and sometimes s distinct demands, whie when N is a mutipe of s, there are aways N s distinct demands (R = [N], R =, s = 0, for the case N. To obtain a cosed-form expression for the achievabe deivery rate, we simpy assume N s + 1 distinct demands at each step. Note that, the more the number of distinct demands at each step, the arger the deivery rate. Therefore R CACC (P in (9 is an upper bound on the actua achievabe deivery rate of CACC. Fig. 1. Deivery rate (R vs. ratio of 2-subfies (r 2,r 3 = = r 10 = 0. C. Aocation of Cache Capacity We can further optimize the cache content distribution P by soving: min R CACC (P N ( N such that p F MF, =1 (17a (17b where the objective is to minimize the achievabe deivery rate under the cache capacity constraint. The probem in (17 can be soved numericay. V. LOWER BOUND In this section, we present a ower bound derived using cutset arguments. Theorem 2. (Cut-set Bound For the caching probem described in Section II, the optima achievabe deivery rate is ower bounded by R (M max p [1:min{N,}] ( p N/p N p N/p s=0 p N/p =1 F+s N/p pm N/p. ( N p N/p s (18a Proof. The proof wi be provided in a onger version of the paper. VI. NUMERICAL RESULTS In this section, we numericay compare the deivery rates of the proposed correation-aware caching schemes CAUC and CACC with the ower bound and the state-of-the-art coded caching scheme from [13], which does not take the content

6 Fig. 2. Deivery rate (R vs. ratio of 10-subfies (r 10, r 2 = = r 10 = 0. correations into account. We refer to the ater scheme as the correation-ignorant coded caching scheme (CICC. We consider N = 10 fies and k = 10 users. Each user is equipped with a cache of size F bits, i.e., M = 1. We denote ( by r the ratio of -subfies among each fie, i.e., r 1 F /F. Note that, we have =1 r = 1. In Fig. IV-C, we assume that the fies have ony pairwise correations, that is, r 3 = = r 10 = 0, and we pot the deivery rate as a function of r 2. Meanwhie, in Fig. VI, we assume that each fie consists of a private part, i.e., 1-subfie, and a common subfie that is shared by a the fies in the ibrary, i.e., 10- subfie, i.e., r 2 = = r 9 = 0. We pot the deivery rate as a function of r 10. We observe in both figures that the deivery rate achieved by the correation-ignorant scheme, CICC, remains the same no matter how high the ratio of common subfies, whie the deivery rates of the correation-aware schemes, CAUC and CACC, decrease as the ratio of the common subfies increases. Obviousy, CACC achieves a ower deivery rate than both CAUC and CICC, since it benefits both from incorporating the correations among the fies as we as coded muticasting. When the ratio of the common subfies is sufficienty arge, even without coded muticasting CAUC achieves a ower deivery rate than CICC. It can aso be observed that the deivery rates of correation-aware schemes decrease faster with the percentage of common subfies in Fig. VI than in Fig. IV-C. That is because the gain from expoiting correation is more pronounced as the common parts are shared among more fies. Whie there is a gap between the cut-set ower bound and the achievabe deivery rate, we note that the gap is smaer in Fig VI, where the eve of commonness is higher. VII. CONCLUSIONS We have studied coded caching taking into account the avaiabe correations among the fies in the ibrary. To capture arbitrary correations, we assume that each fie consists of a number of subfies, each of which is shared by a different subset of fies in the ibrary, and the number of fies that share a certain subfie is defined as its eve of commonness. We proposed both a correation-aware uncoded caching scheme, the optima pacement of which is proven to be caching the subfies with the highest eves of commonness, and a correation-aware coded caching scheme (CACC, the pacement of which is optimized in terms of the achievabe deivery rate. 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