Noisy Broadcast Networks with Receiver Caching

Transcription

1 Noisy Broadcast Networs with Receiver Caching Shirin Saeedi Bidohti, Michèe Wigger and Roy Timo arxiv: v [cs.it] 8 May 06 Abstract We study noisy broadcast networs with oca cache memories at the receivers, where the transmitter can pre-store information even before earning the receivers requests. We mosty focus on pacet-erasure broadcast networs with two disjoint sets of receivers: a set of wea receivers with a-equa erasure probabiities and equa cache sizes and a set of strong receivers with a-equa erasure probabiities and no cache memories. We present ower and upper bounds on the capacity-memory tradeoff of this networ. The ower bound is achieved by a new joint cache-channe coding idea and significanty improves on schemes that are based on separate cache-channe coding. We discuss how this coding idea coud be extended to more genera discrete memoryess broadcast channes and to unequa cache sizes. Our upper bound hods for a stochasticay degraded broadcast channes. For the described pacet-erasure broadcast networ, our ower and upper bounds are tight when there is a singe wea receiver and any number of strong receivers and the cache memory size does not exceed a given threshod. When there are a singe wea receiver, a singe strong receiver, and two fies, then we can strengthen our upper and ower bounds so as they coincide over a wide regime of cache sizes. Finay, we competey characterise the rate-memory tradeoff for genera discrete-memoryess broadcast channes with arbitrary cache memory sizes and arbitrary asymmetric rates when a receivers aways demand exacty the same fie. I. INTRODUCTION We address a one-to-many broadcast communications probem where many users demand fies from a singe server during pea-traffic times periods of high networ congestion. To improve networ performance, the server can pre-pace information in oca cache memories near users at the networ edge. This pre-pacement of information is caed the caching communications phase, and it occurs during off-pea times when the communications rate is not a imiting networ resource. The server typicay does not now in advance which fies the users wi demand, so it can try to cache information that is iey to be usefu for many users during the deivery communications phase the pea-traffic time when the users demand fies from the server. For exampe, researchers at Huawei Laboratories [4] recenty used machine earning techniques to predict user behavior and proactivey cache data to improve user request satisfaction ratios and reduce bachau oads during the deivery phase. The above caching probem is particuary reevant to video-streaming services in mobie networs. Here networ operators pre-pace information in cients caches or, on servers near the cients to improve atency and throughput during pea-traffic times. The networ operator does not now in advance which movies the cients wi request, and thus the cached information cannot depend on the cients specific demands. It is now widey expected that there wi be a nine-fod increase in mobie data traffic by 00, and around 60 percent of this traffic wi be mobie video [5]. Smart data caching strategies, new bandwidth aocations, reduced ce sizes and new radio-access technoogies wi a be needed to meet these growing demands [6]. The information-theoretic aspects of cache-aided communications have received significant attention in recent years [7] [36]. Maddah-Ai and Niesen [7] considered a one-to-many broadcast probem where the receivers have independent caches of equa sizes and the deivery phase taes pace over a noiseess broadcast communications in. They showed that a smart design of the cache contents enabes the server to send coded XOR-ed data during the deivery phase that can simutaneousy meet the demands of mutipe receivers with a singe transmission. This coded caching scheme, by simutaneousy satisfying user demands, aows the server to reduce the deivery rate beyond the obvious oca caching gain the data rate that each receiver can immediatey retrieve from its cache without using coded caching. Intuitivey, this performance improvement occurs because the receivers can profit from other receivers caches, and was thus termed [7] goba caching gain. Improved caching and deivery strategies for the Maddah-Ai and Niesen mode were presented in [8] [0]. Fundamenta converse ower bounds on the tota required deivery rate were presented in [7], [] [6]. It was shown in [7] that the coded caching scheme of Maddah-Ai and Niesen is optima among schemes that have uncoded caching pacement when there are more fies than users. The Maddah-Ai and Niesen mode considers a worst-case scenario, meaning that the goa is to satisfy a possibe user demands. The caching probem has aso been studied in average-case scenarios [4] [6] where the receivers demands foow a given probabiity distribution and the deivery rate is averaged over this demand distribution. S. Saeedi Bidohti is with the Department of Eectrica Engineering at Stanford University, saeedi@stanford.edu. S. Saeedi Bidohti is supported by the Swiss Nationa Science Foundation feowship no M. Wigger is with CNRS, LTCI, Teecom ParisTech, Université Paris-Sacay, 7503 Paris, michee.wigger@teecom-paristech.fr R. Timo is with the Institute for Communications Engineering at the Technica University of Munich, roy.timo@tum.de. R. Timo is supported by the Aexander von Humbodt Foundation. Parts of the materia in this paper have been presented at the IEEE Internationa Symposium on Wireess Communications Systems ISWCS, Bruxees, Begium, August 05 [], and wi be presented at the 06 IEEE Internationa Symposium on Information Theory, Barceona, Spain [] and at the Internationa Symposium on Turbo Codes & Iterative Information Processing, Brest, France, September 06 [3].

2 In contrast to [7], we wi assume in this paper that the deivery phase taes pace over a noisy broadcast channe BC, and we wi see that further goba caching gains can be achieved by joint cache-channe coding. Intuitivey, when the BC is noisy the cache content not ony determines what to transmit but aso how to transmit it. We wi focus on pacet erasure broadcast channes that provide a first order mode of pacet osses in congested networs. The importance of incuding a noisy channe mode for the deivery phase was aso observed in [], [8] [8]. For exampe, [] and [3] iustrate interesting interpay between feedbac or channe state information with caching, and [4] and [5] show that caches at the transmitter-side and receiver-side aow for oad-baancing and interference mitigation in noisy interference networs. The wors in [] [5] focus on the high signa-to-noise ratio regime. Our main interest in this paper is to characterize some of the fundamenta rate-memory tradeoffs for cache-aided broadcast networs; that is, we wish to determine the set of rates at which messages can be reiaby communicated for given cache sizes. We focus on a worst-case worst-demand setup and consider two different communication scenarios. Scenario : We assume that the receivers demands are arbitrary and the messages are a of equa rate. We focus on the pacet-erasure BC iustrated in Figure, and divide the receivers into two sets: A set of w wea receivers with equa arge BC erasure probabiities δ w > 0. These receivers are each equipped with an individua cache of equa memory M. A set of s = w strong receivers with equa sma BC erasure probabiities δ s 0 with δ s δ w. These receivers are not provided with caches. Library: W,W,...,W D Tx X n Pacet Erasure Broadcast Channe Y n Y n Y n Y n w w+ Y n Rx Rx... Rx w Rx w +... Rx Cache Cache...Cache w erasure probabiity s < w nm bits nm bits nm bits erasure probabiity w Fig. : user pacet-erasure BC with w wea and s strong receivers and where the wea receivers have cache memories. This scenario is motivated by previous studies [], [9] that showed the benefit of prioritizing cache pacements near weaer receivers. In practica systems, this means that teecommunications operators with a imited number of caches might first pace caches at houses that are further away from an optica fiber access point. Or, they might pace caches at pico or femto base stations in heterogenous networs that are ocated in areas with notoriousy bad coverage. Scenario aso arises as part of a more compex system mode in which every receiver is equipped with a cache. Suppose, for exampe, that the stronger receivers want to decode additiona data that wi never be demanded by the wea receivers see aso Section VI. This additiona data might represent, for exampe, a higher resoution of a video. A practica soution in this case is to separate transmission of fies from the two ibraries [3] [34]: A first transmission sends the fies that are of interest to a receivers, and a second transmission sends ony fies from the additiona ibrary to the strong receivers. The question is now how to divide the cache memory between the two transmissions. Based on the resuts we obtain in this paper, we propose to assign a the cache memory at the strong receivers to the second transmission, because through a carefu design of the first transmission scheme, the strong receivers can aready benefit from the wea receivers caches without accessing their own cache memories. The fundamenta rate versus cache memory tradeoff of interest in Scenario is the argest rate R at which a messages can be reiaby transmitted in the usua Shannon sense for a given cache size M at the wea receivers. This argest rate is caed the capacity-memory tradeoff and wi be denoted by CM. We present genera achievabe ower and converse upper bounds on the capacity-memory tradeoff CM. Our achievabe bound on CM is based on a joint cache-channe coding scheme that buids on the piggybac coding idea in [39] see Subsection III-D. The basic idea of piggybac coding is to carry messages to strong receivers on the bac of messages to the Here we can assume that bit-eve errors within a pacet are handed on a in-by-in basis using physica ayer error-correction techniques, and pacets arrive at the users prompty or are ost due to, for exampe, buffer overfows. Each user can choose any fie from the server.

3 3 wea receivers. These messages can be carried for free if the server pre-paces appropriate message side information in the wea receivers caches. We wi see that joint cache-channe coding provides substantia gains over separate cache-channe coding with Maddah-Ai and Niesen s coded caching scheme [7] and a capacity achieving scheme for the pacet-erasure BC. For exampe, if the ibrary has D messages of rate R and M is smaer than approximatey DRδw δs w+ s i.e., a sma-memory regime, then CM R 0 + M D γ oca γ goba,sep γ goba,joint. Here R 0 represents the capacity of the networ without caches, and γ oca, γ goba,sep and γ goba,joint are constants depending on the number of receivers and erasure probabiities. The exact expressions of the three constants are presented ater in 34, but they have the foowing interpretations. If we empoy a standard capacity-achieving coding scheme to transmit the parts of the demanded messages that are not aready stored in the intended receivers caches, then we achieve a rate equa to R 0 + M D γ oca. This strategy achieves ony a oca caching gain, and hence the subscript oca for this factor. When some receivers in the networ have no caches s, then γ oca <. A scheme that combines Maddah-Ai and Niesen s coded caching agorithm with a capacity-achieving scheme for the pacet-erasure BC attains the ower bound R 0 + M D γ oca γ goba,sep. The factor γ goba,sep thus describes the goba caching gain obtained by a separate cache-channe coding scheme, and hence the subscript goba,sep. Whenever w >, we have γ goba,sep >. Finay, our joint cache-channe coding scheme achieves the ower bound, and the parameter γ goba,joint describes this scheme s gain over the previous separate cache-channe coding. In other words, the factor γ goba,joint = + w s δ w + w w δ s describes the further goba caching gain that is possibe using our joint cache-channe coding scheme that was not achievabe with the aforementioned separate cache-channe coding scheme. By, the improvement of our joint cache-channe coding scheme over the separate cache-channe coding scheme is not bounded for sma cache sizes. In particuar, it is stricty increasing in the number of strong receivers s. Our genera ower and upper bounds match for w = and M F D δ sδ w δ s s δ w + δ s For the specia case w = w = and D =, we present a refined ower bound on CM as we as a refined upper bound. The idea is to cache aso the XOR of a part of the two messages in the ibrary, simiary to [7, Appendix]. Our refined bounds coincide when w = s = ; D = ; δ w = δ s 3b and w = s = ; D = ; M F δ s + δ w δ s. 3c Scenario : In our second scenario section VIII we aow for genera discrete memoryess broadcast channes DMBCs, arbitrary cache sizes M } =, and non-equa rates of the various messages R,..., R D. However, we impose that each receiver demands exacty the same message. For this scenario we competey characterize the entire rate-memorry tradeoff R,..., R D, M,..., M. The remainder of this paper is organised as foows. In Sections II and III, we state the probem setup and some auxiiary resuts that are hepfu in the design of our joint cache-channe coding scheme. Section IV summarizes our main resuts for the first scenario. We describe and anayze our joint source-cache channe coding scheme in Section V and setch how our scheme can be extended to more genera scenarios with arbitrary cache sizes and arbitrary DMBCs in Section VI. In Section VII, we prove an upper bound on the capacity-memory tradeoff of genera stochasticay degraded BCs with caches at the receivers. Our second scenario is discussed in section VIII. 3a A. Notation II. PROBLEM DEFINITION Random variabes are identified by uppercase etters, e.g. A, their aphabets by matching caigraphic font, e.g. A, and eements of an aphabet by owercase etters, e.g. a A. We aso use uppercase etters for deterministic quantities ie rate R, capacity C, number of users, memory size M, and number of fies in the ibrary D. The Cartesian product of A and A is A A, and the n-fod Cartesian product of A is A n. Vectors are identified by bod font symbos, e.g., a, and matrices by the font A. We use the shorthand notation A n for the tupe A,..., A n. LHS and RHS stand for eft-hand side and right-hand side. Finay, we use the notation W W to denote the bitwise XOR over the binary strings corresponding to the messages W and W, which are assumed to be of equa ength.

4 4 B. Message and channe modes Consider a broadcast channe BC with a singe transmitter and receivers as depicted in Figure. We have two sets of receivers: w wea receivers that statisticay have a bad channe and s = w strong receivers that statisticay have a good channe. The meaning of good and bad channes wi be expained shorty. For convenience of notation, we assume that the first w receivers are wea and the subsequent s receivers are strong, and we define the sets and w :=,..., w } s := w +,..., }. We mode the channe from the transmitter to the receivers by a memoryes pacet-erasure BC with input aphabet and equa output aphabet at a receivers X := 0, } F Y := X }. Here F 0 is a fixed positive integer, and each input symbo x X is an F -bit pacet. The output erasure symbo modes oss of a pacet at a given receiver. Each receiver :=,..., } observes the erasure symbo with a given probabiity δ 0, and it observes an output y equa to the input, y = x, with probabiity δ. The margina transition aws 3 of the memoryess BC are thus described by δ if y = x P[Y = y X = x] = δ if y =. 4 0 otherwise We wi assume throughout that δ i = δw if i w δ s if i s 5 for fixed erasure probabiities 4 0 < δ s δ w. Since δ s δ w, the wea receivers have statisticay worse channes than the strong receivers, hence the distinction between good and bad channes. In the seque, we wi assume that each wea receiver is provided with a cache memory of size nm bits. The strong receivers are not provided with cache memories. We expain shorty how the cache memory at the wea receivers can be expoited. C. Message ibrary and receiver demands The transmitter has access to a ibrary with D messages W,..., W D. 6 These messages are a independent of each other and each of them is uniformy distributed over the message set,..., nr }, where R 0 is the rate of each message and n the bocength of transmission. Each receiver wi demand i.e., request and downoad exacty one of these messages. Let D :=,..., D}. We denote the demand of receiver by d D, the demand of receiver by d D, etc., to indicate that receiver desires message W d, receiver desires message W d, and so on. For most of the time in this manuscript we assume that the demand vector d := d,..., d 7 can tae on any vaue in D. Communication taes pace in two phases: a first caching phase where information is stored in the wea receivers cache memories and a subsequent deivery phase where the demanded messages are deivered to a the receivers. The next two subsections detai these two communication phases. 3 As wi become cear in the foowing, for our probem setup ony this margina transition aw is reevant, but not the joint transition aw. 4 Though, in principe, we aow δ s = δ w, our main interest wi be δ s < δ w.

5 5 D. Caching phase During the first communication phase the transmitter sends caching information V i to each wea receiver i w, who then stores this information in its cache memory. The strong receivers do not tae part in the caching phase. The demand vector d is unnown to the transmitter and receivers during the caching phase, and, therefore, the cached information V i cannot depend on the users specific demands d. Instead, V i is a function of the entire ibrary: for some function where V i := g i W,..., W D i w g i :,..., nr } D V, i w, 8 V :=,..., nm }. The caching phase occurs during a ow-congestion period. We therefore assume that this phase incurs no erasures or other types of errors, and each wea receiver i w can store V i in its cache memory. E. Deivery phase The transmitter is provided with the demand vector d, and it communicates the corresponding messages W d,..., W d over the pacet-erasure BC. The entire demand vector d is assumed to be nown to the transmitter and a receivers 5. Depending on the demand vector d, the transmitter chooses an encoding function and it sends f d :,..., nr } D X n 9 X n = f d W,..., W D, 0 over the pacet-erasure BC. Each receiver,..., } observes Y n according to the memoryess transition aw in 4. Each wea receiver attempts to reconstruct its desired message from it channe output, cache contents and demand vector d. Simiary, each strong receiver attempts to reconstructs its desired message from its channe output and the demand vector d. More formay, ϕi,d Yi n, V i if i w Ŵ i := ϕ i,d Yj n if i a s where and ϕ i,d : Y n V,..., nr } i w b ϕ j,d : Y n,..., nr } j s. c F. Capacity-memory tradeoff An error is said to occur whenever Ŵ W d for some,..., }. For a given demand vector d the probabiity of error is thus [ P e d := P Ŵ W d ]. 3 We consider a worst-case probabiity of error over a feasibe demand vectors: = P e worst := max d D P e d. 4 In definitions 8-4, we sometimes add a superscript n to emphasise the dependency on the bocength n. We say that a rate-memory pair R, M is achievabe if for every ɛ > 0 there exists a sufficienty arge bocength n and worst caching, encoding and decoding functions as in 8, 9 and such that P e < ɛ. The main probem of interest in this paper is to determine the foowing capacity versus cache memory tradeoff. Definition : Given cache memory size M, we define the capacity-memory tradeoff CM as the supremum of a rates R such that the rate-memory pair R, M is achievabe. 5 It taes ony ogd bits to describe the demand vector d. The demand vector can thus be reveaed to a terminas using zero transmission rate.

6 6 G. Trivia and non-trivia memory sizes The capacity-memory tradeoff CM is triviay upper bounded by the capacity of the pacet-erasure BC to the strong receivers ony see Proposition in Section III-A ahead: δs CM F M 0. 5 s For M = DF δ s / s, this upper bound is aso achievabe because the wea receivers can store the entire ibrary in their caches and the transmitter thus needs to ony serve the strong receivers during the deivery phase. Therefore, δs δs CM = F, M DF. 6 We wi henceforth restrict attention to nontrivia cache memories [ M 0, DF δ ] s. s s III. PREVIOUS RELATED RESULTS This section reviews capacity resuts and coding schemes for three reated scenarios that form the basis for our new bounds on the capacity-memory tradeoff CM and the joint cache-channe coding scheme that we present in Section V ahead. s A. Capacity of pacet-erasure BCs Temporariy consider the receiver pacet-erasure BC iustrated in Figure. The BC is characterised by 4 with arbitrary erasure probabiities 0 < δ, δ,..., δ. Suppose that the are no caches i.e., M = 0, and that each receiver wishes to earn an independent message W that is uniformy distributed over the set,..., nr }. Notice that, in contrast to previous sections, messages have different rates, and it is a priori nown which message is intended for which receiver. Let Ŵ denote receiver s reconstruction of message W. W,W,...,W Tx X n Pacet Erasure Broadcast Channe Y n Y n Y n Rx Rx Rx Ŵ Ŵ Ŵ Fig. : Standard -user pacet-erasure BC with arbitrary erasure probabiities and no caches. The capacity region of this standard pacet erasure-bc is achieved by time-sharing capacity-achieving point-to-point codes [37]. The point-to-point capacity of the channe from the transmitter to receiver,..., } is C = δ F. 7 Proposition : The capacity region of the pacet-erasure BC to receivers with erasure probabiities δ, δ,..., δ is the cosure of the set of nonnegative rate-tupes R,..., R that satisfy [37] = R δ F.

7 7 Library: W,W,...,W D Tx X n Rx Rx... Rx Cache Cache... Cache nm bits nm bits nm bits Fig. 3: BC with a common noise-free pipe of rate F to a receivers, which a have a cache memory of nm bits. B. Coded caching over a BC with noise-free common bit-pipe We briefy expain the Maddah-Ai and Niesen s coded caching scheme in [7]. The scheme has two parameters: a positive integer representing the number of users with caches and an index t,..., }. Coded caching appies to the noiseess communication scenario iustrated in Figure 3. This scenario coincides with our origina scenario in Section II in the specia case where w = = and δ w = 0. As in 6, the messages W, W,..., W D depicted in Figure 3 have a common rate R. It wi be convenient to describe coded caching using the foowing methods: Method Ca describing the caching phase for the setup in Figure 3. Method En describing the deivery-phase encoding. Methods De ; =,,..., } describing the deivery-phase decoding at each user. Preiminaries: Let G,..., G t denote the t subsets of,..., } of size t. Spit each message W d into choose t independent submessages, } W d = W d,g : =,...,. t Each of these submessages is of equa rate R sub := R. 8 t Method Ca: This method taes the entire ibrary W,..., W D as an input, and it outputs the cache contents V, V,..., V where V = W d,g : d,..., D} and G },,..., }. 9 In other words, during the caching phase, the tupe W,G, W,G,..., W D,G is stored in the cache memory of every receiver in G. 3 Method En: This method taes the entire ibrary W,..., W D and the demand vector d as inputs, and it outputs WXOR,S : S,..., } and S = t + }, 0 where W XOR,S := s S W ds,s\s}.

8 8 4 Methods De for =,..., : This method taes as inputs the demand vector d; the XOR-messages W S : S} produced by method En; and the cache content V produced by method Ca. It outputs the t -tupe reconstruction Ŵ d := Ŵd,G,..., Ŵd,G, t where W d,g Ŵ d,g = if G s G W ds,g }\s} WXOR,G } if / G. Notice that a the XOR messages on the right are inputs of this method, because they are either part of the cache content V produced by method Ca or part of the XOR messages W XOR,S : S} produced by method En. 5 Anaysis: We now anayse the three methods above for Figure 3. Lemma : Consider the scenario in Figure 3. The XOR messages W XOR,S } produced by method En can be sent over the common noise-free pipe if and ony if the rate of the pipe satisfies F R sub = R t t + t +. 4 Moreover, each receiver,..., } can store the cache content V if, and ony if, the cache memory size satisfies M D R sub = D t R. 5 t Proof: Inequaity 4 foows because there are t+ XOR messages WS of rate R sub and by 8. Inequaity 5 foows because each V produced by method Ca, consists of D t messages of rate Rsub, see 9. C. Separate cache and channe coding for pacet-erasure BCs and the proof of Proposition 6 Starting from Maddah-Ai and Niesen s coded-caching scheme one readiy obtains a separation-based coding scheme for the pacet-erasure BC with caches described in Section II. Detais are as foows. Choose = w and an arbitrary t,..., w }. For the caching phase: Appy Method Ca to the ibrary W,..., W D ; tae the resuting V,..., V w ; and store each V the cache memories of receiver. The deivery-phase transmitter proceeds in two steps: T: The transmitter appies method En to the ibrary W, W,..., W D and demand vector d w := d, d,..., d w. T: The transmitter sends the XOR-messages produced in step T together with the messages that are demanded by receivers in s using a capacity-achieving scheme for the pacet-erasure BC. The strong receivers decode their intended messages using an optima decoding method for the pacet-erasure BC. The wea receivers decode in two steps: R: Each wea receiver uses an optima decoder for the pacet-erasure BC to recover a XOR-messages generated by method En. R: It appies method De to the XOR messages decoded in step R. This separate cache and channe coding scheme can be easiy anaysed using Lemma and Proposition, and from this anaysis one obtains Proposition 6 in Section IV. D. Piggybac coding for BCs with message side information Consider the two-user BC with message side-information [40], [4] iustrated in Figure 4. The transmitter observes two independent messages W and W of rates R and R. Message W is intended for receiver and message W for receiver. Suppose that receiver is given W prior to communications and the BC has arbitrary transition probabiities P YY X. The capacity region of this BC with message side-information at receiver can be derived 6 from [4, Thm. 3]. We now present a specific random coding scheme for the BC in Figure 4, which we ca piggybac coding. Code Construction: Fix a distribution P X on the input aphabet of the BC, a sma ɛ > 0 and a arge bocength n. Randomy generate a codeboo C with nr nr codewords of ength n by independenty picing each entry of each codeword using P X. We pace the codewords into a matrix with nr coumns and nr rows, and denote the codeword in coumn w and row w by x n w, w. Figure 5 setches the codeboo: Each dot represents a codeword; message W determines the coumn of the codeword to pic and W determines the row. The codeboo is given to the transmitter and both receivers. 6 ramer and Shamai assume in [4, Thm. 3] that receiver not ony needs to decode message W but aso W. However, since receiver has message W as side information, this additiona requirement is not a imitation and the two setups have identica capacity regions. 3

9 9 W,W Tx X n Broadcast Channe P YY X W Y n Rx Rx Y n Ŵ Ŵ Fig. 4: Standard two-user BC with message side-information at receiver. W =4 W W Fig. 5: Codeboo C where the dots represent the codewords x n w, w arranged in a matrix array. Coumns encode message W and rows encode message W. Receiver nows the vaue of message W and thus can restrict its decoding to a singe row of the codeboo. Encoding: Given that the transmitter wishes to send messages W = w and W = w, it transmits the codeword x n w, w over the channe. 3 Decoding at receiver the receiver with side information: Since receiver nows W = w, it can restrict attention to the w -th row of the codeword. For exampe if W = 4, then it needs ony to consider the codewords dots that ie in the highighted row of Figure 5. Given that receiver observes the channe outputs Y n = y n, it oos for a unique index ŵ,..., nr } satisfying x n ŵ, w, y n T n ɛ P XY, where Tɛ n P XY denotes the typica set as defined in [4]. If there is no such index ŵ, then receiver decares an error. 4 Decoding at receiver the receiver without side information: Receiver wi attempt to decode both messages W and W. Given that it observes channe outputs Y n = y n, it oos for the unique pair of indices ŵ, ŵ 0,..., nr } 0,..., nr } satisfying x n ŵ, ŵ, y n T n ɛ P XY. If there is no such pair of indices, then receiver decares an error. 5 Anaysis & discussion: By the covering emma [4] and because receiver restricts attention to a singe row in the codeboo, the probabiity of decoding error at receiver, P[Ŵ W ], tends to 0 as n whenever R < IX; Y. Moreover, by this covering emma and because receiver decodes both messages, the probabiity of error at receiver, P[Ŵ W ], tends to 0 as n whenever R + R < IX; Y. 6b We have the foowing proposition. 6a

10 0 R F F F R Fig. 6: The gray area depicts the rate-region achieved with piggybac coding over a two-user pacet-erasure BC with message side-information at receiver when receiver has smaer erasure probabiity than receiver, i.e., δ < δ. The dashed ine indicates the border of the capacity region without message side-information. The figure shows that the message side-information aows to piggybac information to receiver without reducing the achievabe rate to receiver. Proposition 3: For a DMBC P YY X with message side-information at receiver shown in Figure 4, piggybac coding achieves a nonnegative rate-pairs R, R satisfying 6 for some channe input distribution P X. Speciaising this proposition to the pacet-erasure BC, we obtain the foowing. Proposition 4: For the two-user F bit pacet-erasure BC, with erasure probabiities δ, δ and message side information at receiver, piggybac coding achieves a rate-pairs R, R that satisfy max R δ F, R + R δ F }. 7 The achievabe region in Proposition 4 is iustrated in Figure 6. This region covers the entire capacity region of the two-user F -bit pacet-erasure BC with message side-information at receiver whenever δ < δ. Remar : The piggybac coding scheme does not achieve the entire capacity region of an arbitrary DMBC with message side-information at receiver. Consider, for exampe, a degenerate DMBC in which the channe to receiver is useess i.e., P Y X has 0 point-to-point capacity and receiver s channe P Y X has positive capacity. Over such a channe the piggybac coding achieves no positive rates because 6b constrains the sum-rate to 0. It is cear that with a different scheme positive rates R > 0 can be achieved. The optima coding scheme spits message W into W,p, W,c and combines piggybac coding with superposition coding: in the coud center it sends W,c, W as in piggybac coding, and in the sateite it sends W,p. In this section we assume that the demand vector IV. NEW RESULTS FOR ARBITRARY DEMANDS d can tae on every vaue in D. Our main resuts are a genera ower bound and a genera upper bound on the capacity-memory tradeoff CM. The bounds are tight in certain regimes of M when w =. We further present improved ower and upper bounds for the specia case s = w = and D =. These bounds match except for a sma regime of M s. A. Genera ower bounds Define w + rate-memory pairs R t, M t ; t = 0,,..., w + } as foows: i ii For each t,..., w }: R 0 := F w + s, M 0 := 0; 8 δ w δ s F δ w + w t + δ w δ s t s δ w R t := w t + + w t, δ w δ s δ w + s t t + s δ w δ s D M t := R t t + w t+ δ w δ s ; 9 w t s δ w

11 iii R w+ := F δs s, M w+ := DF δs s. 30 Theorem 5 Lower bound: The upper convex hu of the w + rate-memory pairs R t, M t ; t = 0,,..., w + } in 8 30 forms a ower bound on the capacity-memory tradeoff: CM upper hu R t, M t : t = 0,..., w + }. 3 Proof outine: The pair R 0, M 0 = 0 corresponds to the case without caches, and the achievabiity of R 0 foows from the usua capacity resut for pacet-erasure BCs, see Proposition in the previous section III-A. Achievabiity of the pair R w+, M w+ foows from 6. The remaining pairs R t, M t, t =,..., w, are more interesting and are achieved by the joint cache-channe coding scheme in section V ahead. The upper convex hu of R t, M t ; t = 0,,..., w + }, finay, is achieved by time-sharing. To better iustrate the strength of our ower bound, and thus of our joint cache-channe coding scheme in section V, consider the foowing separation-based ower bound: Proposition 6: The upper convex hu of the rate-memory pairs R t,sep, M t,sep ; t = 0,..., w } } is achievabe, where R t,sep := F M t,sep := D t w R t,sep. w t t + δ w + s δ s, 3a Proof: For t = 0, no cache memory is used and the achievabiity of R 0,sep foows simpy by the standard capacity region of the pacet-erasure BC, Proposition in section III-A. For t,..., w } achievabiity of the pair R t,sep, M t,sep can be shown by triviay combining the Maddah-Ai & Niesen coded caching [7, Agorithm ] with a capacity-achieving scheme for the pacet-erasure BC without caching, see the previous section III-C. Of specia interest is the regime of sma cache size M. In this regime, Theorem 5 speciaizes as foows: Coroary 6.: For sma cache memory sizes, M M, the capacity-memory tradeoff is ower bounded as where R 0 is defined in 8 and 3b CM R 0 + M D γ oca γ goba,sep γ goba,joint, M M, 33 γ oca := w δ s w δ s + s δ w, 34a γ goba,sep := + w, 34b γ goba,joint := + w s δ w + w w δ s. 34c If in the above ower bound one repaces the product γ goba,sep γ goba,joint by, then one obtains the ower bound that corresponds to a coding scheme with ony oca caching gain. If ony the factor γ joint is repaced by, then one obtains the ower bound impied by Proposition 6. The factor γ goba,sep is thus due to the separation-based Maddah-Ai & Niesen coded caching idea. In contrast, the ast factor γ goba,joint is due to our joint cache-channe coding scheme. Notice that this factor γ goba,joint is unbounded when one increases the number of strong receivers s or more generay the ratio s δw w δ s. B. Genera upper bound We now present our upper bound. Define for each w 0,..., w } w R w M := F + s + wm δ w δ s D Theorem 7 Upper bound: The capacity-memory tradeoff CM is upper bounded as CM min R w M. 35 w 0,..., w} Proof: In section VII we derive an upper bound on the capacity-memory tradeoff for a genera degraded BC with arbitrary cache sizes at the receivers, see Theorem 9. We then speciaize this upper bound to pacet-erasure BCs with arbitrary cache sizes and erasure probabiities in Coroary 9., and we show how the upper bound in 35 is obtained from this coroary. We numericay compare our upper and ower bounds on the capacity-memory tradeoff in Figures 7 and 8.

12 Capacity CM Joint Cache-Channe Coding Theorem 5 Upper Bound of Theorem 8 Separate Cache-Channe Coding Proposition Memory M Fig. 7: Bounds on the capacity-memory tradeoff CM for w = 4, s = 6, D = 50, δ w = 0.8, δ s = 0., and F = Capacity CM Joint Cache-Channe Coding Theorem 5 Upper Bound of Theorem 8 Separate Cache-Channe Coding Proposition Memory M Fig. 8: Bounds on capacity-memory tradeoff CM for w = s = 0, D = 50, δ w = 0.8, δ s = 0., and F = 50. C. Specia case of w = We first evauate our bounds for a setup with a singe wea receiver and any number of strong receivers. Let Γ := F δ s s δ w δ s s δ w + δ s, 36 Γ := δ s s F, 37 Γ 3 := F δ s s δ s s δ w + δ s. 38 Notice that 0 Γ Γ 3 Γ. From Theorems 5 and 7 we obtain the foowing coroary.

13 Capacity CM Joint Cache-Channe Coding Theorem 5 Upper Bound of Theorem 7 Separate Cache-Channe Coding Proposition Memory M Fig. 9: Bounds on the capacity-memory tradeoff for w =, s = 0, D =, δ w = 0.8, δ s = 0., and F = 0. Coroary 7.: If w = the capacity-memory tradeoff is ower bounded by: F δ w δ s CM + M s δ w+ δ s D, if M D [0, Γ ] F δs + s + M +, if M sd D Γ, Γ ], and upper bounded by: CM F δ w δ s + M s δ w+ δ s D, if M D [0, Γ 3] F δs s if M D Γ 3, Γ ]. Figure 9 shows these two bounds and the bound in Proposition 6 for w =, s = 0, D =, D = 0, δ w = 0.8, δ s = 0., and F = 0. We identify two regimes. In the first regime 0 M D Γ, the cache memory aows reducing the rate R to each receiver by M D. This is the same performance as when a naive uncoded caching strategy is used in a setup where a s + receivers have cache memories of rate M. In the first regime, our joint cache-channe coding scheme thus enabes a receivers to profit from the singe cache memory and provides the best possibe goba caching gain. In the second regime Γ < M D Γ the gains are not as significant as in the first regime, but increasing the cache size sti resuts in an improved performance. This won t be the case for M D > Γ. In the first regime, 0 M D Γ, our joint cache-channe coding scheme of section V achieves the capacity-memory tradeoff CM. D. Specia case w = s = and D = For this specia case we present tighter upper and ower bounds on CM. These new bounds meet for a arge range of cache memory sizes M. Let Γ := F δ w + δ s δ w δ s, 4 δ w + δ s Γ := F δ s + δ w δ s Theorem 8: If w = s = and D =, the capacity-memory tradeoff is upper bounded as: F δw δs δ + M w+ δ s, if M [0, Γ ] CM F 3 δ s δ w + M 3, if M Γ, Γ ] F δ s if M Γ, Γ ]. 43

14 4 8 B 7 A 6 Capacity CM Joint Cache-Channe Coding Theorem 8 Upper Bound of Theorem 8 Separate Cache-Channe Coding Proposition Memory M Fig. 0: Bounds on the capacity-memory tradeoff for w =, s =, D =, δ w = 0.8, δ s = 0., and F = 0. and ower bounded as: δw δs F δ w + δ + M s, M [0, Γ] δ CM s 3 δ s δ F δs + M, M w Γ, Γ ] 44 M F δ s Γ, Γ ]. Proof: Lower bound 44 coincides with the upper convex hu of the three rate-memory pairs: R 0, M 0 in 8; R, M in 9; and F δ s, Γ. Achievabiity of the former two pairs foows from Theorem 5. Achievabiity of the ast pair is proved in appendix D. The upper bound is proved in appendix E. Figure 0 shows the bounds of Theorem 8 for δ w = 0.8, δ s = 0., and F = 0. Coroary 8.: The minimum cache size M for which communication is possibe at the maximum rate F δ s is Γ. Upper and ower bounds of Theorem 8 coincide in the case of equa erasure probabiities δ w = δ s : Coroary 8.: If w = s =, D = and δ w = δ s = δ: F CM = δ + M, if M [ 0, F δ] F δ if M F δ, Γ ] 45. Proof: It foows from Theorem 8 because for δ w = δ s : Γ = Γ = F δ, and in the regime M bound 44 speciaises to CM F δ+ M. V. A JOINT CACHE-CHANNEL SCHEME FOR ARBITRARY DEMANDS We describe a joint cache-channe coding scheme parameterised by Γ, Γ ] ower t,..., w }. 46 We show in subsection V-D that, for parameter t, this scheme achieves the rate-memory pair R t, M t in 9. A. Preparations For each d,..., D}, spit message W d into two parts: of rates W d = W t d, W t d R t = R + w t + δw δ s 48a t s δ w 47

15 5 where R t + R t = R. R t t s = R + w t + δ w 48b δ w δ s B. Caching phase The first step is to cache information about both parts of each message in 47 at the wea receivers. Specificay, we first appy Method Ca with = w and t = t to messages W t,..., W t D. We then appy Method Ca with = w and t = t to messages W t,..., W t D. In the foowing, we wi use the superscript t to identify the outputs of Methods Ca, En, and De i with t = t. Simiary, we wi use the superscript t to identify these outputs for t = t. We use the notation that we introduced in Section III-B. Consider an arbitrary wea receiver i w. The tota cache content at this receiver is C. Deivery phase V i = V t = i W t V t i d,g t W t : d,..., D} and G t d,g t } : d,..., D} and G t }. 49 The deivery phase taes pace in three subphases consisting of β n, β n, and β 3 n channe uses, where β, β, β 3 0 and β + β + β 3 =. 50 Deivery subphase : Here we ony consider the wea receivers, and we communicate the t parts of their demanded messages see 47 using separate source and channe coding. The strong receivers wi not participate in this subphase. The transmitter proceeds in two steps: T: The transmitter appies Method En with = w and t = t to demand vector d,..., d w and to messages W t d i : i w }. Let } W t XOR,S : S,..., w}, S = t + denote the output of Method En. T: The transmitter uses a capacity-achieving code for the pacet-erasure BC to send the XORs in 5 to the wea receivers. Each wea receiver i w decodes in two steps: R: Receiver i recovers a transmitted XOR messages in 5 using an appropriate channe decoder. R: Receiver i appies method De i with = w and t = t to demand vector d,..., d w and to the XOR messages produced in step R. For i w, et Ŵ t d i = Ŵdi,G t, Ŵd i,g t,..., Ŵd i,g t w t denote the output produced by De i. Deivery subphase : Here we consider a receivers. To the strong receivers: We communicate the t parts of their demanded messages. To the wea receivers: We communicate the t parts of their demanded messages. Both communications wi be done simutaneousy using joint cache-channe coding via piggybac coding. The transmitter proceeds in two steps: T: The transmitter appies Method En with = w and t = t to demand vector d,..., d w and messages W t d i : i w }. Method En outputs an XOR message for each size-t subset of,..., w }, for exampe, see. We denote these XOR messages by 7 } W t w : =,...,. 53 XOR,G t t 7 The messages in 53 have the superscript t, because they correspond to the output of Method En with the parameter t = t. In contrast, G t } have superscript t because they correspond to subsets of size t = t. 5 5

16 6 T: Time-sharing is performed over w t different periods, where each period is associated with a size-t subset of,,..., w }. Consider the -th subset G t. First reca that the subset G t of wea receivers has } W t : j s d j,g t stored as side information in their cache memories. The transmitter uses piggybac coding to send } W t : j s to a strong receivers s and the XOR message to a wea receivers in G t. d j,g t W t XOR,G t Each strong receiver j s performs piggybac decoding for the receiver without side-information for a w t transmission periods. It forms the w t -tupe estimatḙ W t t d j := Ŵ Each wea receiver i w proceeds in two steps: R: Receiver i considers each subset G t d j,g t it decodes the XOR message W t XOR,G t channe outputs of the period associated with G t 54a 54b,..., Ŵ t, j d j,g t s. 55 w t of size t to which it beongs i.e., each G t,..., w } such that G t i, and by appying piggybac decoding for the receiver with side-information to the. R: Receiver i then appies Method De i with = w and t = t to demand vector d,..., d w, the XOR messages decoded in step R and its cache content V i. Let Ŵ t d i, i w 56 denote the output of Method De i. 3 Deivery subphase 3: Here we consider ony the strong receivers, and we wi communicate the remaining t parts of their demanded messages. The wea receivers wi not participate in this subphase. The transmitter communicates } W t d j : j s to the strong receivers using a capacity-achieving code for the pacet-erasure BC. Each receiver uses an optima decoding method to produce the estimate Ŵ t d j, j s Fina decoding: Each receiver,..., } outputs D. Anaysis Ŵ d = Ŵ t d, Ŵ t d. 58 Fix t,..., w }. We show that the above scheme achieves the rate-memory pair R t, M t in 9. Caching phase: By 5, our caching strategy requires a cache memory size of M = R t D t + R t D t w w D = R t + w t + δw δ s. 59 w t s δ w We now anayse the probabiity of decoding error. We present conditions under which the probabiity that the estimates produced in Subphases 3, 5, 55, 56, and 57 are not equa to the corresponding submessages in 47 tends to 0 as n. Deivery subphase : Proposition combined with Lemma and 48b, prove that the probabiity that the estimates in 5 are incorrect tends to 0 as n, whenever R w t t+ < β ts δw w t+ δ w δ s F δ w

17 7 3 Deivery subphase : Consider a singe period with the transmission of message in 54. Since a wea receivers and a strong receivers are statisticay equivaent, the probabiity that the estimates in 55 and 56 are incorrect is at most t s times arger than the probabiity of error in 55 and 56 for a singe wea and a singe strong receiver. By Coroary 4 and Lemma, this atter probabiity of error tends to 0 and thus aso the origina probabiity of error tends to 0 as n, whenever max R t w t+ t F δ w, Rt w t+ t F δ s + R t s } < β. 6 By our choice 48 the two terms in the maximization are equa, and thus by 48a we concude that the probabiity of producing an error in 55 or 56 tends to 0 as n, whenever + w t+ t s R w t+ t δw δs δ w F δ w < β. 6 4 Deivery subphase 3: Proposition combined with 48a prove that the probabiity of producing a wrong guess in 57 tends to 0 as n, whenever R s < β w t+ t s δw δs δ w F δ s 5 Overa scheme: Combining 60, 6, and 63 and using 50, after some agebraic manipuations, we see that the probabiity of decoding error tends to 0 as n, whenever R < F δ w w t+ t + w t+ t s δw δs δ w. + w t t+ s δw δs δ δ w + w s δ s Together with 59, this proves achievabiity of the rate-memory pair R t, M t in 9. VI. EXTENSIONS OF OUR JOINT CACHE-CHANNEL CODING SCHEME The scenario in Section II aowed for a compact exposition of our new joint cache-channe coding idea. This idea however extends aso to more genera scenarios. In the foowing subsections we present some ideas. A. Wea receivers have different erasure probabiities: For simpicity, assume that the wea receivers are ordered so that δ δ... δ w hods. The scheme of section V may be modified as foows: For each XOR message sent in deivery phase, set the rate of the codeboo to be equa to the capacity of the weaest receiver to whom the XOR message is intended. B. Strong receivers have different erasure probabiities: For simpicity, assume that the strong receivers are ordered so that δ w+ δ w+... δ. We spit the set of strong receivers into a set of moderatey strong receivers w +,..., j and a set of very strong receivers j,...,, where j is chosen depending on the various erasure probabiities and cache sizes. We now time-share two schemes whose engths need to be optimized: In the first period, a standard capacity-achieving coding scheme for the pacet-erasure BC is used to serve ony the moderatey strong receivers w +,..., j. In the second period, our joint cache-channe coding scheme is used to serve a other receivers. C. Some weaer receivers do not have caches: We time-share two schemes whose engths need to be optimized. In the first period, a standard capacity-achieving code for the pacet-erasure BC is used to serve the wea receivers without caches. In the second period, our joint cache-channe coding scheme is used to serve a other receivers. D. Wea receivers have different cache sizes: For simpicity, assume that the wea receivers are ordered so that M M... M w, where M i denotes the cache memory size at receiver i w. We time-share w schemes of equa ength. In period i w, we treat the wea receivers i +,..., w assuming that they have no zero cache memories and we treat the wea receivers,..., i assuming that they have cache memories of size M i M i+. We thus appy the coding scheme that we setched in the previous subsection.

18 8 E. Strong receivers have cache memories For simpicity, assume that a strong receivers in s have cache memories of equa size M > M s > 0. We time-share two schemes. In the first period of ength n Ms M we suppose that the strong receivers have no caches: we thus use our joint cache channe coding scheme. In the second period of ength n Ms M we suppose that a receivers in the networ have cache memory M: we appy separate cache-channe coding combining Maddah-Ai & Niesen coded caching over a receivers with capacity-achieving code for the pacet-erasure BC. F. Strong receivers have cache memories and are served additiona data and The strong receivers have cache memories of size M s as in the previous subsection. There are two ibraries now: ibrary A: ibrary B: Each wea receiver i w demands ony fie W A d i fies W A,..., W A D fies W B,..., W B D. from ibrary A, whereas each strong receiver demands a fie W A d j from ibrary A and a fie W B d j from ibrary B. We time-share two schemes whose engths need to be optimized. In the first period, we suppose that the strong receivers have no cache memories and use our joint cache-channe coding scheme for ibrary A. In the second period, we ony serve the strong receivers from ibrary B. To this end, we appy separate cache-channe coding combining Maddah-Ai & Niesen coded caching over a receivers with capacity-achieving code for the pacet-erasure BC. In the scheme that we propose, the caches at the strong receivers are used ony to cache messages from ibrary B, but not from ibrary A. The idea is that when the strong receivers are sufficienty strong, then our joint cache-channe coding scheme is as performant as if a receivers in the networ had caches. One might therefore choose to dedicate the caches at the strong receivers entirey to the transmission of fies from ibrary B. G. Genera DMBCs Pacet-erasure BCs are simper than genera BCs because time-sharing of optima point-to-point codes for various receivers achieves capacity without caches. For our joint cache-channe coding scheme to be effective on more genera DMBCs, we wi have to partiay repace time-sharing by superposition coding and more generay Marton coding. More specificay, we use superposition coding and superpose the codewords sent in deivery subphase 3 on the piggybac codeword sent in deivery subphase and on the codewords sent in deivery subphase. When there is no cear notion of weaer and stronger, i.e., the channe is neither degraded, ess noisy, more capabe, or essentiay ess noisy, then we use Marton coding where the piggybac codewords serve as coud centers and the other codewords as sateites. VII. UPPER BOUND FOR GENERAL DEGRADED BCS UNDER ARBITRARY DEMANDS We consider a more genera setup for the upper bound, where each receiver i has a cache of size M i, and where the broadcast channe is a discrete memoryess degraded BC with input aphabet X and equa output aphabets Y,..., Y. The joint transitiona aw of the memoryess BC is given by P YY Y Xy,..., y x. We assume that the BC is degraded, i.e., the transition aw satisfies the Marov chain X Y Y Y. 64 For our probem setup, ony the margina transition aw is reevant. Therefore our upper bound hods aso for stochasticay degraded BC, i.e., for transition aws P Y,...,Y X for which there exists a conditiona probabiity distribution P Y Y, P Y3 Y,..., P Y Y that satisfies P YY Y Xy,..., y x = P Y Xy x P Y Y y y... P Y Y y y 65 for a x, y, y,..., y X Y Y Y. Note that the pacet-erasure BC that we study fas in the cass of stochasticay degraded BCs. worst The ibrary and the probabiity of worst-case error P e is defined as before. A rate-memory tupe R, M,..., M is said achievabe if for every ɛ > 0 there exists a sufficienty arge bocength n and caching, encoding and decoding functions as in 8 such that P worst e < ɛ. The capacity-memory tradeoff CM,..., M is defined as the supremum over a rates R so that R, M,..., M are achievabe. For each S, et R sym,s denote the argest equa-rate that is achievabe over a BC with receivers in S when there are no cache memories. We prove the foowing upper bound on CM,..., M :