The Exact Rate Memory Tradeoff for Small Caches with Coded Placement
11 The Exact Rate Memory Tradeoff for SmallCaches with Coded Placement
Vijith Kumar K P, Brijesh Kumar Rai and Tony Jacob
Abstract
The idea of coded caching was introduced by Maddah-Ali and Niesen who demonstrated theadvantages of coding in caching problems. To capture the essence of the problem, they introducedthe ( ๐, ๐พ ) canonical cache network in which ๐พ users with independent caches of size ๐ request ๏ฌlesfrom a server that has ๐ ๏ฌles. Among other results, the caching scheme and lower bounds proposedby them led to a characterization of the exact rate memory tradeoff when ๐ โฅ ๐๐พ ( ๐พ โ ) . These lowerbounds along with the caching scheme proposed by Chen et al. led to a characterization of the exactrate memory tradeoff when ๐ โค ๐พ . In this paper we focus on small caches where ๐ โ (cid:2) , ๐๐พ (cid:3) andderive new lower bounds. For the case when (cid:6) ๐พ + (cid:7) โค ๐ โค ๐พ and ๐ โ (cid:2) ๐พ , ๐๐พ ( ๐ โ ) (cid:3) , our lower boundsdemonstrate that the caching scheme introduced by Gรณmez-Vilardebรณ is optimal and thus extend thecharacterization of the exact rate memory tradeoff. For the case โค ๐ โค (cid:6) ๐พ + (cid:7) , we show that the newlower bounds improve upon the previously known lower bounds. Index Terms
Coded caching, coded placement, exact rate memory tradeoff, lower bounds.
I. I
NTRODUCTION
Content distribution networks use memory distributed across the network, known as caches,to reduce the peak time data traf๏ฌc by keeping copies of ๏ฌle fragments near the end-users. Thesetechniques, known as caching techniques, generally operate in two phases. In the ๏ฌrst phase,called the placement phase, the server ๏ฌlls the caches with fragments of ๏ฌles available in theserver. In the second phase, called the delivery phase, the server broadcasts a set of packetsto meet each userโs requests, aided by the caches available near to the user. Maddah-Ali andNiesen, in their seminal work [1], noted that traditional caching techniques fail to exploit themulticast opportunity available in such networks. To address this limitation, they introduced thenotion of coded caching and proposed a scheme to demonstrate that coding reduces the peak
February 10, 2021 DRAFT a r X i v : . [ c s . I T ] F e b data traf๏ฌc load over traditional uncoded caching schemes. They introduced the canonical ( ๐, ๐พ ) cache network where the server has ๐ ๏ฌles { ๐ , . . . , ๐ ๐ } and is communicating with ๐พ users { ๐ , . . . , ๐ ๐พ } through a common shared error-free link. Here, each user ๐ ๐ is equipped withan isolated cache ๐ ๐ of size ๐ โ [ , ๐ ] as shown in Fig. 1. During the placement phase, the ๐ ๐ . . . ๐ ๐ โ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐พ ๐ ๐พ . . . ๐ FilesServer ๐ CachesUsers User 1 User 2 User ๐พ Fig. 1: The ( ๐, ๐พ ) cache networkserver ๏ฌlls each userโs cache without knowing their future demands. Let the usersโ requests berepresented by d = ( ๐ ๐ , . . . , ๐ ๐ ๐พ ) , where ๐ ๐ ๐ is the ๏ฌle requested by user ๐ ๐ . During thedelivery phase, when demand d is revealed, the server broadcasts a set of packets ๐ d of size ๐ d ( ๐ ) so that users can obtain their requested ๏ฌle, aided by their cache contents. The quantity ๐ d ( ๐ ) is called the load experienced by the network. The fundamental issue in a caching schemeis to decide what to store in each userโs cache and, accordingly, what to broadcast to ful๏ฌll eachuserโs demand such that the load experienced by the shared link is minimized. Variations ofthis problem have been introduced for decentralized networks [2], hierarchical networks [3],multiple servers [4], heterogeneous networks [6], D2D networks [8], and shared cache networks[7]. Issues of privacy [5] and data shuf๏ฌing [9] have also been studied in this setup.As a result of the inherent symmetry of the problem of coded caching for the canonical ( ๐, ๐พ ) cache network, all demands related to each other through a permutation are naturally groupedtogether. This leads to the consideration of symmetric caching schemes which were shown byTian [10] to operate at the same rate. Thus, we consider only the class of symmetric cachingschemes in this paper. Consider a demand d , where the user ๐ ๐ requires the ๏ฌle ๐ ๐ ๐ , d = ( ๐ ๐ , . . . , ๐ ๐ ๐พ ) (1) DRAFT February 10, 2021
Let ๐ ( . ) be a permutation operation de๏ฌned over the set { , . . . , ๐พ } and ๐ โ ( . ) be its inverse.Now consider another demand ๐ d , which is obtained by permuting the ๏ฌles requested by theusers, ๐ d = ( ๐ ๐ ๐ โ ( ) , . . . , ๐ ๐ ๐ โ ( ๐พ ) ) . (2)In the demand ๐ d , the user ๐ ๐ ( ๐ ) requires the ๏ฌle ๐ ๐ ๐ . In response to the demand ๐ d , the serverbroadcasts a set of packets ๐ ๐ d . For a symmetric caching scheme, we have [10] ๐ป ( ๐ ๐ ๐ , ๐ ๐ ( ๐ ) , ๐ ๐ d ) = ๐ป ( ๐ ๐ ๐ , ๐ ๐ , ๐ d ) (3)Consider the demands where each of the ๐ ๏ฌles is required by at least one user (and hence ๐ โค ๐พ ). The set of all such demands is denoted by D and the corresponding rate is denoted by ๐ ( ๐ ) , where ๐ ( ๐ ) = max { ๐ d ( ๐ ) | d โ D } . (4)For the ( ๐, ๐พ ) cache network with cache size ๐ , the memory rate pair ( ๐, ๐ ) is said to beachievable if there is a scheme with ๐ ( ๐ ) โค ๐ . For a such a scheme, we have ๐ป ( ๐ ๐ ) โค ๐ (5) ๐ป ( ๐ d ) โค ๐ (6) ๐ป ( ๐ ๐ , ๐ d ) = ๐ป ( ๐ ๐ ๐ , ๐ ๐ , ๐ d ) , (7) ๐ป ( ๐ , . . . , ๐ ๐ , ๐ ๐ , ๐ d ) = ๐ป ( ๐ , . . . , ๐ ๐ ) , (8)where (5) follows from the fact that size of each cache is ๐ , (6) follows from the fact that forany demand in D the size of ๐ d is at most ๐ ( ๐ ) โค ๐ , (7) follows from the fact that the ๏ฌle ๐ d ๐ can be computed from ๐ d and ๐ ๐ by the user ๐ ๐ , and (8) follows from the fact that ๐ ๐ and ๐ d are functions of ๏ฌles { ๐ , . . . , ๐ ๐ } . For a given cache size ๐ , the smallest ๐ such that ( ๐, ๐ ) is achievable is called the exact rate memory tradeoff denoted by ๐ โ ( ๐ ) = min { ๐ : ( ๐, ๐ ) is achievable } (9)Maddah-Ali and Niesen in [1] proposed a coding scheme with an uncoded placement phaseand a coded delivery phase for the demands in D and demonstrated that the rate achievedby the proposed scheme is within a multiplicative gap of 12 from the optimal rate using cutset arguments. Several caching schemes were proposed in [11]โ[19] to improve upon the rateachieved by the scheme proposed in [1]. Despite several lower bounds on the achievable rates February 10, 2021 DRAFT
Caching Scheme Cache Size ( ๐ ) Rate Memory Tradeoff ConditionChen et al. [11] (cid:2) , ๐พ (cid:3) ๐ โ ( ๐ ) = ๐ โ ๐ ๐ ๐ โค ๐พ Gรณmez-Vilardebรณ [14] (cid:2) ๐ , ๐ โ (cid:3) ๐ โ ( ๐ ) = ๐ โ ๐ โ ( ๐ โ ) ๐ ๐ = ๐พ Vijith et al. [16], [17] (cid:104) ๐ โ โ ๐ , ๐ โ (cid:105) ๐ โ ( ๐ ) = ๐ + ๐ โ ๐ โ ๐ ๐ = ๐พ Maddah-Ali& Niesen [1] (cid:104) ๐ ( ๐พ โ ) ๐พ , ๐ (cid:105) ๐ โ ( ๐ ) = โ ๐ ๐ -Yu et al. [13] [ , ๐ ] ๐ ( ๐ ) = ๐ ๐ + ( ๐ ๐ โ ๐ ๐ + ) (cid:16) ๐ โ ๐๐พ ๐ (cid:17) where ๐ ( ๐ ) = ๐พ ๐ถ ๐ + โ ๐พ โ ๐ ๐ถ ๐ + ๐พ ๐ถ ๐ and ๐ โ { , . . . , ๐พ } Optimal amonguncoded prefetchingschemesThis paper (cid:104) ๐พ , ๐๐พ ( ๐ โ ) (cid:105) ๐ โ ( ๐ ) = ๐พ ๐ โ ๐พ โ ( ๐ โ ) ๐ (cid:6) ๐พ + (cid:7) โค ๐ โค ๐พ TABLE I: Previous work in coded cachingbeing proposed in [20]โ[24], the nature of the exact rate memory tradeoff is still elusive, exceptfor the ( ๐, ) cache network. The schemes proposed in [1], [11] provide a characterization ofthe exact rate memory tradeoff when ๐ โ (cid:2) , ๐พ (cid:3) โช (cid:104) ๐ ( ๐พ โ ) ๐พ , ๐ (cid:105) . For the special case of ๐ = ๐พ ,the schemes proposed in [14], [17] provide a characterization of the exact rate memory tradeoffwhen ๐ โ (cid:104) ๐ , ๐ ( ๐ โ ) (cid:105) โช (cid:2) ๐ โ โ ๐ , ๐ โ (cid:3) . In a surprising result, Yu et al. [13] showed theexistence of a universal scheme among caching schemes with an uncoded placement phase.These results are summarised in TABLE I. In this paper, we consider the ( ๐, ๐พ ) cache networkwhere ๐ โค ๐พ and ๐ โ (cid:2) ๐พ , ๐๐พ (cid:3) and derive a set of new lower bounds for the demands in D .The contributions of this paper are as follows: โข For the case (cid:6) ๐พ + (cid:7) โค ๐ โค ๐พ , we derive a new lower bound and obtain a characterizationof the exact rate memory tradeoff when ๐ โค ๐๐พ ( ๐ โ ) . โข For the case โค ๐ โค (cid:6) ๐พ + (cid:7) , we derive a new lower bound which improves upon thepreviously known lower bounds.Throughout this paper we use [ ๐ฟ ] to represent the set { , , . . . , ๐ฟ } , and ๐ [ ๐ฟ ] to represent theset { ๐ , ๐ , . . . , ๐ ๐ฟ } . II. E XAMPLE NETWORKS
In this section, we consider two examples to motivate the results we present in the paper. The ( , ) network is an example for the case (cid:6) ๐พ + (cid:7) โค ๐ โค ๐พ and the ( , ) network is an example DRAFT February 10, 2021 for the case โค ๐ โค (cid:6) ๐พ + (cid:7) . A. Case I: The ( , ) cache network Here, users { ๐ , ๐ , ๐ , ๐ } are connected to a server with three ๏ฌles { ๐ด, ๐ต, ๐ถ } (each of size ๐น bits). Each user ๐ ๐ has a cache ๐ ๐ of size ๐ ๐น bits. For a demand d , we have: Lemma 1.
For the ( , ) cache network, achievable memory rate pairs ( ๐, ๐ ) must satisfy theconstraint ๐ + ๐ โฅ Proof.
We have, ๐ + ๐ ( ๐ ) โฅ ๐ป ( ๐ ) + ๐ป ( ๐ ) + ๐ป ( ๐ ) + ๐ป ( ๐ ) + ๐ป ( ๐ ( ๐ด,๐ต,๐ถ,๐ด ) ) + ๐ป ( ๐ ( ๐ต,๐ถ,๐ด,๐ด ) ) + ๐ป ( ๐ ( ๐ถ,๐ด,๐ด,๐ต ) ) ( ๐ ) โฅ ๐ป ( ๐ , ๐ , ๐ ( ๐ด,๐ต,๐ถ,๐ด ) ) + ๐ป ( ๐ , ๐ , ๐ ( ๐ด,๐ต,๐ถ,๐ด ) ) + ๐ป ( ๐ , ๐ , ๐ ( ๐ต,๐ถ,๐ด,๐ด ) ) + ๐ป ( ๐ , ๐ , ๐ ( ๐ถ,๐ด,๐ด,๐ต ) ) ( ๐ ) = ๐ป ( ๐ด, ๐ต, ๐ , ๐ , ๐ ( ๐ด,๐ต,๐ถ,๐ด ) ) + ๐ป ( ๐ด, ๐ต, ๐ , ๐ , ๐ ( ๐ด,๐ต,๐ถ,๐ด ) ) + ๐ป ( ๐ด, ๐ต, ๐ , ๐ , ๐ ( ๐ต,๐ถ,๐ด,๐ด ) )+ ๐ป ( ๐ด, ๐ต, ๐ , ๐ , ๐ ( ๐ถ,๐ด,๐ด,๐ต ) ) ( ๐ ) โฅ ๐ป ( ๐ด, ๐ต, ๐ , ๐ ( ๐ด,๐ต,๐ถ,๐ด ) ) + ๐ป ( ๐ด, ๐ต, ๐ , ๐ , ๐ , ๐ ( ๐ด,๐ต,๐ถ,๐ด ) ) + ๐ป ( ๐ด, ๐ต, ๐ , ๐ , ๐ ( ๐ต,๐ถ,๐ด,๐ด ) )+ ๐ป ( ๐ด, ๐ต, ๐ , ๐ , ๐ ( ๐ถ,๐ด,๐ด,๐ต ) )โฅ ๐ป ( ๐ด, ๐ต, ๐ , ๐ , ๐ ) + ๐ป ( ๐ด, ๐ต, ๐ , ๐ , ๐ ( ๐ต,๐ถ,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ต, ๐ , ๐ ( ๐ด,๐ต,๐ถ,๐ด ) )+ ๐ป ( ๐ด, ๐ต, ๐ , ๐ , ๐ ( ๐ถ,๐ด,๐ด,๐ต ) ) ( ๐ ) โฅ ๐ป ( ๐ด, ๐ต, ๐ , ๐ , ๐ , ๐ ( ๐ต,๐ถ,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ต, ๐ , ๐ ) + ๐ป ( ๐ด, ๐ต, ๐ , ๐ ( ๐ด,๐ต,๐ถ,๐ด ) )+ ๐ป ( ๐ด, ๐ต, ๐ , ๐ , ๐ ( ๐ถ,๐ด,๐ด,๐ต ) ) ( ๐ ) = ๐ป ( ๐ด, ๐ต, ๐ถ, ๐ , ๐ , ๐ , ๐ ( ๐ต,๐ถ,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ต, ๐ , ๐ ( ๐ด,๐ต,๐ถ,๐ด ) ) + ๐ป ( ๐ด, ๐ต, ๐ , ๐ )+ ๐ป ( ๐ด, ๐ต, ๐ , ๐ , ๐ ( ๐ถ,๐ด,๐ด,๐ต ) ) ( ๐ ) = ๐ป ( ๐ด, ๐ต, ๐ถ ) + ๐ป ( ๐ด, ๐ต, ๐ , ๐ ( ๐ด,๐ต,๐ถ,๐ด ) ) + ๐ป ( ๐ด, ๐ต, ๐ , ๐ ) + ๐ป ( ๐ด, ๐ต, ๐ , ๐ , ๐ ( ๐ถ,๐ด,๐ด,๐ต ) ) ( ๐ ) โฅ ๐ป ( ๐ด, ๐ต, ๐ถ ) + ๐ป ( ๐ด, ๐ต, ๐ , ๐ ( ๐ด,๐ต,๐ถ,๐ด ) ) + ๐ป ( ๐ด, ๐ต, ๐ ) + ๐ป ( ๐ด, ๐ต, ๐ , ๐ , ๐ , ๐ ( ๐ถ,๐ด,๐ด,๐ต ) ) ( ๐ ) = ๐ป ( ๐ด, ๐ต, ๐ถ ) + ๐ป ( ๐ด, ๐ต, ๐ , ๐ ( ๐ด,๐ต,๐ถ,๐ด ) ) + ๐ป ( ๐ด, ๐ต, ๐ ) + ๐ป ( ๐ด, ๐ต, ๐ถ, ๐ , ๐ , ๐ , ๐ ( ๐ถ,๐ด,๐ด,๐ต ) ) ( ๐ ) = ๐ป ( ๐ด, ๐ต, ๐ถ ) + ๐ป ( ๐ด, ๐ต, ๐ , ๐ ( ๐ด,๐ต,๐ถ,๐ด ) ) + ๐ป ( ๐ด, ๐ต, ๐ )โฅ ๐ป ( ๐ด, ๐ต, ๐ถ ) + ๐ป ( ๐ด, ๐ต, ๐ ( ๐ด,๐ต,๐ถ,๐ด ) ) + ๐ป ( ๐ด, ๐ต, ๐ ) February 10, 2021 DRAFT ( ๐ ) = ๐ป ( ๐ด, ๐ต, ๐ถ ) + ๐ป ( ๐ด, ๐ต, ๐ ( ๐ด,๐ด,๐ต,๐ถ ) ) + ๐ป ( ๐ด, ๐ต, ๐ ) ( ๐ ) โฅ ๐ป ( ๐ด, ๐ต, ๐ถ ) + ๐ป ( ๐ด, ๐ต ) + ๐ป ( ๐ด, ๐ต, ๐ , ๐ ( ๐ด,๐ด,๐ต,๐ถ ) ) ( ๐ ) = ๐ป ( ๐ด, ๐ต, ๐ถ ) + ๐ป ( ๐ด, ๐ต ) + ๐ป ( ๐ด, ๐ต, ๐ถ, ๐ , ๐ ( ๐ด,๐ด,๐ต,๐ถ ) ) ( ๐ ) = ๐ป ( ๐ด, ๐ต, ๐ถ ) + ๐ป ( ๐ด, ๐ต ) โฅ , where ( ๐ ) follows from (6) and (5), ( ๐ ) follows from the submodularity property of entropy, ( ๐ ) follows from (7), ( ๐ ) follows from (8), ( ๐ ) follows from (3). (cid:3) The above result improves upon the previous results from [1], [14], [22] and is summarisedin TABLE II and Fig. 2.
Memory Rate [14] Lower Bound [1], [22] New Lower Bound โค ๐ โค โ ๐ ๐ โฅ max (cid:110) ( โ ๐ ) , (cid:16) โ ๐ (cid:17)(cid:111) ๐ โฅ โ ๐ TABLE II: Rate memory tradeoff for the ( , ) cache network / / / / Cache size ๐ R a t e ๐ ( , ) ( , )( , ) New Rate Memory TradeoffRate Memory Tradeoff [1], [11]Fig. 2: Rate memory tradeoff for the ( , ) cache network DRAFT February 10, 2021
B. Case II: The ( , ) cache network Here, users { ๐ , ๐ , ๐ , ๐ } are connected to a server with ๏ฌles { ๐ด, ๐ต } (each of size ๐น bits).Each user ๐ ๐ has cache ๐ ๐ of size ๐ ๐น bits. For a demand d , we have: Lemma 2.
For the ( , ) cache network, achievable memory rate pairs ( ๐, ๐ ) must satisfy theconstraint ๐ + ๐ โฅ . (10) Proof.
We have, ๐ + ๐ โฅ ๐ป ( ๐ ) + ๐ป ( ๐ ) + ๐ป ( ๐ ) + ๐ป ( ๐ ) + ๐ป ( ๐ ( ๐ด,๐ต,๐ด,๐ด ) ) + ๐ป ( ๐ ( ๐ต,๐ด,๐ด,๐ด ) ) + ๐ป ( ๐ ( ๐ด,๐ด,๐ต,๐ด ) ) ( ๐ ) โฅ ๐ป ( ๐ , ๐ ( ๐ด,๐ต,๐ด,๐ด ) ) + ๐ป ( ๐ , ๐ ( ๐ด,๐ต,๐ด,๐ด ) ) + ๐ป ( ๐ , ๐ ( ๐ด,๐ต,๐ด,๐ด ) ) + ๐ป ( ๐ , ๐ ( ๐ต,๐ด,๐ด,๐ด ) ) + ๐ป ( ๐ , ๐ ( ๐ต,๐ด,๐ด,๐ด ) )+ ๐ป ( ๐ , ๐ ( ๐ด,๐ด,๐ต,๐ด ) ) + ๐ป ( ๐ ) + ๐ป ( ๐ ) ( ๐ ) = ๐ป ( ๐ด, ๐ , ๐ ( ๐ด,๐ต,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ , ๐ ( ๐ด,๐ต,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ , ๐ ( ๐ด,๐ต,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ , ๐ ( ๐ต,๐ด,๐ด,๐ด ) )+ ๐ป ( ๐ด, ๐ , ๐ ( ๐ต,๐ด,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ , ๐ ( ๐ด,๐ด,๐ต,๐ด ) ) + ๐ป ( ๐ ) + ๐ป ( ๐ ) ( ๐ ) โฅ ๐ป ( ๐ด, ๐ , ๐ , ๐ , ๐ ( ๐ด,๐ต,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ ( ๐ด,๐ต,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ , ๐ , ๐ ( ๐ต,๐ด,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ ( ๐ต,๐ด,๐ด,๐ด ) )+ ๐ป ( ๐ด, ๐ , ๐ ( ๐ด,๐ด,๐ต,๐ด ) ) + ๐ป ( ๐ ) + ๐ป ( ๐ ) ( ๐ ) = ๐ป ( ๐ด, ๐ , ๐ , ๐ , ๐ ( ๐ด,๐ต,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ , ๐ , ๐ ( ๐ต,๐ด,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ , ๐ ( ๐ด,๐ด,๐ต,๐ด ) ) + ๐ป ( ๐ด, ๐ ( ๐ด,๐ด,๐ด,๐ต ) )+ ๐ป ( ๐ด, ๐ ( ๐ด,๐ต,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ ( ๐ต,๐ด,๐ด,๐ด ) ) + ๐ป ( ๐ ) + ๐ป ( ๐ ) ( ๐ ) โฅ ๐ป ( ๐ด, ๐ , ๐ , ๐ , ๐ ( ๐ด,๐ต,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ , ๐ , ๐ ( ๐ต,๐ด,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ , ๐ ( ๐ด,๐ด,๐ต,๐ด ) ) + ๐ป ( ๐ด, ๐ ( ๐ด,๐ด,๐ด,๐ต ) )+ ๐ป ( ๐ด, ๐ , ๐ ( ๐ด,๐ต,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ , ๐ ( ๐ต,๐ด,๐ด,๐ด ) ) ( ๐ ) = ๐ป ( ๐ด, ๐ , ๐ , ๐ , ๐ ( ๐ด,๐ต,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ , ๐ , ๐ ( ๐ต,๐ด,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ , ๐ ( ๐ด,๐ด,๐ต,๐ด ) ) + ๐ป ( ๐ด, ๐ ( ๐ด,๐ด,๐ด,๐ต ) )+ ๐ป ( ๐ด, ๐ต, ๐ , ๐ ( ๐ด,๐ต,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ต, ๐ , ๐ ( ๐ต,๐ด,๐ด,๐ด ) ) ( ๐ ) = ๐ป ( ๐ด, ๐ , ๐ , ๐ , ๐ ( ๐ด,๐ต,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ , ๐ , ๐ ( ๐ต,๐ด,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ , ๐ ( ๐ด,๐ด,๐ต,๐ด ) ) + ๐ป ( ๐ด, ๐ ( ๐ด,๐ด,๐ด,๐ต ) )+ ๐ป ( ๐ด, ๐ต ) ( ๐ ) โฅ ๐ป ( ๐ด, ๐ , ๐ , ๐ , ๐ ( ๐ด,๐ต,๐ด,๐ด ) , ๐ ( ๐ต,๐ด,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ , ๐ ) + ๐ป ( ๐ด, ๐ , ๐ ( ๐ด,๐ด,๐ต,๐ด ) ) + ๐ป ( ๐ด, ๐ ( ๐ด,๐ด,๐ด,๐ต ) )+ ๐ป ( ๐ด, ๐ต ) ( ๐ ) = ๐ป ( ๐ด, ๐ต, ๐ , ๐ , ๐ , ๐ ( ๐ด,๐ต,๐ด,๐ด ) , ๐ ( ๐ต,๐ด,๐ด,๐ด ) ) + ๐ป ( ๐ด, ๐ , ๐ ) + ๐ป ( ๐ด, ๐ , ๐ ( ๐ด,๐ด,๐ต,๐ด ) ) + ๐ป ( ๐ด, ๐ ( ๐ด,๐ด,๐ด,๐ต ) )+ ๐ป ( ๐ด, ๐ต ) February 10, 2021 DRAFT ( ๐ ) = ๐ป ( ๐ด, ๐ต ) + ๐ป ( ๐ด, ๐ , ๐ ) + ๐ป ( ๐ด, ๐ , ๐ ( ๐ด,๐ด,๐ต,๐ด ) ) + ๐ป ( ๐ด, ๐ ( ๐ด,๐ด,๐ด,๐ต ) ) ( ๐ ) โฅ ๐ป ( ๐ด, ๐ต ) + ๐ป ( ๐ด, ๐ , ๐ , ๐ ( ๐ด,๐ด,๐ต,๐ด ) ) + ๐ป ( ๐ด, ๐ ) + ๐ป ( ๐ด, ๐ ( ๐ด,๐ด,๐ด,๐ต ) ) ( ๐ ) = ๐ป ( ๐ด, ๐ต ) + ๐ป ( ๐ด, ๐ต, ๐ , ๐ , ๐ ( ๐ด,๐ด,๐ต,๐ด ) ) + ๐ป ( ๐ด, ๐ ) + ๐ป ( ๐ด, ๐ ( ๐ด,๐ด,๐ด,๐ต ) ) ( ๐ ) = ๐ป ( ๐ด, ๐ต ) + ๐ป ( ๐ด, ๐ ) + ๐ป ( ๐ด, ๐ ( ๐ด,๐ด,๐ด,๐ต ) ) ( ๐ ) โฅ ๐ป ( ๐ด, ๐ต ) + ๐ป ( ๐ด, ๐ , ๐ ( ๐ด,๐ด,๐ด,๐ต ) ) + ๐ป ( ๐ด ) ( ๐ ) = ๐ป ( ๐ด, ๐ต ) + ๐ป ( ๐ด, ๐ต, ๐ , ๐ ( ๐ด,๐ด,๐ด,๐ต ) ) + ๐ป ( ๐ด ) ( ๐ ) = ๐ป ( ๐ด, ๐ต ) + ๐ป ( ๐ด ) โฅ , where ( ๐ ) follows from the submodularity property of entropy, ( ๐ ) follows from (7), ( ๐ ) follows from (3), ( ๐ ) follows from (8). (cid:3) The above result improves upon the previous results from [1], [14], [22] and is summarisedin TABLE III and Fig. 3.
Memory Rate [14] Lower Bound [1], [22] New Lower Bound โค ๐ โค โ ๐ ๐ โฅ โ ๐ ๐ โฅ โ ๐ TABLE III: Rate memory tradeoff for the ( , ) cache network Remark 1.
It should be noted that, for the ( , ) cache network, the bound ๐ + ๐ โฅ isalready mentioned in [10]. We present the proof above, which can be extended to the ( ๐, ๐พ ) cache network. III. N
EW LOWER BOUNDS
In this section, we derive new lower bounds on the rate memory tradeoff for the ( ๐, ๐พ ) cachenetwork where ๐ โค ๐พ and cache size ๐ โ (cid:2) ๐พ , ๐๐พ (cid:3) . The key ideas we employ are identities (7),(8) and the properties of symmetric caching schemes stated in (3). As in Section II, we considertwo cases, namely (cid:6) ๐พ + (cid:7) โค ๐ โค ๐พ and โค ๐ โค (cid:6) ๐พ + (cid:7) . DRAFT February 10, 2021 / / / Cache size ๐ R a t e ๐ ( , ) ( , )( , )( , )( , ) New Lower BoundKnown Lower Bound [1], [22]Rate Memory Tradeoff [1], [11]Known Achievable Rate [10]Fig. 3: Rate memory tradeoff for the ( , ) cache network A. Case I: (cid:6) ๐พ + (cid:7) โค ๐ โค ๐พ Consider the demand d = ( ๐ , ๐ , . . . , ๐ ๐ , ๐ , ๐ , . . . , ๐ ๐พ โ ๐ ) (11)Demands { d ๐ : 2 โค ๐ โค ๐พ } , are obtained from the demand d by cyclic left shifts as shownin TABLE IV. For the demand d ๐ , let ๐ d ๐ denote the set of packets broadcast by the server.Consider the user index ๐ de๏ฌned as ๐ = ๏ฃฑ๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃณ ๐ + โ ๐, for โค ๐ โค ๐๐พ + ๐ + โ ๐, for ๐ + โค ๐ โค ๐พ (12)It can be noted that in demand d ๐ , the user ๐ ๐ requires the ๏ฌle ๐ ๐ . For S โ { ๐ , . . . , ๐ ๐พ } , let ๐ S denote the cache contents of all the users in set S .The following lemma are easy to obtain: Lemma 3.
For S , T โ { ๐ , . . . , ๐ ๐พ } \ { ๐ ๐ } , we have the identity ๐ป ( ๐ [ ๐ โ ] , ๐ S , ๐ ๐ ) + ๐ป ( ๐ [ ๐ โ ] , ๐ T , ๐ d ๐ ) โฅ ๐ป ( ๐ [ ๐ โ ] , ๐ S โฉ T ) + ๐, February 10, 2021 DRAFT0
Users d . . . d ๐ . . . d ๐ d ๐ + . . . d ๐ + ๐ . . . d ๐พ A ๐ ๏ฃฑ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃณ ๐ ๐ . . . ๐ ๐ . . . ๐ ๐ ๐ . . . ๐ ๐ . . . ๐ ๐พ โ ๐ ๐ ๐ . . . ๐ ๐ + . . . ๐ ๐ . . . ๐ ๐ + . . . ๐ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ๐ ๐พ โ ๐ โ ๐ + ๐ ๐พ โ ๐ โ ๐ + . . . ๐ ๐พ โ ๐ . . . ๐ ๐พ โ ๐ โ ๐ ๐ ๐พ โ ๐ โ ๐ + . . . ๐ ๐พ โ ๐ . . . ๐ ๐พ โ ๐ โ ๐ C ๐ ๏ฃฑ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃณ ๐ ๐พ โ ๐ โ ๐ + ๐ ๐พ โ ๐ โ ๐ + . . . ๐ ๐พ โ ๐ + . . . ๐ ๐พ โ ๐ โ ๐ + ๐ ๐พ โ ๐ โ ๐ + . . . ๐ . . . ๐ ๐พ โ ๐ โ ๐ โ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ๐ ๐พ โ ๐ ๐ ๐พ โ ๐ . . . ๐ ๐พ โ ๐ + ๐ โ . . . ๐ ๐พ โ ๐ โ ๐ ๐พ โ ๐ . . . ๐ ๐ โ . . . ๐ ๐พ โ ๐ โ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ๐ ๐ โ ๐ ๐ ๐ โ ๐ . . . ๐ ๐ โ . . . ๐ ๐ โ ๐พ โ ๐ โ ๐ ๐ โ ๐พ โ ๐ . . . ๐ ๐ โ ๐พ โ . . . ๐ ๐ โ ๐ โ ๐ ๐ โ ๐ + ๐ ๐ โ ๐ + . . . ๐ ๐ . . . ๐ ๐ โ ๐พ โ ๐ ๐ ๐ โ ๐พ โ ๐ + . . . ๐ ๐ โ ๐พ . . . ๐ ๐ โ ๐ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ๐ ๐ ๐ ๐ . . . ๐ ๐ โ . . . ๐ ๐ โ ๐พ ๐ ๐ โ ๐พ + . . . ๐ ๐ โ ๐พ + ๐ . . . ๐ ๐ โ E ๏ฃฑ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃณ ๐ ๐ + ๐ . . . ๐ ๐ . . . ๐ ๐ โ ๐พ + ๐ ๐ โ ๐พ + . . . ๐ ๐ โ ๐พ + ๐ + . . . ๐ ๐ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ๐ ๐พ โ ๐ + ๐ ๐พ โ ๐ โ ๐ + . . . ๐ ๐พ โ ๐ . . . ๐ ๐ โ ๐ ๐ ๐ โ ๐ + . . . ๐ ๐ . . . ๐ ๐พ โ ๐ โ ๐ B ๐ ๏ฃฑ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃณ ๐ ๐พ โ ๐ + ๐ ๐พ โ ๐ โ ๐ + . . . ๐ . . . ๐ ๐ โ ๐ + ๐ ๐ โ ๐ + . . . ๐ . . . ๐ ๐พ โ ๐ โ ๐ + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ๐ ๐พ ๐ ๐พ โ ๐ . . . ๐ ๐ โ . . . ๐ ๐ โ ๐ ๐ . . . ๐ ๐ โ . . . ๐ ๐พ โ ๐ โ TABLE IV: The set of demands { d ๐ : 1 โค ๐ โค ๐พ } Proof.
We have, ๐ป ( ๐ [ ๐ โ ] , ๐ S , ๐ ๐ ) + ๐ป ( ๐ [ ๐ โ ] , ๐ T , ๐ d ๐ ) ( ๐ ) โฅ ๐ป ( ๐ [ ๐ โ ] , ๐ S โฉ T ) + ๐ป ( ๐ [ ๐ โ ] , ๐ S โช T , ๐ ๐ , ๐ d ๐ ) ( ๐ ) = ๐ป ( ๐ [ ๐ โ ] , ๐ S โฉ T ) + ๐ป ( ๐ [ ๐ โ ] , ๐ ๐ , ๐ S โช T , ๐ ๐ , ๐ d ๐ ) ( ๐ ) = ๐ป ( ๐ [ ๐ โ ] , ๐ S โฉ T ) + ๐ป ( ๐ [ ๐ ] ) = ๐ป ( ๐ [ ๐ โ ] , ๐ S โฉ T ) + ๐ where ( ๐ ) follows from the submodularity property of entropy, ( ๐ ) follows from (7), ( ๐ ) follows from (8). (cid:3) Lemma 4.
For a sequence of sets S ๐ โ { ๐ , . . . , ๐ ๐พ } \ { ๐ ๐ } , such that S ๐ = S ๐ + โช { ๐ ๐ + } , we DRAFT February 10, 20211 have ๐ป ( ๐ [ ๐ โ ] , ๐ S ๐ ) + ๐ โ๏ธ ๐ = ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ S ๐ , ๐ d ๐ ) โฅ( ๐ โ ๐ ) ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ S ๐ ) Proof.
We have, ๐ป ( ๐ [ ๐ โ ] , ๐ S ๐ )+ ๐ โ๏ธ ๐ = ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ S ๐ , ๐ d ๐ ) = ๐ป ( ๐ [ ๐ โ ] , ๐ S ๐ ) + ๐ป ( ๐ [ ๐ โ ] , ๐ S ๐ + , ๐ d ๐ + ) + ๐ โ๏ธ ๐ = ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ S ๐ , ๐ d ๐ ) ( ๐ ) = (cid:104) ๐ป ( ๐ [ ๐ โ ] , ๐ S ๐ + , ๐ ๐ + ) + ๐ป ( ๐ [ ๐ โ ] , ๐ S ๐ + , ๐ d ๐ + ) (cid:105) + ๐ โ๏ธ ๐ = ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ S ๐ , ๐ d ๐ ) ( ๐ ) โฅ ๐ + (cid:104) ๐ป ( ๐ [ ๐ โ ] , ๐ S ๐ + ) + ๐ป ( ๐ [ ๐ โ ] , ๐ S ๐ + , ๐ d ๐ + ) (cid:105) + ๐ โ๏ธ ๐ = ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ S ๐ , ๐ d ๐ ) ( ๐ ) โฅ ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ S ๐ + ) + ๐ป ( ๐ [ ๐ โ ] , ๐ S ๐ + , ๐ d ๐ + ) + ๐ โ๏ธ ๐ = ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ S ๐ , ๐ d ๐ ) ( ๐ ) โฅ ( ๐ โ ๐ ) ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ S ๐ ) where ( ๐ ) follows from de๏ฌnition of set S ๐ , ( ๐ ) follows from Lemma 3 with S โช { ๐ ๐ } = S ๐ + and T = S ๐ + , ( ๐ ) follows from Lemma 3 with S โช { ๐ ๐ } = S ๐ + and T = S ๐ + , ( ๐ ) follows from repeated use of Lemma 3 with S โช { ๐ ๐ } = S ๐ and T = S ๐ + for ๐ + โค ๐ โค ๐ . (cid:3) In a similar fashion, for a sequence of sets T ๐ โ { ๐ , . . . , ๐ ๐พ } \ { ๐ ๐ } , such that T ๐ + = T ๐ โช { ๐ ๐ } ,we can obtain ๐ป ( ๐ [ ๐ โ ] , ๐ T ๐ , ๐ ๐ ) + ๐ โ๏ธ ๐ = ๐ ๐ป ( ๐ [ ๐ โ ] , ๐ T ๐ , ๐ d ๐ ) โฅ ( ๐ โ ๐ + ) ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ T ๐ ) (13)For โค ๐ โค ๐ , consider the sets of users as shown below: February 10, 2021 DRAFT2
Set Users Number Files Requested in Demand d ๐ A ๐ ๐ , . . . , ๐ ๐ โ ๐ ๐ โ ๐ ๐ ๐ , . . . , ๐ ๐ โ B ๐ ๐ ๐พ + โ ๐ , . . . , ๐ ๐พ ๐ โ ๐ , . . . , ๐ ๐ โ C ๐ ๐ ๐พ + โ ๐ โ ๐ , . . . , ๐ ๐ โ ๐ ๐ โ ๐พ โ ๐ ๐พ โ ๐ + , . . . , ๐ ๐ โ E ๐ ๐ + , . . . , ๐ ๐พ ๐พ โ ๐ ๐ , . . . , ๐ ๐พ โ ๐ These sets are also indicated in TABLE IV. Note that A ๐ = B = C ๐ = ๐ (14) A ๐ + โช { ๐ ๐ + } = A ๐ (15) B ๐ โช { ๐ ๐ + ๐ } = B ๐ + (16) B ๐พ โ ๐ โช { ๐ ๐พ } = B ๐พ โ ๐ + = E (17) A ๐ โฉ C ๐ = C ๐ (18) B ๐ โฉ E = ๏ฃฑ๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃณ B ๐ when โค ๐ โค ๐พ โ ๐ E when ๐พ โ ๐ + โค ๐ โค ๐ (19)It can be noted that in the demands d ๐ and d ๐ + ๐ , the users in set B ๐ are requesting for thesame set of ๏ฌles { ๐ , . . . , ๐ ๐ โ } (for โค ๐ โค ๐ ). Thus, from (3) we have ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ , ๐ d ๐ ) = ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ , ๐ d ๐ + ๐ ) (20)Note that | A ๐ โช B ๐ | = | C ๐ โช E | = ๐ โ . Thus, we have ( ๐ โ ) ๐ + ๐ โฅ ๐ป ( ๐ A ๐ โช B ๐ ) + ๐ป ( ๐ d ๐ ) โฅ ๐ป ( ๐ A ๐ โช B ๐ , ๐ d ๐ ) (21)Similarly, ( ๐ โ ) ๐ + ๐ โฅ ๐ป ( ๐ C ๐ โช E ) + ๐ป ( ๐ d ๐ ) โฅ ๐ป ( ๐ C ๐ โช E , ๐ d ๐ ) (22)Now, we have the following result: Theorem 1.
For the ( ๐, ๐พ ) cache network, when (cid:6) ๐พ + (cid:7) โค ๐ โค ๐พ , achievable memory rate pairs ( ๐, ๐ ) must satisfy the constraint ๐พ ( ๐ โ ) ๐ + ๐พ ๐ โฅ ๐พ ๐ โ . DRAFT February 10, 20213
Proof.
We have, ๐พ ( ๐ โ ) ๐ + ๐พ ๐ = ๐พ โ ๐ โ๏ธ ๐ = (cid:104) ( ๐ โ ) ๐ + ๐ + ( ๐ โ ) ๐ + ๐ (cid:105) + ๐ โ๏ธ ๐ = ๐พ โ ๐ + (cid:104) ( ๐ โ ) ๐ + ๐ (cid:105) ( ๐ ) โฅ ๐พ โ ๐ โ๏ธ ๐ = (cid:104) ๐ป ( ๐ A ๐ โช B ๐ , ๐ d ๐ ) + ๐ป ( ๐ C ๐ โช E , ๐ d ๐ ) (cid:105) + ๐ โ๏ธ ๐ = ๐พ โ ๐ + ๐ป ( ๐ A ๐ โช B ๐ , ๐ d ๐ ) ( ๐ ) = ๐พ โ ๐ โ๏ธ ๐ = (cid:104) ๐ป ( ๐ [ ๐ โ ] , ๐ A ๐ โช B ๐ , ๐ d ๐ ) + ๐ป ( ๐ [ ๐ โ ] , ๐ C ๐ โช E , ๐ d ๐ ) (cid:105) + ๐ โ๏ธ ๐ = ๐พ โ ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ A ๐ โช B ๐ , ๐ d ๐ ) ( ๐ ) โฅ ๐พ โ ๐ โ๏ธ ๐ = (cid:104) ๐ป ( ๐ [ ๐ โ ] , ๐ A ๐ โช E , ๐ d ๐ ) + ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ โช C ๐ , ๐ d ๐ ) (cid:105) + ๐ โ๏ธ ๐ = ๐พ โ ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ A ๐ โช B ๐ , ๐ d ๐ )โฅ (cid:34) ๐ป ( ๐ [ ๐ โ ] , ๐ A โช E ) + ๐พ โ ๐ โ๏ธ ๐ = ๐ป ( ๐ [ ๐ โ ] , ๐ A ๐ โช E , ๐ d ๐ ) (cid:35) + ๐ โ๏ธ ๐ = ๐พ โ ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ A ๐ โช B ๐ , ๐ d ๐ )+ ๐พ โ ๐ โ๏ธ ๐ = ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ โช C ๐ , ๐ d ๐ ) ( ๐ ) โฅ ( ๐พ โ ๐ โ ) ๐ + (cid:34) ๐ป ( ๐ [ ๐ โ ] , ๐ A ๐พ โ ๐ โช E ) + ๐ โ๏ธ ๐ = ๐พ โ ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ A ๐ โช B ๐ , ๐ d ๐ ) (cid:35) + ๐พ โ ๐ โ๏ธ ๐ = ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ โช C ๐ , ๐ d ๐ ) ( ๐ ) = ( ๐พ โ ๐ โ ) ๐ + (cid:34) ๐ป ( ๐ [ ๐ โ ] , ๐ A ๐พ โ ๐ โช B ๐พ โ ๐ + ) + ๐ โ๏ธ ๐ = ๐พ โ ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ A ๐ โช B ๐ , ๐ d ๐ ) (cid:35) + ๐พ โ ๐ โ๏ธ ๐ = ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ โช C ๐ , ๐ d ๐ ) ( ๐ ) โฅ ( ๐ โ ) ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ A ๐ โช B ๐พ โ ๐ + ) + ๐พ โ ๐ โ๏ธ ๐ = ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ โช C ๐ , ๐ d ๐ ) ( ๐ ) = ( ๐ โ ) ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐พ โ ๐ + ) + ๐พ โ ๐ โ๏ธ ๐ = ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ โช C ๐ , ๐ d ๐ )โฅ( ๐ โ ) ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐พ โ ๐ + ) + ๐พ โ ๐ โ๏ธ ๐ = ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ , ๐ d ๐ ) ( โ ) = ( ๐ โ ) ๐ + (cid:34) ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐พ โ ๐ + ) + ๐พ โ ๐ โ๏ธ ๐ = ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ , ๐ d ๐ + ๐ ) (cid:35) ( ๐ ) โฅ ( ๐พ โ ) ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ B ) ( ๐ ) = ( ๐พ โ ) ๐ + ๐ป ( ๐ [ ๐ โ ] ) โฅ ๐พ ๐ โ February 10, 2021 DRAFT4 where ( ๐ ) follows from (21) and (22), ( ๐ ) follows from (7) and de๏ฌnition of sets A ๐ , B ๐ , C ๐ and E , ( ๐ ) follows from the submodularity property of entropy, and the facts that A ๐ โฉ C ๐ = C ๐ and B ๐ โฉ E = B ๐ for โค ๐ โค ๐พ โ ๐ (refer (18) and (19)), ( ๐ ) follows from Lemma 4 with S ๐ = A ๐ โช E , ๐ = , ๐ = ๐พ โ ๐ and (15), ( ๐ ) follows from (17) ( ๐ ) follows from Lemma 4 with S ๐ = A ๐ โช B ๐ , ๐ = ๐พ โ ๐ , ๐ = ๐ and (15), ( ๐ ) follows from (14), ( โ ) follows from (20), ( ๐ ) follows from (13) with T ๐ = B ๐ , ๐ = , ๐ = ๐พ โ ๐ and (16). (cid:3) B. Case II: โค ๐ โค (cid:6) ๐พ + (cid:7) Consider the demand d = ( ๐ , ๐ , . . . , ๐ ๐ , ๐ , ๐ , . . . , ๐ ๐ โ , ๐ , ๐ , . . . , ๐ ) (23)Demands { d ๐ : 2 โค ๐ โค ๐พ } , are obtained from the demand d by cyclic left shifts as shown inTABLE V.Consider the user index ๐ de๏ฌned as ๐ = ๏ฃฑ๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃณ ๐ + โ ๐, for โค ๐ โค ๐๐พ + ๐ + โ ๐, for ๐ + โค ๐ โค ๐พ (24)It can be noted that in demand d ๐ , the user ๐ ๐ requires the ๏ฌle ๐ ๐ . The following lemma iseasy to obtain. Lemma 5.
Let A , B , C โ { ๐ , . . . , ๐ ๐พ } be such that in demand d ๐ , every user in B requests the๏ฌle ๐ and users in C together request all the ๏ฌles in { ๐ , . . . , ๐ ๐ } . We have ๐ป ( ๐ [ ๐ โ ] , ๐ A , ๐ d ๐ ) + โ๏ธ ๐ โ B ๐ป ( ๐ ๐ )+ | B | ๐ป ( ๐ d ๐ )+ | B | ๐ป ( ๐ C ) โฅ ๐ป ( ๐ [ ๐ โ ] , ๐ A โช B , ๐ d ๐ )+ | B | ๐ DRAFT February 10, 20215
Users d . . . d ๐ . . . d ๐ d ๐ + . . . d ๐ + ๐ . . . d ๐ โ d ๐ . . . d ๐ . . . d ๐พ J ๐ ๏ฃฑ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃณ A ๐ ๏ฃฑ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃณ ๐ ๐ . . . ๐ ๐ . . . ๐ ๐ ๐ . . . ๐ ๐ . . . ๐ ๐ โ ๐ . . . ๐ . . . ๐ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ๐ ๐ โ ๐ ๐ ๐ โ ๐ . . . ๐ ๐ โ . . . ๐ ๐ โ ๐ โ ๐ ๐ โ ๐ . . . ๐ ๐ โ . . . ๐ ๐ . . . ๐ . . . ๐ ๐ โ ๐ ๐ โ ๐ + ๐ ๐ โ ๐ + . . . ๐ ๐ . . . ๐ ๐ โ ๐ ๐ ๐ โ ๐ + . . . ๐ . . . ๐ ๐ . . . ๐ . . . ๐ ๐ โ ๐ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F ๐ ๏ฃฑ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃณ ๐ ๐ + ๐ . . . ๐ ๐ . . . ๐ ๐ . . . ๐ . . . ๐ ๐ . . . ๐ . . . ๐ ๐ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ๐ ๐ โ ๐ ๐ ๐ โ ๐ . . . ๐ ๐ โ . . . ๐ ๐ . . . ๐ . . . ๐ ๐ . . . ๐ . . . ๐ ๐ โ ๐ โ G ๐ ๏ฃฑ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃณ ๐ ๐ โ ๐ + ๐ ๐ โ ๐ + . . . ๐ . . . ๐ ๐ . . . ๐ . . . ๐ ๐ . . . ๐ . . . ๐ ๐ โ ๐ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ๐ ๐พ โ ๐ + ๐ . . . ๐ . . . ๐ ๐ . . . ๐ . . . ๐ ๐ . . . ๐ . . . ๐ S ๐ ๏ฃฑ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃณ ๐ ๐พ โ ๐ + ๐ . . . ๐ . . . ๐ ๐ . . . ๐ . . . ๐ ๐ . . . ๐ . . . ๐ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ๐ ๐พ + ๐ โ ๐ + ๐ . . . ๐ . . . ๐ ๐ . . . ๐ . . . ๐ ๐ . . . ๐ ๐ . . . ๐ P ๐ ๏ฃฑ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃณ ๐ ๐พ + ๐ โ ๐ + ๐ . . . ๐ . . . ๐ ๐ . . . ๐ . . . ๐ ๐ . . . ๐ . . . ๐ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ๐ ๐พ + ๐ โ ๐ ๐ . . . ๐ . . . ๐ ๐ . . . ๐ . . . ๐ ๐ . . . ๐ ๐ โ . . . ๐ Q ๐ ๏ฃฑ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃณ ๐ ๐พ + ๐ โ ๐ + ๐ . . . ๐ . . . ๐ ๐ . . . ๐ . . . ๐ ๐ . . . ๐ . . . ๐ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ๐ ๐พ โ ๐ + ๐ . . . ๐ . . . ๐ ๐ โ ๐ ๐ ๐ โ ๐ + . . . ๐ ๐ . . . ๐ ๐ โ ๐ ๐ ๐ โ ๐ + . . . ๐ . . . ๐ B ๐ ๏ฃฑ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃณ ๐ ๐พ โ ๐ + ๐ . . . ๐ . . . ๐ ๐ โ ๐ + ๐ ๐ โ ๐ + . . . ๐ . . . ๐ ๐ โ ๐ + ๐ ๐ โ ๐ + . . . ๐ . . . ๐ K ๐ ๏ฃฑ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃณ ๐ ๐พ โ ๐ + ๐ . . . ๐ . . . ๐ ๐ โ ๐ + ๐ ๐ โ ๐ + . . . ๐ . . . ๐ ๐ โ ๐ + ๐ ๐ โ ๐ + . . . ๐ . . . ๐ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ๐ ๐พ ๐ . . . ๐ ๐ โ . . . ๐ ๐ โ ๐ ๐ . . . ๐ ๐ โ . . . ๐ ๐ โ ๐ ๐ โ . . . ๐ . . . ๐ TABLE V: Demand set { d ๐ : 1 โค ๐ โค ๐พ } Proof.
We have, ๐ป ( ๐ [ ๐ โ ] , ๐ A ,๐ d ๐ ) + โ๏ธ ๐ โ B ๐ป ( ๐ ๐ )+ | B | ๐ป ( ๐ d ๐ )+ | B | ๐ป ( ๐ C ) = ๐ป ( ๐ [ ๐ โ ] , ๐ A , ๐ d ๐ ) + โ๏ธ ๐ โ B (cid:104) ๐ป ( ๐ ๐ ) + ๐ป ( ๐ d ๐ ) (cid:105) + | B | ๐ป ( ๐ C ) ( ๐ ) โฅ ๐ป ( ๐ [ ๐ โ ] , ๐ A , ๐ d ๐ ) + โ๏ธ ๐ โ B ๐ป ( ๐ ๐ , ๐ d ๐ )+ | B | ๐ป ( ๐ C ) ( ๐ ) = ๐ป ( ๐ [ ๐ โ ] , ๐ A , ๐ d ๐ ) + โ๏ธ ๐ โ B ๐ป ( ๐ , ๐ ๐ , ๐ d ๐ )+ | B | ๐ป ( ๐ C ) ( ๐ ) โฅ ๐ป ( ๐ [ ๐ โ ] , ๐ A , ๐ d ๐ ) + (cid:104) ๐ป ( ๐ , ๐ B , ๐ d ๐ ) + (| B | โ ) ๐ป ( ๐ , ๐ d ๐ ) (cid:105) + | B | ๐ป ( ๐ C ) February 10, 2021 DRAFT6 ( ๐ ) โฅ ๐ป ( ๐ [ ๐ โ ] , ๐ A โช B , ๐ d ๐ ) + ๐ป ( ๐ , ๐ A โฉ B , ๐ d ๐ ) + (| B | โ ) ๐ป ( ๐ , ๐ d ๐ )+ | B | ๐ป ( ๐ C )โฅ ๐ป ( ๐ [ ๐ โ ] , ๐ A โช B , ๐ d ๐ )+ | B | (cid:104) ๐ป ( ๐ , ๐ d ๐ ) + ๐ป ( ๐ C ) (cid:105) ( ๐ ) โฅ ๐ป ( ๐ [ ๐ โ ] , ๐ A โช B , ๐ d ๐ )+ | B | ๐ป ( ๐ , ๐ C , ๐ d ๐ ) ( ๐ ) = ๐ป ( ๐ [ ๐ โ ] , ๐ A โช B , ๐ d ๐ )+ | B | ๐ป ( ๐ [ ๐ ] , ๐ C , ๐ d ๐ ) ( ๐ ) = ๐ป ( ๐ [ ๐ โ ] , ๐ A โช B , ๐ d ๐ )+ | B | ๐ป ( ๐ [ ๐ ] )โฅ ๐ป ( ๐ [ ๐ โ ] , ๐ A โช B , ๐ d ๐ )+ | B | ๐ where ( ๐ ) follows form the submodularity property of entropy, ( ๐ ) follows from (7) and the de๏ฌnition of B , ( ๐ ) follows from (7) and the de๏ฌnition of C , ( ๐ ) follows from (8). (cid:3) For โค ๐ โค ๐ , consider the sets of users as shown below: Set Users Number Files Requested in Demand d ๐ A ๐ ๐ , . . . , ๐ ๐ โ ๐ ๐ โ ๐ ๐ ๐ , . . . , ๐ ๐ โ B ๐ ๐ ๐พ โ ๐ + , . . . , ๐ ๐พ ๐ โ ๐ . . . , ๐ ๐ โ F ๐ ๐ ๐ + , . . . , ๐ ๐ โ ๐ ๐ โ ๐ ๐ ๐ , . . . , ๐ ๐ โ G ๐ ๐ ๐ โ ๐ + , . . . , ๐ ๐พ โ ๐ + ๐พ โ ๐ + ๐ J ๐ ๐ , . . . , ๐ ๐ โ ๐ + ๐ โ ๐ + ๐ ๐ . . . , ๐ ๐ K ๐ ๐ ๐พ โ ๐ + . . . ๐ ๐พ ๐ โ ๐ . . . , ๐ ๐ โ These sets are also indicated in TABLE V. Let I ๐ = J ๐ โช K ๐ (25) L ๐ = A ๐ โช B ๐ โช F ๐ โช G ๐ (26)Note that A ๐ = B = F ๐ = K = K = ๐ (27) L ๐ + โช { ๐ ๐ + } = L ๐ (28) B ๐ โช { ๐ ๐ + ๐ } = B ๐ + (29) DRAFT February 10, 20217
It can be noted that in the demands d ๐ and d ๐ + ๐ , users in the set B ๐ are requesting for thesame set of ๏ฌles { ๐ , . . . , ๐ ๐ โ } (for โค ๐ โค ๐ ). Thus, from (3) we have ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ , ๐ d ๐ ) = ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ , ๐ d ๐ + ๐ ) (30)Note that | A ๐ โช B ๐ | = | B ๐ โช F ๐ | = ๐ โ . Thus, we have ( ๐ โ ) ๐ + ๐ โฅ ๐ป ( ๐ A ๐ โช B ๐ ) + ๐ป ( ๐ d ๐ ) โฅ ๐ป ( ๐ A ๐ โช B ๐ , ๐ d ๐ ) (31)Similarly, ( ๐ โ ) ๐ + ๐ โฅ ๐ป ( ๐ B ๐ โช F ๐ ) + ๐ป ( ๐ d ๐ ) โฅ ๐ป ( ๐ B ๐ โช F ๐ , ๐ d ๐ ) (32)We can now obtain the following lemma: Lemma 6.
The sets B ๐ and L ๐ , de๏ฌned as above, satisfy ( ๐ ( ๐พ โ ๐ + ) โ ๐ + ) ๐ + ( ๐ ( ๐พ โ ๐ + ) โ ) ๐ โฅ ๐ป ( ๐ [ ๐ โ ] , ๐ L ๐ ) + ๐ โ โ๏ธ ๐ = ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ , ๐ d ๐ + ๐ ) + ๐ ( ( ๐พ โ ๐ + ) ๐ โ ) Proof.
We have, ( ๐ ( ๐พ โ ๐ + ) โ ๐ + ) ๐ + ( ๐ ( ๐พ โ ๐ + ) โ ) ๐ = ๐ โ๏ธ ๐ = (cid:104) ( ๐ โ ) ๐ + ๐ + ( ๐พ โ ๐ + ) ๐ + ( ๐พ โ ๐ + ) ( ๐ + ( ๐ โ ) ๐ ) (cid:105) + ๐ โ โ๏ธ ๐ = [( ๐ โ ) ๐ + ๐ ] ( ๐ ) โฅ ๐ โ๏ธ ๐ = (cid:20) ๐ป ( ๐ A ๐ โช B ๐ , ๐ d ๐ ) + โ๏ธ ๐ โ G ๐ ๐ป ( ๐ ๐ )+ | G ๐ | ๐ป ( ๐ d ๐ )+ | G ๐ | ๐ป ( ๐ I ๐ ) (cid:21) + ๐ โ โ๏ธ ๐ = ๐ป ( ๐ B ๐ โช F ๐ , ๐ d ๐ ) ( ๐ ) = ๐ โ๏ธ ๐ = (cid:20) ๐ป ( ๐ [ ๐ โ ] , ๐ A ๐ โช B ๐ , ๐ d ๐ ) + โ๏ธ ๐ โ G ๐ ๐ป ( ๐ ๐ )+ | G ๐ | ๐ป ( ๐ d ๐ )+ | G ๐ | ๐ป ( ๐ I ๐ ) (cid:21) + ๐ โ โ๏ธ ๐ = ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ โช F ๐ , ๐ d ๐ ) ( ๐ ) โฅ ๐ โ๏ธ ๐ = (cid:20) ๐ป ( ๐ [ ๐ โ ] , ๐ A ๐ โช B ๐ โช G ๐ , ๐ d ๐ )+ | G ๐ | ๐ (cid:21) + ๐ โ โ๏ธ ๐ = ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ โช F ๐ , ๐ d ๐ ) = ๐ป ( ๐ [ ๐ โ ] , ๐ A ๐ โช B ๐ โช G ๐ , ๐ d ๐ ) + ๐ โ โ๏ธ ๐ = (cid:104) ๐ป ( ๐ [ ๐ โ ] , ๐ A ๐ โช B ๐ โช G ๐ , ๐ d ๐ ) + ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ โช F ๐ , ๐ d ๐ ) (cid:105) + ๐ โ๏ธ ๐ = | G ๐ | ๐ February 10, 2021 DRAFT8 ( ๐ ) โฅ ๐ป ( ๐ [ ๐ โ ] , ๐ A ๐ โช B ๐ โช G ๐ โช F ๐ , ๐ d ๐ ) + ๐ โ โ๏ธ ๐ = (cid:104) ๐ป ( ๐ [ ๐ โ ] , ๐ A ๐ โช B ๐ โช G ๐ โช F ๐ , ๐ d ๐ ) + ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ , ๐ d ๐ ) (cid:105) +( ๐พ โ ๐ + ) ๐ ( ๐ ) = ๐ โ๏ธ ๐ = ๐ป ( ๐ [ ๐ โ ] , ๐ L ๐ , ๐ d ๐ ) + ๐ โ โ๏ธ ๐ = ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ , ๐ d ๐ ) + ( ๐พ โ ๐ + ) ๐ โฅ (cid:34) ๐ป ( ๐ [ ๐ โ ] , ๐ L ) + ๐ โ๏ธ ๐ = ๐ป ( ๐ [ ๐ โ ] , ๐ L ๐ , ๐ d ๐ ) (cid:35) + ๐ โ โ๏ธ ๐ = ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ , ๐ d ๐ ) + ( ๐พ โ ๐ + ) ๐ ( ๐ ) โฅ (cid:104) ( ๐ โ ) ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ L ๐ ) (cid:105) + ๐ โ โ๏ธ ๐ = ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ , ๐ d ๐ ) + ( ๐พ โ ๐ + ) ๐ ( ๐ ) = ๐ป ( ๐ [ ๐ โ ] , ๐ L ๐ ) + ๐ โ โ๏ธ ๐ = ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ , ๐ d ๐ + ๐ ) + ๐ ( ( ๐พ โ ๐ + ) ๐ โ ) where ( ๐ ) follows from (31) and (32), ( ๐ ) follows from (7) and de๏ฌnition of set A ๐ , B ๐ , and F ๐ , ( ๐ ) follows from Lemma 5 with A = A ๐ โช B ๐ , B = G ๐ and C = I ๐ , ( ๐ ) follows from (27) the submodularity property of entropy, ( ๐ ) follows from the de๏ฌnition of L ๐ , ( ๐ ) follows from Lemma 4 with S ๐ = L ๐ , ๐ = , ๐ = ๐ and (28), ( ๐ ) follows from (30). (cid:3) Now, for ๐ โค ๐ โค ๐พ , consider another sets of users as shown below: Set Users Number Files Requested in Demand d ๐ P ๐ ๐ ๐พ + ๐ + โ ๐ , . . . , ๐ ๐พ + ๐ โ ๐ ๐ โ ๐ , . . . , ๐ ๐ โ Q ๐ ๐ ๐พ + ๐ + โ ๐ , . . . , ๐ ๐พ ๐ โ ๐ ๐ S ๐ ๐ ๐พ โ ๐ + , . . . , ๐ ๐พ + ๐ โ ๐ + ๐ โ ๐ , . . . , ๐ ๐ These sets are also indicated in TABLE V. Let T ๐ = P ๐ โช Q ๐ (33)Note that Q ๐ = S ๐ = ๐ (34) T ๐ + โช { ๐ ๐ + } = T ๐ (35) DRAFT February 10, 20219 T ๐พ โช { ๐ ๐พ } = L ๐ (36) B ๐ โ โช { ๐ ๐ โ } = B ๐ = T ๐ (37)Note that | P ๐ | = ๐ โ . Thus, we have ( ๐ โ ) ๐ + ๐ โฅ ๐ป ( ๐ P ๐ ) + ๐ป ( ๐ d ๐ ) โฅ ๐ป ( ๐ P ๐ , ๐ d ๐ ) (38)The following lemma is easy to obtain: Lemma 7.
The set T ๐ , as de๏ฌned above, satisfy ( ๐พ โ ๐ + ) (cid:20) (cid:0) ๐ ( ๐พ โ ๐ + ) โ (cid:1) ๐ + ( ๐พ โ ๐ + ) ๐ (cid:21) โฅ ๐พ โ๏ธ ๐ = ๐ ๐ป ( ๐ [ ๐ โ ] , ๐ T ๐ , ๐ d ๐ )+ ( ๐พ โ ๐ + ) ( ๐พ โ ๐ ) ๐ Proof.
We have, ( ๐พ โ ๐ + ) (cid:20) (cid:0) ๐ ( ๐พ โ ๐ + ) โ (cid:1) ๐ + ( ๐พ โ ๐ + ) ๐ (cid:21) = ๐พ โ๏ธ ๐ = ๐ (cid:104) ( ( ๐ โ ) ๐ + ๐ ) + ( ๐ โ ๐ ) ๐ + ( ๐ โ ๐ ) ( ๐ + ( ๐ โ ) ๐ ) (cid:105) ( ๐ ) โฅ ๐พ โ๏ธ ๐ = ๐ ๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฐ ๐ป ( ๐ P ๐ , ๐ d ๐ ) + โ๏ธ ๐ โ Q ๐ ๐ป ( ๐ ๐ )+ | Q ๐ | ๐ป ( ๐ d ๐ )+ | Q ๐ | ๐ป ( ๐ S ๐ ) ๏ฃน๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃป ( ๐ ) = ๐พ โ๏ธ ๐ = ๐ ๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฐ ๐ป ( ๐ [ ๐ โ ] , ๐ P ๐ , ๐ d ๐ ) + โ๏ธ ๐ โ Q ๐ ๐ป ( ๐ ๐ )+ | Q ๐ | ๐ป ( ๐ d ๐ )+ | Q ๐ | ๐ป ( ๐ S ๐ ) ๏ฃน๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃป ( ๐ ) โฅ ๐พ โ๏ธ ๐ = ๐ (cid:104) ๐ป ( ๐ [ ๐ โ ] , ๐ P ๐ โช Q ๐ , ๐ d ๐ )+ | Q ๐ | ๐ (cid:105) ( ๐ ) = ๐พ โ๏ธ ๐ = ๐ ๐ป ( ๐ [ ๐ โ ] , ๐ T ๐ , ๐ d ๐ ) + ๐พ โ๏ธ ๐ = ๐ ( ๐ โ ๐ ) ๐ = ๐พ โ๏ธ ๐ = ๐ ๐ป ( ๐ [ ๐ โ ] , ๐ T ๐ , ๐ d ๐ ) + ( ๐พ โ ๐ + ) ( ๐พ โ ๐ ) ๐ where ( ๐ ) follows from (38) and the fact that Q ๐ = ๐ , ( ๐ ) follows from de๏ฌnition of sets P ๐ , Q ๐ and (7), ( ๐ ) follows from Lemma 5 with A = P ๐ , B = Q ๐ and C = S ๐ , ( ๐ ) follows from the de๏ฌnition of T ๐ . February 10, 2021 DRAFT0 (cid:3)
Using the above lemma, we can obtain the following result:
Theorem 2.
For the ( ๐, ๐พ ) cache network, when โค ๐ < (cid:6) ๐พ + (cid:7) , achievable memory rate pairs ( ๐, ๐ ) must satisfy the constraint ๐พ ( ๐ ( ๐พ + ) โ ( ๐ + )) ๐ + ๐พ ( ๐พ + โ ๐ ) ๐ โฅ ๐ ๐พ ( ๐พ โ ๐ + ) โ Proof.
We have, ๐พ ( ๐ ( ๐พ + ) โ ( ๐ + )) ๐ + ๐พ ( ๐พ + โ ๐ ) ๐ = (cid:104) ( ๐ ( ๐พ โ ๐ + ) โ ๐ + ) ๐ + ( ๐ ( ๐พ โ ๐ + ) โ ) ๐ (cid:105) + (cid:20) ( ๐พ โ ๐ + ) (cid:18) (cid:0) ๐ ( ๐พ โ ๐ + ) โ (cid:1) ๐ + ( ๐พ โ ๐ + ) ๐ (cid:19) (cid:21) ( ๐ ) โฅ (cid:34) ๐ป ( ๐ [ ๐ โ ] , ๐ L ๐ ) + ๐ โ โ๏ธ ๐ = ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ , ๐ d ๐ + ๐ ) + ๐ ( ( ๐พ โ ๐ + ) ๐ โ ) (cid:35) + ๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฐ ๐พ โ๏ธ ๐ = ๐ ๐ป ( ๐ [ ๐ โ ] , ๐ T ๐ , ๐ d ๐ ) + ( ๐พ โ ๐ + ) ( ๐พ โ ๐ ) ๐ ๏ฃน๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃป ( ๐ ) = ๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฐ ๐ป ( ๐ [ ๐ โ ] , ๐ T ๐พ , ๐ ๐พ ) + ๐พ โ๏ธ ๐ = ๐ ๐ป ( ๐ [ ๐ โ ] , ๐ T ๐ , ๐ d ๐ ) ๏ฃน๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃป + ๐ โ โ๏ธ ๐ = ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ , ๐ d ( ๐ + ๐ ) )+ ( ๐พ ( ๐พ โ ๐ + ) + ๐ โ ) ๐ ( ๐ ) โฅ (cid:104) ( ๐พ โ ๐ + ) ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ T ๐ ) (cid:105) + ๐ โ โ๏ธ ๐ = ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ , ๐ d ( ๐ + ๐ ) ) + ( ๐พ ( ๐พ โ ๐ + ) + ๐ โ ) ๐ ( ๐ ) = (cid:34) ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ โ , ๐ ๐ โ ) + ๐ โ โ๏ธ ๐ = ๐ป ( ๐ [ ๐ โ ] , ๐ B ๐ , ๐ d ( ๐ + ๐ ) ) (cid:35) + ( ๐พ ( ๐พ โ ๐ + ) โ ๐ ) ๐ ( ๐ ) โฅ (cid:104) ( ๐ โ ) ๐ + ๐ป ( ๐ [ ๐ โ ] , ๐ B ) (cid:105) + ( ๐พ ( ๐พ โ ๐ + ) โ ๐ ) ๐ ( ๐ ) = ๐ป ( ๐ [ ๐ โ ] ) + ( ๐พ ( ๐พ โ ๐ + ) โ ) ๐ โฅ ๐ ๐พ ( ๐พ โ ๐ + ) โ where DRAFT February 10, 20211 ( ๐ ) follows from Lemma 6 and Lemma 7, ( ๐ ) follows from (36), ( ๐ ) follows from (13) with T ๐ = T ๐ , ๐ = ๐ , ๐ = ๐พ and (35), ( ๐ ) follows from (37), ( ๐ ) follows from (13) with T ๐ = B ๐ , ๐ = , ๐ = ๐ โ and (29), ( ๐ ) follows from (27). (cid:3) IV. C
OMPARISON WITH PREVIOUS BOUNDS
In [1], Maddah-Ali and Niesen derived a lower bound on achievable rates using cut setarguments, which was further improved in [20]โ[24]. A comparison between these lower boundsand the new lower bounds in Section III, at cache size ๐ = ๐๐พ ( ๐ โ ) , is given in TABLE VI. It can Lower Bound Case I: (cid:6) ๐พ + (cid:7) โค ๐ โค ๐พ Case II: โค ๐ โค (cid:6) ๐พ + (cid:7) Cut Set bound [1] ๐ โ ๐ ( ๐ โ ) ๐พ ๐ โ ๐ ( ๐ โ ) ๐พ Ghasemi &Ramamoorthy [21] ๐ โ ๐ ( ๐ โ ) ๐พ ๐ โ ๐ ( ๐ โ ) ๐พ Ajaykrishnan et al. [20] ๐ โ ๐ ( ๐ โ ) ๐พ ๐ โ ๐ ( ๐ โ ) ๐พ Wang et al. [23] ๐ โ ๐ ( ๐ โ ) ๐พ ๐ โ ๐ ( ๐ โ ) ๐พ Yu et al. [24] ๐ โ ๐ ( ๐ โ ) ๐พ + ๐พ ( ๐ โ ) (cid:16) ๐ โ ๐พ + ๐พ๐ (cid:17) ๐ โ ๐ ( ๐ โ ) ๐พ Sengupta et al. [22] ๐ โ ๐ ( ๐ โ ) ๐พ + ๐พ ( ๐ โ ) (cid:16) ๐ โ ๐พ + ๐พ๐ (cid:17) ๐ โ ๐ ( ๐ โ ) ๐พ New lower bound ๐ โ ๐ ( ๐ โ ) ๐พ + ๐พ ( ๐ โ ) ๐ โ ๐ ๐พ ( ๐ โ ) + ๐พ ( ๐ โ ) ( ๐พ + โ ๐ ) TABLE VI: Comparison with previous lower bounds for ๐ = ๐๐พ ( ๐ โ ) be noted that the new bounds improve upon the previous ones. For the ( ๐, ๐พ ) cache network,the scheme proposed by Gรณmez-Vilardebรณ in [14] achieves memory rate pairs ( ๐, ๐ ๐บ ) = (cid:18) ๐, ๐พ ๐ โ ๐พ โ ( ๐ โ ) ๐ (cid:19) , for ๐ โ (cid:2) ๐พ , ๐ ( ๐ โ ) ๐พ (cid:3) . From Theorem 1, when (cid:6) ๐พ + (cid:7) โค ๐ โค ๐พ ,we have that all achievable memory rate pairs satisfy the constraint ๐ โฅ ๐พ ๐ โ ๐พ โ ( ๐ โ ) ๐ = ๐ ๐บ ( ๐ ) Thus we have:
February 10, 2021 DRAFT2
Theorem 3.
For the ( ๐, ๐พ ) cache network, when (cid:6) ๐พ + (cid:7) โค ๐ โค ๐พ , the exact rate memorytradeoff is given by ๐ โ ( ๐ ) = ๐พ ๐ โ ๐พ โ ( ๐ โ ) ๐ (39) where ๐ โ (cid:104) ๐พ , ๐๐พ ( ๐ โ ) (cid:105) . Remark 2.
In [14], with the help of the lower bounds derived in [22] and [24], Gรณmez-Vilardebรณshowed that when ๐พ = ๐ and ๐ = ๐ โ , his scheme is optimal. We extend his result to the casewhere (cid:6) ๐พ + (cid:7) โค ๐ โค ๐พ in Theorem 3. V. C
ONCLUSIONS
In this paper we considered the canonical ( ๐, ๐พ ) cache network where ๐ โค ๐พ and ๐ โ (cid:2) , ๐๐พ (cid:3) .We derived a new set of lower bounds on the achievable rate when each ๏ฌle in the server isrequested by at least one user. Using these lower bounds, we showed that when (cid:6) ๐พ + (cid:7) โค ๐ โค ๐พ the scheme proposed in [14] is optimal for ๐ โ (cid:104) ๐พ , ๐๐พ ( ๐ โ ) (cid:105) . For the case โค ๐ โค (cid:6) ๐พ + (cid:7) , thenew lower bound was shown to improve upon the previous lower bounds, but a matching schemeis still not known. The work presented forms another step in the attempt to ๏ฌnd a characterizationof the exact rate memory tradeoff for coded caching and is illustrated in Fig. 4.Cache size ๐ R a t e ๐ ๐พ ๐ ( ๐พ โ ) ๐พ๐๐พ ( ๐ โ ) New Rate Memory TradeoffRate Memory Tradeoff [1], [11]Fig. 4: Rate memory tradeoff for the ( ๐, ๐พ ) cache network when (cid:6) ๐พ + (cid:7) < ๐ โค ๐พ DRAFT February 10, 20213 R EFERENCES [1] M. A. Maddah-Ali and U. Niesen, โFundamental limits of caching,โ
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