On Optimal Load-Memory Tradeoff of Cache-Aided Scalar Linear Function Retrieval
Kai Wan, Hua Sun, Mingyue Ji, Daniela Tuninetti, Giuseppe Caire
aa r X i v : . [ c s . I T ] J a n On Optimal Load-Memory Tradeoff ofCache-Aided Scalar Linear Function Retrieval
Kai Wan,
Member, IEEE,
Hua Sun,
Member, IEEE,
Mingyue Ji,
Member, IEEE,
Daniela Tuninetti,
Senior Member, IEEE, and Giuseppe Caire,
Fellow, IEEE
Abstract
Coded caching has the potential to greatly reduce network traffic by leveraging the cheap andabundant storage available in end-user devices so as to create multicast opportunities in the deliveryphase. In the seminal work by Maddah-Ali and Niesen (MAN), the shared-link coded caching problemwas formulated, where each user demands one file (i.e., single file retrieval). This paper generalizes theMAN problem so as to allow users to request scalar linear functions of the files. This paper proposesa novel coded delivery scheme that, based on MAN uncoded cache placement, is shown to allow forthe decoding of arbitrary scalar linear functions of the files (on arbitrary finite fields). Interestingly,and quite surprisingly, it is shown that the load for cache-aided scalar linear function retrieval dependson the number of linearly independent functions that are demanded, akin to the cache-aided single-fileretrieval problem where the load depends on the number of distinct file requests. The proposed schemeis optimal under the constraint of uncoded cache placement, in terms of worst-case load, and within afactor 2 otherwise. The key idea of this paper can be extended to all scenarios which the original MANscheme has been extended to, including demand-private and/or device-to-device settings.
A short version of this paper was submitted to the IEEE 2020 International Symposium on Information Theory, Los Angeles,California, USA.K. Wan and G. Caire are with the Electrical Engineering and Computer Science Department, Technische Universität Berlin,10587 Berlin, Germany (e-mail: [email protected]; [email protected]). The work of K. Wan and G. Caire was partiallyfunded by the European Research Council under the ERC Advanced Grant N. 789190, CARENET.H. Sun is with the Department of Electrical Engineering, University of North Texas, Denton, TX 76203 (email:[email protected]).M. Ji is with the Electrical and Computer Engineering Department, University of Utah, Salt Lake City, UT 84112, USA(e-mail: [email protected]). The work of M. Ji was supported in part by NSF Awards 1817154 and 1824558.D. Tuninetti is with the Electrical and Computer Engineering Department, University of Illinois at Chicago, Chicago, IL60607, USA (e-mail: [email protected]). The work of D. Tuninetti was supported in part by NSF Award 1910309.
Index Terms
Coded caching; uncoded cache placement; linear scalar function retrieval.
I. I
NTRODUCTION
Information theoretic coded caching was originally proposed by Maddah-Ali and Niesen(MAN) in [1] for the shared-link caching systems containing a server with a library of N equal-length files, which is connected to K users through a noiseless shared-link. Each user can store M files in their local cache. Two phases are included in the MAN caching system: i) cacheplacement phase: content is pushed into each cache without knowledge of future demands; ii)delivery phase: each user demands one file, and according to the cache contents, the serverbroadcasts coded packets to all the users. The objective is to minimize the transmitted load (i.e.,number of transmitted bits normalized by the length of a single file) to satisfy the all the user’demands.The MAN coded caching scheme proposed in [1], uses a combinatorial design in the placementphase (referred to as MAN cache placement), such that in the delivery phase binary multicastmessages (referred to as MAN multicast messages) can simultaneously satisfy the demands ofusers. Under the constraint of uncoded cache placement (i.e., each user directly caches a subset ofthe library bits), the MAN scheme can achieve the minimum worst-case load among all possibledemands when N ≥ K [2]. On the observation that if if there are files demanded multiple times,some MAN multicast messages can be obtained as a binary linear combination of other MANmulticast messages, Yu, Maddah-Ali, and Avestimehr (YMA) proposed an improved deliveryscheme in [3]. The YMA delivery, with MAN placement, achieves the minimum worst-caseload under the constraint of uncoded cache placement. The cost of uncoded cache placementcompared to coded cache placement was proved in [4] to be at most .MAN coded caching [1] has been extended to numerous models, such as Device-to-Device(D2D) caching systems [5], private coded caching [6], [7], coded distributed computing [8],and coded data shuffling [9]–[11] – just to name a few. A common point of these models isthat each user requests one file – some allow for users to request (the equivalent of) multiplefiles [8]–[13] which however does not change much the nature of the problem. In general, linearand multivariate polynomial operations are widely used fundamental primitives for buildingthe complex queries that support on-line analytics and data mining procedures. For example, linear operations are critical in modern deep learning and artificial intelligence algorithms,where matrix-matrix or matrix-vector multiplications are at the core of iterative optimizationalgorithms; while algebraic polynomial queries naturally arise in engineering problems such asthose described by differential equations and distributed machine learning algorithms [14], [15].In those scenarios, it may be too resource-consuming (in terms of bandwidth, or execution time,or storage space) to download locally all the input variables in order to compute the desiredoutput value. Instead, it is desirable to directly download the result of the desired output function.This paper studies the fundamental tradeoff between local storage and network load when usersare interested in retrieving a function of the dataset available at the server.The question we ask in this paper is, compared to the original MAN caching problem, whetherthe optimal worst-case load is increased when the users are allowed to request scalar linearfunctions of the files – the first non-trivial extension of the MAN single-file-retrieval problem,on the way to understand the problem of retrieving general functions. The original MAN shared-link caching problem in [1] where each user request one file is a special case of the formulatedshared-link cache-aided scalar linear function retrieval problem.In addition to the novel problem formulation, our main results are as follows: • Achievable scheme for demanded functions on the binary field.
We start by consideringthe case of scalar linear functions on F . Based on the YMA delivery, which uses an“interference cancellation” idea on F , we propose a novel delivery scheme whose key ideais to deliver only the largest set of linearly independent functions, while the remaining onescan be reconstructed by proper linear combinations of those already retrieved. This can bethought of as the generalization of the idea to only deliver the files requested by the “leaderusers” in the YMA delivery. • Generalization to demanded functions on arbitrary finite field.
We then generalize theproposed scheme to the case where the demands are scalar linear functions on an arbitraryfinite field F q . To the best of our knowledge, even for the originally MAN coded cachingproblem, no caching scheme is known in the literature for arbitrary finite fields. Comparedto the YMA delivery scheme, we use different encoding (based on a finite field interferencealignment idea) and decoding procedures that work on an arbitrary finite field. Interestingly, the achieved load by the proposed scheme only depends on the number oflinearly independent functions that are demanded, akin to the YMA’s cache-aided single-file retrieval scheme where the load depends on the number of distinct file requests. • Optimality.
On observation that the converse bound for the original MAN caching problemin [2], [3] is also a converse in the considered cache-aided function retrieval problem, weprove that the proposed scheme achieves the optimal worst-cast load under the constraint ofuncoded cache placement. Moreover, the achieved worst-case load of the proposed schemeis also proved to be order optimal in general within a factor of .From the results in this paper, we can answer the question we asked at the beginning ofthis paper: the optimal worst-case load under the constraint of uncoded cache placementis not increased when the users are allowed to request scalar linear functions of the files.A. Paper Organization The rest of this paper is organized as follows. Section II formulates the cache-aided functionretrieval problem and introduces some related results in the literature. Section III provides anddiscusses the main results in this paper. Section IV and Section V describe the proposed achiev-able caching schemes on the binary field and on arbitrary finite field, respectively. Section VIconcludes the paper. Some of the proofs are given in the Appendices.
B. Notation Convention
Calligraphic symbols denote sets, bold symbols denote vectors, and sans-serif symbols denotesystem parameters. We use | · | to represent the cardinality of a set or the length of a vector; [ a : b ] := { a, a + 1 , . . . , b } and [ n ] := [1 , , . . . , n ] ; ⊕ represents bit-wise XOR; E [ · ] representsthe expectation value of a random variable; [ a ] + := max { a, } ; a ! = a × ( a − × . . . × represents the factorial of a ; F q represents a finite field with order q ; rank q ( A ) represents therank of matrix A on field F q ; det ( A ) represents the determinant matrix A ; A S , V represents thesub-matrix of A by selecting from A , the rows with indices in S and the columns with indicesin V . we let (cid:0) xy (cid:1) = 0 if x < or y < or x < y . In this paper, for each set of integers S , wesort the elements in S in an increasing order and denote the i th smallest element by S ( i ) , i.e., S (1) < . . . < S ( |S| ) . II. S YSTEM M ODEL AND R ELATED R ESULTS
A. System Model A ( K , N , M , q ) shared-link cache-aided scalar linear function retrieval problem is defined asfollows. A central server has access to a library of N files. The files are denoted as F , . . . , F N . Each file has B independent and uniformly distributed symbols over a finite field F q , for someprime-power q . The central server is connected to K users through an error-free shared-link.Each user is equipped with a cache that can store up to MB symbols, where M ∈ [0 , N ] .The system operates in two phases. Cache Placement Phase.
During the cache placement phase, each user stores informationabout the N files in its local cache without knowledge of future users’ demands, that is, thereexist placement functions φ k , k ∈ [ K ] , such that φ k : [ F q ] BN → [ F q ] BM , (1)We denote the content in the cache of user k ∈ [ K ] by Z k = φ k ( F , . . . , F N ) . Delivery Phase.
During the delivery phase, each user requests one scalar linear function of thefiles. The demand of user k ∈ [ K ] is represented by the row vector y k = ( y k, , . . . , y k, N ) ∈ [ F q ] N ,which means that user k wants to retrieve y k, F + . . . + y k, N F N . We denote the demand matrixof all users by D = [ y ; . . . ; y K ] ∈ [ F q ] KN . (2)Given the demand matrix D , the server broadcasts the message X = ψ ( D , F , . . . , F N ) to eachuser k ∈ [ K ] , where the encoding function ψ is such that ψ : [ F q ] KN × [ F q ] BN → [ F q ] BR , (3)for some non-negative R . Decoding.
Each user k ∈ [ K ] decode its desired function from ( D , Z k , X ) . In other words,there exist decoding functions ξ k , k ∈ [ K ] , such that ξ k : [ F q ] KN × [ F q ] BM × [ F q ] BR → [ F q ] B , (4) ξ k ( D , Z k , X ) = y k, F + . . . + y k, N F N . (5) Objective.
For a given memory size M ∈ [0 , N ] , our objective is to determine the minimumworst-case load among all possible demands, defined as the smallest R such that there existplacement functions φ k , k ∈ [ K ] , encoding function ψ , and decoding functions ξ k , k ∈ [ K ] , satisfying all the above constraints. The optimal load is denoted as R ⋆ .If each user directly copies some symbols of the N files into its cache, the cache placementis said to be uncoded . The minimum worst-case load under the constraint of uncoded cacheplacement is denoted by R ⋆ u . B. Review of the MAN [1] and YMA [3] Coded Caching Schemes
In the following, we review the MAN and YMA coded caching schemes, which are on thebinary field F , for the shared-link caching problem, where each user requests one file. MAN Scheme:
File Split.
Let t ∈ [0 : K ] . Partition each file F i , i ∈ [ N ] , into (cid:0) K t (cid:1) equal-lengthsubfiles denoted as F i = { F i, W : W ⊆ [ K ] , |W| = t } . (6) Placement Phase.
User k ∈ [ K ] caches F i, W , i ∈ [ N ] , if k ∈ W . Hence, each user caches N (cid:0) K − t − (cid:1) subfiles, each of which contains B ( K t ) symbols, which requires M = N t K . (7) Delivery Phase.
User k ∈ [ K ] requests the file with index d k ∈ [ N ] . The server then broadcaststhe following MAN multicast messages : for each
S ⊆ [ K ] where |S| = t + 1 , the server sends W S = ⊕ k ∈S F d k , S\{ k } . (8) Decoding.
The multicast message W S in (8) is useful to each user k ∈ S , since this usercaches all subfiles contained by W S except for the desired subfile F d k , S\{ k } . Considering allmulticast messages, each user can recover all uncached subfiles and thus recover its demandedfile. Load.
The achieved memory-load tradeoff of the MAN scheme is the lower convex envelopof the following points ( M , R ) = N t K , (cid:0) K t +1 (cid:1)(cid:0) K t (cid:1) ! , ∀ t ∈ [0 : K ] . (9) YMA Scheme:
File splitting and cache placement are as for the MAN scheme.
Delivery Phase.
The main idea of the YMA delivery is that, when a file is demanded bymultiple users, some MAN multicast messages in (8) can be obtained as a linear combinationsof others. Thus the load of the MAN scheme in (9) can be further reduced by removing theredundant MAN multicast messages. More precisely, for each demanded file, randomly chooseone user among all users demanding this file and designate it as the “leader user” for this file.Let D := ∪ k ∈ [ K ] { d k } be the set of all distinct files that are demanded, and L be the set of |D| leader users. The server only sends those multicast message W S in (8) that are useful for theleader users, that is, if S ∩ L 6 = ∅ , thus saving (cid:0) K −|D| t +1 (cid:1) transmissions. Decoding.
Clearly, all leaders users can decode their demanded files as per the MAN scheme.The non-leader users appear to miss the multicast message W A for each A ⊆ [ K ] where A∩L = ∅ and |A| = t + 1 . It was proved in [3] that ⊕ F∈ F B W B\F = 0 , (10)where B = A ∪ L , and F B is the family of subsets F ⊆ B , where each file in D is requested byexactly one user in F . The key observation is that in ⊕ F∈ F B W B\F each involved subfile appearsexactly twice (i.e., contained into two MAN multicast messages) , whose contribution on F isthus zero. From (10), we have W A = ⊕ F∈ F B : F6 = L W B\F . (11)In other words, the multicast message W A can be reconstructed by all users from the deliveryphase. Load.
The YMA scheme requires the load of (cid:0) K t +1 (cid:1) − (cid:0) K −|D| t +1 (cid:1)(cid:0) K t (cid:1) , (12)if the set of the demanded files is D . The worst-case load is attained for |D| = min( N , K ) ,thus the achieved memory-load tradeoff of the YMA scheme is the lower convex envelop of thefollowing points ( M , R ) = N t K , (cid:0) K t +1 (cid:1) − (cid:0) K − min( N , K ) t +1 (cid:1)(cid:0) K t (cid:1) ! , ∀ t ∈ [0 : K ] . (13)III. M AIN R ESULTS AND D ISCUSSION
In this section, we summarize the main results in this paper.The proposed caching scheme in Section IV (for q = 2 ) and Section V (for general prime-power q ), achieves the following load. In this paper, A ‘appears’ in a linear combination means that in the linear combination, there exists some term in the linearcombination including A . A linear combination ‘contains’ B means that in the linear combination, the total coefficient of B is not . For example, we say A appears in the linear combination ( A ⊕ B ) ⊕ ( A ⊕ C ) , but the linear combination does notcontain A . Theorem 1 (Achievability) . For the ( K , N , M , q ) shared-link cache-aided scalar linear functionretrieval problem, the YMA load in (13) is an achievable worst-case load. More precisely, forcache size M = N t K , with t ∈ [0 : K ] , and for demand matrix D , the load R ( D ) := (cid:0) K t +1 (cid:1) − (cid:0) K − rank q ( D ) t +1 (cid:1)(cid:0) K t (cid:1) (14) is achievable. The worst-case load is attained by rank q ( D ) = min( N , K ) . (cid:3) Remark 1 (Dependance on the rank of the demand matrix) . The load in (14) is a generalizationof the load in (12) achieved by the YMA scheme. More precisely, if each user k ∈ [ K ] requestsone file (i.e., y k ∈ [0 : 1] N with a unit norm), rank q ( D ) is exactly the number of demanded files,and thus the proposed scheme achieves the load in (12) as the YMA scheme. Interestingly, theload of the proposed scheme only depends on the rank of the demand matrix of all users, insteadof on the specifically demanded functions. (cid:3) Remark 2 (High-level ideas to derive the load in Theorem 1) . We partition the “symbolpositions" set [ B ] as follows [ B ] = {I W : W ⊆ [ K ] , |W| = t } such that |I W | = B / (cid:18) K t (cid:19) . (15)Then, with a Matlab-inspired notation, we let F i, W = F i ( I W ) , ∀W ⊆ [ K ] : |W| = t, ∀ i ∈ [ N ] , (16)representing the set of symbols of F i whose position is in I W . As in the MAN placement, user k ∈ [ K ] caches F i, W if k ∈ W . By doing so, any scalar linear function is naturally partitionedinto “blocks” as follows y k, F + . . . + y k, N F N = { y k, F ( I W ) + . . . + y k, N F N ( I W ) | {z } := B k, W is the W -th block of the k -th demanded function : W ⊆ [ K ] , |W| = t } . (17)Some blocks of the demanded functions can thus be computed based on the cache contentavailable at each user while the remaining ones need to be delivered by the server. With thisspecific file split (and corresponding MAN cache placement), we operate the MAN deliveryscheme over the blocks instead of over the subfiles ; more precisely, instead of (8) we transmit W S = X k ∈S α S ,k B k, S\{ k } , ∀S ⊆ [ K ] : |S| = t + 1 , (18) for some α S ,k ∈ F q \ { } and where B k, W was defined in (17). Clearly, this scheme achievesthe same load as in (9) (and works on any finite field and any α S ,k ∈ F q \ { } ).The questions is, whether with (18) we can do something similar to the YMA delivery scheme.More specifically,1) what is a suitable definition of the leader user set L ;2) what is a suitable choice of α S ,k ’s in (18); and3) assuming we only send the multicast messages in (18) that are useful for the leader users(i.e., W S where S ⊆ [ K ] , |S| = t + 1 , and S ∩ L 6 = ∅ ), what is the counterpart of (11);here for each A ⊆ [ K ] where |A| = t + 1 and A ∩ L = ∅ , we seek W A = X S⊆ [ K ]: |S| = t +1 , S∩L6 = ∅ β A , S W S . (19)The novelty of our scheme lays in the answers to these questions as follows:1) we first choose rank q ( D ) leaders (the set of leader users is denoted by L ), where thedemand matrix of the leaders is full-rank.2) When q = 2 (i.e., on the binary field), lets α S ,k = 1 . When q is a prime-power, the proposedscheme in Section V separates the demanded blocks by the leaders and non-leaders in W S in (18) as X k ∈S α S ,k B k, S\{ k } = X k ∈S∩L α S ,k B k , S\{ k } + X k ∈S\L α S ,k B k , S\{ k } ; (20)we then alternate the coefficients of the desired blocks by the leaders (i.e., users in S ∩ L )between +1 and − , i.e., the coefficient of the desired block of the first leader is +1 , thecoefficient of the desired block of the second leader is − , the coefficient of the desiredblock of the third leader is +1 , etc; similarly, we alternate the coefficients of the desiredblocks by the non-leaders (i.e., users in S \ L ) between +1 and − .
3) With the above encoding scheme, we can compute the decoding coefficients (as β A , S in (19)) such that (19) holds for each A ⊆ [ K ] where |A| = t + 1 and A ∩ L = ∅ . In otherwords, each user can recover all multicast messages W S where S ⊆ [ K ] and |S| = t + 1 ,and thus it can recover its desired function. This type of code was originally proposed in [16] for the private function retrieval problem, where there is a memory-lessuser aiming to retrieval a scalar linear function of the files in the library from multiple servers (each server can access to thewhole library), while preserving the demand of this user from each server. (cid:3) Since the setting where each user demands one file is a special case of the considered cache-aided scalar linear function retrieval problem, the converse bounds in [2]–[4] for the originalshared-link coded caching problem is also a converse in our considered problem, thus we have:
Theorem 2 (Optimality) . For the ( K , N , M , q ) shared-link cache-aided scalar linear functionretrieval problem, under the constraint of uncoded cache placement, the optimal worst-caseload-memory tradeoff is the lower convex envelop of ( M , R ⋆ u ) = N t K , (cid:0) K t +1 (cid:1) − (cid:0) K − min { K , N } t +1 (cid:1)(cid:0) K t (cid:1) ! , ∀ t ∈ [0 : K ] . (21) Moreover, the achieved worst-case load in (21) is optimal within a factor of in general. (cid:3) Remark 3 (Extensions) . We discuss three extensions of the proposed caching scheme in Theo-rem 1 in the following.
Optimal average load under uncoded and symmetric cache placement.
We define uncodedand symmetric cache placement as follows, which is a generalization of file split in (15)-(16).We partition the “symbol positions" set [ B ] as [ B ] = {I W : W ⊆ [ K ] } , (22)and let F i, W = F i ( I W ) as in (16). Each user k ∈ [ K ] caches F i, W if k ∈ W .Hence, in the delivery phase, user k needs to recover B k, W (defined in (17)) where W ⊆ [ K ] \ { k } . By directly using [4, Lemma 2] in the caching converse bound under uncoded cacheplacement in [2], [3], we can prove that the proposed caching scheme in Theorem 1 achievesthe minimum average load over uniform demand distribution under the constraint of uncodedand symmetric cache placement cross files. Corollary 1. [Optimal average load] For the ( K , N , M , q ) shared-link cache-aided scalar linearfunction retrieval problem, under the constraint of uncoded and symmetric cache placement, theminimum average load over uniform demand distribution is the lower convex envelop of ( M , R ) = N t K , E D " (cid:0) K t +1 (cid:1) − (cid:0) K − rank q ( D ) t +1 (cid:1)(cid:0) K t (cid:1) , ∀ t ∈ [0 : K ] . (23)Notice that an uncoded and asymmetric cache placement can be treated as a special case ofthe inter-file coded cache placement in the originally MAN caching problem. It is one of the on-going works to derive the converse bound under the constraints of uncoded cache placementfor the considered cache-aided function retrieval problem. Device-to-Device (D2D) cache-aided scalar linear function retrieval.
Coded caching wasoriginally used in Device-to-Device networks in [5], where in the delivery phase each userbroadcasts packets as functions of its cached content and the users’ demands, to all other users.The authors in [17] extended the YMA scheme to D2D networks by dividing the D2D networksinto K shared-link networks, and used the YMA scheme in each shared-link network. Hence,when users request scalar linear functions, we can use the same method as in [17] to dividethe D2D networks into K shared-link networks, and then use the proposed caching scheme inTheorem 1 in each shared-link network. Corollary 2. [D2D cache-aided scalar linear function retrieval] For the ( K , N , M , q ) D2D cache-aided scalar linear function retrieval problem, the minimum worse-case load is upper boundedby the lower convex envelop of ( M , R ) = N t K , max D (cid:0) K − t (cid:1) − K P k ∈ [ K ] (cid:0) K − − rank q ( D [ K ] \{ k } ) t (cid:1)(cid:0) K − t − (cid:1) ! , ∀ t ∈ [ K ] . (24) Cache-aided private scalar linear function retrieval.
For the successful decoding of the pro-posed scheme in Theorem 1, users need to be aware of the demands of other users, which is notprivate. To preserve the privacy of the demand of each user against other users, we can generatevirtual users as in [6], such that each of all possible demanded functions (the total number ofpossible demanded functions is N ′ := q N − q − ) is demanded exactly K times. Thus there are totally N ′ K real or virtual users in the system. Then the proposed scheme in Theorem 1 can be usedto satisfy the demands of all real or virtual users. Since each user cannot distinguish other realusers from virtual users, the resulting scheme does not leak any information on the demands ofreal users. Corollary 3. [Cache-aided private scalar linear function retrieval] For the ( K , N , M , q ) D2Dcache-aided scalar linear function retrieval problem, the minimum load is upper bounded by thelower convex envelop of ( M , R ) = t N ′ K N , (cid:0) N ′ K t +1 (cid:1) − (cid:0) N ′ K − N t +1 (cid:1)(cid:0) N ′ K t (cid:1) ! , ∀ t ∈ [ N ′ K ] . (25) (cid:3) IV. A
CHIEVABLE S CHEME IN T HEOREM FOR q = 2 In the following, we describe the proposed scheme when the demands are scalar linearfunctions on F . We start with the following example. A. Example
Consider the ( K , N , M , q ) = (6 , , , shared-link cache-aided scalar linear function retrievalproblem, where t = KM / N = 2 . In the cache placement, each file is partitioned into (cid:0) K t (cid:1) = 15 equal-length subfiles. We use the file split in (15)-(16), resulting in the demand split in (17).In the delivery phase, we assume thatuser demands F ;user demands F ;user demands F ;user demands F ⊕ F ;user demands F ⊕ F ;user demands F ⊕ F ⊕ F ;i.e., the demand matrix is D = . (26)On the observation that rank ( D ) = 3 , we choose users as leaders, where the demand matrixof these leaders is also full-rank. Here, we choose L = [3] . Encoding.
For each set
S ⊆ [ K ] where |S| = t + 1 = 3 , we generate a multicast message with α S ,k = 1 in (18). Hence, we have W { , , } = F , { , } ⊕ F , { , } ⊕ F , { , } ; (27a) W { , , } = F , { , } ⊕ F , { , } ⊕ ( F , { , } ⊕ F , { , } ); (27b) W { , , } = F , { , } ⊕ F , { , } ⊕ ( F , { , } ⊕ F , { , } ); (27c) W { , , } = F , { , } ⊕ F , { , } ⊕ ( F , { , } ⊕ F , { , } ⊕ F , { , } ); (27d) W { , , } = F , { , } ⊕ F , { , } ⊕ ( F , { , } ⊕ F , { , } ); (27e) W { , , } = F , { , } ⊕ F , { , } ⊕ ( F , { , } ⊕ F , { , } ); (27f) W { , , } = F , { , } ⊕ F , { , } ⊕ ( F , { , } ⊕ F , { , } ⊕ F , { , } ); (27g) W { , , } = F , { , } ⊕ ( F , { , } ⊕ F , { , } ) ⊕ ( F , { , } ⊕ F , { , } ); (27h) W { , , } = F , { , } ⊕ ( F , { , } ⊕ F , { , } ) ⊕ ( F , { , } ⊕ F , { , } ⊕ F , { , } ); (27i) W { , , } = F , { , } ⊕ ( F , { , } ⊕ F , { , } ) ⊕ ( F , { , } ⊕ F , { , } ⊕ F , { , } ); (27j) W { , , } = F , { , } ⊕ F , { , } ⊕ ( F , { , } ⊕ F , { , } ); (27k) W { , , } = F , { , } ⊕ F , { , } ⊕ ( F , { , } ⊕ F , { , } ); (27l) W { , , } = F , { , } ⊕ F , { , } ⊕ ( F , { , } ⊕ F , { , } ⊕ F , { , } ); (27m) W { , , } = F , { , } ⊕ ( F , { , } ⊕ F , { , } ) ⊕ ( F , { , } ⊕ F , { , } ); (27n) W { , , } = F , { , } ⊕ ( F , { , } ⊕ F , { , } ) ⊕ ( F , { , } ⊕ F , { , } ⊕ F , { , } ); (27o) W { , , } = F , { , } ⊕ ( F , { , } ⊕ F , { , } ) ⊕ ( F , { , } ⊕ F , { , } ⊕ F , { , } ); (27p) W { , , } = F , { , } ⊕ ( F , { , } ⊕ F , { , } ) ⊕ ( F , { , } ⊕ F , { , } ); (27q) W { , , } = F , { , } ⊕ ( F , { , } ⊕ F , { , } ) ⊕ ( F , { , } ⊕ F , { , } ⊕ F , { , } ); (27r) W { , , } = F , { , } ⊕ ( F , { , } ⊕ F , { , } ) ⊕ ( F , { , } ⊕ F , { , } ⊕ F , { , } ); (27s) W { , , } = ( F , { , } ⊕ F , { , } ) ⊕ ( F , { , } ⊕ F , { , } ) ⊕ ( F , { , } ⊕ F , { , } ⊕ F , { , } ) . (27t) Delivery.
The server broadcasts W S for each S ⊆ [ K ] where |S| = t + 1 = 3 and S ∩ L 6 = ∅ .In other words, the server broadcasts all the multicast messages in (27) except for W { , , } . Decoding.
We show that the untransmitted multicast message W { , , } can be reconstructed bythe transmitted multicast messages. For each set of users B ⊆ [ K ] , we define V B as the family ofsubsets V ⊆ B , where |V| = |L| and rank ( D V ) = |L| . It can be seen that V B is the generalizationof F B defined in the YMA scheme described in Section II-B. When B = L ∪ { , , } = [6] ,we have V [6] = n { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } o . (28) From the above definition, we focus on the following sum of multicast messages ⊕ V∈ V [6] W [6] \V = 0 , (29)where (29) is because on the LHS of (29), among all subfiles F i, W where i ∈ [3] , W ⊆ [6] ,and |W| = 2 , the coefficient of each of F , { , } , F , { , } , F , { , } , F , { , } , F , { , } , F , { , } is , F , { , } appears times and other subfiles appear times. Hence, the sum is equivalent to on F . Notice that in the YMA delivery scheme, the coefficient of each subfile appearing in thesum ⊕ F∈ F B W B\F is .We can write (29) as W { , , } = ⊕ V∈ V [6] : V6 = L W [6] \V . (30)In other words, the untransmitted multicast message W { , , } can be reconstructed by the trans-mitted multicast messages. Thus each user can recover all the multicast messages in (27), andthen recover its desired function. Performance.
In total we transmit (cid:0) K t +1 (cid:1) − (cid:0) K − rank ( D ) t +1 (cid:1) = (cid:0) (cid:1) − (cid:0) (cid:1) = 19 multicast messages,each of which contains B bits. Hence, the transmitted load is , which coincides with theoptimal worst-case load in Theorem 2. B. General Description
We use the file split in (15)-(16), resulting in the demand split in (17).In the delivery phase, the demand matrix D is revealed where each element in D is either or . Among the K users we first choose rank ( D ) leaders (assume the set of leaders is L = {L (1) , . . . , L ( |L| ) } ), where |L| = rank ( D L ) = rank ( D ) . (31) Encoding.
We focus on each set
S ⊆ [ K ] where |S| = t + 1 , and generate the multicastmessage in (18) with α S ,k = 1 . Delivery.
The server broadcasts W S for each S ⊆ [ K ] where |S| = t + 1 and S ∩ L 6 = ∅ . Decoding.
For each set of users
B ⊆ [ K ] , recall that V B is the family of subsets V ⊆ B , where |V| = |L| and rank ( D V ) = |L| . We now consider each set
A ⊆ [ K ] where |A| = t + 1 and A ∩ L = ∅ , and focus on the binary sum ⊕ V∈ V B W B\V , (32) where B = L ∪ A . A subfile F i, W appears in the sum (32) if and only if W ⊆ B and thereexists some user k ∈ B \ W such that rank ( D B\ ( W∪{ k } ) ) = |L| (i.e., D B\ ( W∪{ k } ) is full-rank)and y k,i = 0 . We then provide the following Lemma, proved in Appendix A. Lemma 1. If F i, W appears in the sum (32) , the number of multicast messages in the sum whichcontains F i, W is even. (cid:3) From Lemma 1, it can be seen that each subfile in the sum (32) appears an even number oftimes, and thus the coefficient of this subfile in the sum is , which leads to W A = ⊕ V∈ V B : V6 = L W B\V . (33)In other words, W A can be reconstructed by the transmitted multicast messages.As a result, each user k can recover each multicast message W S where S ⊆ [ K ] and |S| = t +1 ,and thus it can decode its desired function. Performance.
In total, we transmit (cid:0) K t +1 (cid:1) − (cid:0) K − rank ( D ) t +1 (cid:1) multicast messages, each of whichcontains B ( K t ) bits. Hence, the transmitted load is (cid:0) K t +1 (cid:1) − (cid:0) K − rank ( D ) t +1 (cid:1)(cid:0) K t (cid:1) . (34)For the worst-case demands where rank ( D ) is full-rank, we have rank ( D ) = min { K , N } ,and we achieve the worst-case load in (21).V. A CHIEVABLE S CHEME IN T HEOREM FOR GENERAL PRIME - POWER q In the following, we generalize the proposed caching scheme in Section IV to the case wherethe demands are scalar linear functions on arbitrary finite field F q . All the operations in theproposed scheme are on F q . We again start with an example. A. Example
Consider the ( K , N , M , q ) = (5 , , / , q ) shared-link cache-aided scalar linear function re-trieval problem, where q is a prime-power. In this case, we have t = KM / N = 1 . Hence, in thecache placement, each file is partitioned into (cid:0) K t (cid:1) = 5 equal-length subfiles. We use the file splitin (15)-(16), resulting in the demand split in (17). Notice that in the YMA scheme for the original MAN caching problem, each subfile in (10) is contained by two multicastmessages in (10). Hence, Lemma 1 is also a generalization of [3, Lemma 1] for the YMA scheme. In the delivery phase, we assume thatuser demands F ;user demands F ;user demands F ;user demands y , F + y , F + y , F ;user demands y , F + y , F + y , F ;i.e., the demand matrix is D = y , y , y , y , y , y , ∈ [ F q ] × . (35)We choose the set of leaders L = [3] , since rank q ( D [3] ) = 3 .Each user k ∈ [ K ] should recover each block B k, W = y k, F , W + y k, F , W + y k, F , W in thedelivery phase, where W ∈ [5] \ { k } and |W| = 1 . Encoding.
For each set
S ⊆ [ K ] where |S| = t + 1 = 2 , recall that the multicast messages aregiven in (18) and we separate it as W S = X k ∈S∩L α S ,k B k , S\{ k } + X k ∈S\L α S ,k B k , S\{ k } . (36)We first alternate the coefficients (either or − ) of the desired blocks of the leaders in S ,and then alternate the coefficients (either or − ) of the desired blocks of the non-leadersin S . For example, if S = { , } , we have W { , } = F , { } − F , { } ; if S = { , } , we have W { , } = F , { } +( y , F , { } + y , F , { } + y , F , { } ) ; if S = { , } , we have W { , } = ( y , F , { } + y , F , { } + y , F , { } ) − ( y , F , { } + y , F , { } + y , F , { } ) . With this, we can list all the multicastmessages as W { , } = F , { } − F , { } ; (37a) W { , } = F , { } − F , { } ; (37b) W { , } = F , { } + ( y , F , { } + y , F , { } + y , F , { } ); (37c) W { , } = F , { } + ( y , F , { } + y , F , { } + y , F , { } ); (37d) W { , } = F , { } − F , { } ; (37e) W { , } = F , { } + ( y , F , { } + y , F , { } + y , F , { } ); (37f) W { , } = F , { } + ( y , F , { } + y , F , { } + y , F , { } ); (37g) W { , } = F , { } + ( y , F , { } + y , F , { } + y , F , { } ); (37h) W { , } = F , { } + ( y , F , { } + y , F , { } + y , F , { } ); (37i) W { , } = ( y , F , { } + y , F , { } + y , F , { } ) − ( y , F , { } + y , F , { } + y , F , { } ) . (37j) Delivery.
The server broadcasts W S for each S ⊆ [ K ] where |S| = t + 1 = 2 and S ∩ L 6 = ∅ .In other words, the server broadcasts all the multicast messages in (37) except for W { , } . Decoding.
We first show the untransmitted multicast message W { , } can be reconstructed bythe transmitted multicast messages. More precisely, we aim to choose the decoding coefficients β { , } , S ∈ F q for each S ⊆ [ K ] where |S| = t + 1 and S ∩ L 6 = ∅ , such that W { , } = X S⊆ [ K ]: |S| = t +1 , S∩L6 = ∅ β { , } , S W S . (38)Since on the RHS of (38) F , { } only appears in W { , } and on the LHS of (38) the coefficientof F , { } is − y , , in order to have the same coefficient of F , { } on both sides of (38), we let β { , } , { , } = − y , = − det ([ y , ]) . (39)Similarly, we let β { , } , { , } = − y , = − det ([ y , ]) , (40)such that the coefficients of F , { } on both sides of (38) are the same; let β { , } , { , } = − y , = − det ([ y , ]) , (41)such that the coefficients of F , { } on both sides of (38) are the same; let β { , } , { , } = y , = det ([ y , ]) , (42)such that the coefficients of F , { } on both sides of (38) are the same; let β { , } , { , } = y , = det ([ y , ]) , (43)such that the coefficients of F , { } on both sides of (38) are the same; let β { , } , { , } = y , = det ([ y , ]) , (44) such that the coefficients of F , { } on both sides of (38) are the same.Next we focus F , { } , which appears in W { , } and W { , } . Since β { , } , { , } = − y , and β { , } , { , } = y , , the coefficient of F , { } on the RHS of (38) is y , β { , } , { , } + y , β { , } , { , } = 0 . (45)Similarly, the coefficient of F , { } on the RHS of (38), which appears in W { , } and W { , } , is . The coefficient of F , { } on the RHS of (38), which appears in W { , } and W { , } , is ,Now we focus on F , { } , which appears in W { , } , W { , } , and W { , } . Since β { , } , { , } = − y , and β { , } , { , } = y , , in order to let the coefficient of F , { } on the RHS of (38) be , we let β { , } , { , } = y , y , − y , y , = det ([ y , , y , ; y , , y , ]) . (46)In addition, F , { } appears in W { , } , W { , } , and W { , } . The coefficient of F , { } on the RHSof (38) is − β { , } , { , } + y , β { , } , { , } + y , β { , } , { , } = 0 . (47)Similarly, we let β { , } , { , } = y , y , − y , y , = det ([ y , , y , ; y , , y , ]) , (48)such that the coefficients of F , { } and F , { } on the RHS of (38) are . We let β { , } , { , } = y , y , − y , y , = det ([ y , , y , ; y , , y , ]) , (49)such that the coefficients of F , { } and F , { } on the RHS of (38) are .With the above choice of decoding coefficients, on the RHS of (38), the coefficients of allthe subfiles which is not contained by W { , } are . In addition, the coefficients of each subfilecontained by W { , } are the same on both sides of (38). Thus we prove (38). In conclusion, eachuser can recover all multicast messages in (37), and then recover its demanded function. Performance.
In total we transmit (cid:0) K t +1 (cid:1) − (cid:0) K − rank q ( D ) t +1 (cid:1) = (cid:0) (cid:1) − (cid:0) (cid:1) = 9 multicast messages,each of which contains B symbols. Hence, the transmitted load is , which coincides with theoptimal worst-case load in Theorem 2. B. General Description
We use the file split in (15)-(16), resulting in the demand split in (17).In the delivery phase, after the demand matrix D is revealed, among the K users we firstchoose rank q ( D ) leaders (assume the set of leaders is L = {L (1) , . . . , L ( |L| ) } ), where |L| = rank q ( D L ) = rank q ( D ) . (50)For each i ∈ [ |L| ] , we also define that the leader index of leader L ( i ) is i .From (50), we can represent the demands of non-leaders by the linear combinations of thedemands of leaders. More precisely, we define F ′ i := y L ( i ) , F + . . . + y L ( i ) , N F N , ∀ i ∈ [ |L| ] , (51)and represent the demand of each user k ∈ [ K ] by y k, F + . . . + y k, N F N = x k, F ′ + . . . + x k, |L| F ′|L| . (52)Clearly, for each leader L ( i ) where i ∈ [ |L| ] , x L ( i ) is an |L| -dimension unit vector where the i th element is . The transformed demand matrix D ′ is defined as follows, D ′ = [ x , , . . . , x , |L| ; . . . ; x K , , . . . , x K , |L| ] . (53)In addition, for each i ∈ [ |L| ] and each W ⊆ [ K ] where |W| = t , we define F ′ i, W := y L ( i ) , F , W + . . . + y L ( i ) , N F N , W , (54)refer F ′ i, W to as a transformed subfile , and refer B ′ k, W = x k, F ′ , W + . . . + x k, |L| F ′|L| , W to as a transformed block . Encoding.
For each
S ⊆ [ K ] , we denote the set of leaders in S by L S := S ∩ L , (55)and the set of non-leaders in S by N S := S \ L . (56)We also denote the leader indices of leaders in S byInd S := { i ∈ [ |L| ] : L ( i ) ∈ S} , (57) For example, if L = { , , } and S = { , , } , we have L S = { , } , N S = { } , andInd S = { , } .Now we focus on each set S ⊆ [ K ] where |S| = t + 1 , and generate the multicast message W S = X i ∈ [ |L S | ] ( − i − B ′L S ( i ) , S\{L S ( i ) } + X j ∈ [ |N S | ] ( − j − B ′N S ( j ) , S\{N S ( j ) } . (58)The construction of W S can be explained as follows. • The coefficient of each transformed block in W S is either or − . • We divide the transformed blocks in W S into two groups, demanded by leaders and non-leaders, respectively. We alternate the sign (i.e., the coefficient or − ) of each transformedblock demanded by leaders, and alternate the sign of each transformed block demanded bynon-leaders, respectively. We then sum the resulting summations of these two groups. • For each i ∈ [ |L S | , by the construction in (51), we have B ′L S ( i ) , S\{L S ( i ) } = F ′ Ind S ( i ) , S\{L S ( i ) } . Delivery.
The server broadcasts W S for each S ⊆ [ K ] where |S| = t + 1 and S ∩ L 6 = ∅ . Decoding.
We consider each set
A ⊆ [ K ] where |A| = t + 1 and A ∩ L = ∅ .We define that the non-leader index of non-leader A ( i ) is i , where i ∈ [ t + 1] . For each S ⊆ A ∪ L , recall that Ind S defined in (57) represents the leader indices of leaders in S andthat N S defined in (56) represents the set of non-leaders in S . By definition, we have N S ⊆ A .In addition, with a slight abuse of notation we denote the non-leader indices of non-leaders in A \ S by Ind S = { i ∈ [ t + 1] : A ( i ) / ∈ S} . (59)For example, if A = { , , } and S = { , , } , we have Ind S = { , } .For any set X and any number y , we define Tot ( X ) as the sum of the elements in X , i.e.,Tot ( X ) := X i ∈|X | X ( i ); . (60)For example, if X = { , , , } , we have Tot ( X ) = 1 + 3 + 4 + 5 = 13 .Recall that A S , V represents the sub-matrix of A by selecting from A , the rows with indicesin S and the columns with indices in V . It will be proved in Appendix B that W A = X S⊆A∪L : |S| = t +1 , S6 = A β A , S W S , (61) β A , S = ( − Tot ( Ind S ) det ( D ′A\S , Ind S ) . (62)In other words, each user k ∈ [ K ] can recover all messages W S where S ⊆ [ K ] and |S| = t + 1 .For each desired transformed block B ′ k, W , where W ⊆ ([ K ] \ { k } ) and |W| = t , user k canrecover it in W W∪{ k } , because it knows all the other transformed blocks in W W∪{ k } . Hence, user k can recover x k, F ′ + . . . + x k, |L| F ′|L| , which is identical to its demand. Performance.
In total, we transmit (cid:0) K t +1 (cid:1) − (cid:0) K − rank q ( D ) t +1 (cid:1) multicast messages, each of whichcontains B ( K t ) symbols. Hence, the transmitted load is (cid:0) K t +1 (cid:1) − (cid:0) K − rank q ( D ) t +1 (cid:1)(cid:0) K t (cid:1) . (63)For the worst-case demands where rank q ( D ) is full-rank, we have rank q ( D ) = min { K , N } ,and we achieve the worst-case load in (21).VI. C ONCLUSIONS
In this paper, we introduced a novel problem, cache-aided function retrieval, which is ageneralization of the classic coded caching problem and allows users to request scalar linearfunctions of files. We proposed a novel scheme for the demands functions on arbitrary finitefield. The proposed scheme was proved to be optimal under the constraint of uncoded cacheplacement. In addition, for any demand, the achieved load only depends on the rank of thedemand matrix. From the results in this paper, we showed that compared to the original MANcaching problem, the optimal worst-case load of coded caching under the constraint of uncodedcache placement, is not increased when users request scalar linear functions.Further works include the extension of the proposed caching scheme to the case where thedemanded functions are non-linear or vectorial, and finding novel caching schemes for the cache-aided function retrieval problem with coded cache placement.A
PPENDIX AP ROOF OF L EMMA k ∈ B \ W satisfyingthe following constraints is even,1) Constraint 1: rank ( D B\ ( W∪{ k } ) ) = |L| ;2) Constraint 2: y k,i = 0 . We assume that user k satisfies the above constraints. Hence, D B\ ( W∪{ k } ) is full-rank, and y k ,i = 0 . We let Y = {Y (1) , . . . , Y ( |L| ) } = B \ ( W ∪ { k } ) .In the following, we operate a linear space transformation. More precisely, we let G j = y Y ( j ) [ F ; . . . ; F N ] , ∀ j ∈ [ |L| ] . (64)From (64), we can re-write the demand of each user Y ( j ) as G j = y ′ j [ G ; . . . ; G |L| ] , where y ′ j is the |L| -dimension unit vector whose j th element is . The transformed demandmatrix of the users in Y is D ′Y = [ y ′ ; . . . ; y ′|L| ] , which is an identity matrix.In addition, we can also re-write the demand of user k as y ′ [ G ; . . . ; G |L| ] , where y ′ is an |L| -dimension vector on F . Notice that if the p th element in y ′ is and G p contains F i , F i appears one time in y ′ [ G ; . . . ; G |L| ] . Since y k ,i = 0 , it can be seen that y ′ [ G ; . . . ; G |L| ] contains F i . Hence, the number of p ∈ [ |L| ] where the p th element in y ′ is and G p contains F i , is odd. For each of such p , if we replace the p th row of D ′Y by y ′ , the resulting matrix isstill full-rank, because the p th element in y ′ is . Since the resulting matrix is full-rank, it canbe seen that D B\ ( W∪{Y ( p ) } ) is also full-rank. In addition, since G p contains F i , we can see that y Y ( p ) ,i = 0 . Hence, user Y ( p ) also satisfies the two constraints. Moreover, for any s ∈ [ |L| ] , ifthe s th element in y ′ is not , user Y ( s ) does not satisfy Constraint 1; if G s does not contain F s , user Y ( s ) does not satisfy Constraint 2.As a result, besides user k , the number of users in B \ W satisfying the two constraints isodd. In conclusion, by taking user k into consideration, the number of users in B \ W satisfyingthe two constraints is even. Thus Lemma 1 is proved.A
PPENDIX BP ROOF OF (61)We focus on one set of non-leaders
A ⊆ [ K ] where |A| = t + 1 and A ∩ L = ∅ . For any positive integer n , Perm ( n ) represents the set of all permutations of [ n ] . For any set X and any number y , we define Card ( X , y ) as the number of elements in X which is smallerthan y , i.e., Card ( X , y ) := |{ i ∈ X : i < y }| . (65)For example, if X = { , , , } and y = 4 , we have Card ( X , y ) = |{ , }| = 2 .Our objective is to prove W A = X S⊆A∪L : |S| = t +1 , S6 = A β A , S W S , (66) β A , S = X u =( u ,...,u | Ind S| ) ∈ Perm ( | Ind S | ) ( − Tot ( Ind S )+ P i ∈ [ | Ind S| ] Card ([ | Ind S | ] \{ u ,...,u i } ,u i ) Y i ∈ [ | Ind S | ] x A (cid:0) Ind S ( u i ) (cid:1) , Ind S ( i ) , (67)where (67) is obtained from expand the determinant in (62). Let us go back to the illustratedexample in Section V-A, where we choose L = [3] . When A = { , } and S = { , } , fromthe definition in (57) we have Ind S = [2] and from the definition in (59) we have Ind S = [2] .In addition, Perm ( | Ind S | ) = Perm (2) { (1 , , (2 , } . Hence, when u = ( u , u ) = (1 , , in (67)we have the term ( − Tot ([2])+
Card ([2] \{ } , Card ([2] \{ , } , x , x , = ( − Tot ([2])+
Card ([2] \{ } , Card ([2] \{ , } , y , y , = y , y , , (68)where (68) is because in the example we have F i = F ′ i for each i ∈ [ N ] , and thus x k = y k foreach k ∈ [ K ] . Similarly, when u = (2 , , in (67) we have the term ( − Tot ([2])+
Card ([2] \{ } , Card ([2] \{ , } , x , x , = ( − Tot ([2])+
Card ([2] \{ } , Card ([2] \{ , } , y , y , = − y , y , . (69)Hence, in (67) we have β { , } , { , } = y , y , − y , y , , which coincides (46) in the illustratedexample.By the definition of W S in (58), it is obvious to check that in (66), there only exist thetransformed subfiles F i, W where i ∈ [ N ] , |W| ⊆ ( A ∪ L ) , and |W| = t . Now we divide such transformed subiles into hierarchies, where we say a transformed subfile F ′ i, W appearing in (66)is in Hierarchy h ∈ [0 : t ] , if |W ∩ L| = h . In addition, on the LHS of (66), only transformedsubfiles in Hierarchy exist.We consider the following three cases,1) Case 1: F ′ i, W is in Hierarchy . In Appendix B-A, we will prove that the coefficient of F ′ i, W on the RHS of (66) is equal to the coefficient of F ′ i, W on the LHS of (66).2) Case 2: F ′ i, W is in Hierarchy h > and L ( i ) ∈ W . In Appendix B-B, we will prove thatthe coefficient of F ′ i, W on the RHS of (66) is .3) Case 3: F ′ i, W is in Hierarchy h > and L ( i ) / ∈ W . In Appendix B-C, we will prove thatthe coefficient of F ′ i, W on the RHS of (66) is .Hence, after proving the above three cases, (66) can be directly derived.In the illustrated example in Section V-A, since F i = F ′ i for each i ∈ [ N ] , it can be seen that F ′ i, W = F i, W for each i ∈ [ N ] , |W| ⊆ [ K ] , and |W| = t . For each subfile F i, W , it is in one of thefollowing three cases,1) Case 1: F i, W is in Hierarchy . In this case, we have the subfiles F i, { } , F i, { } for i ∈ [3] .2) Case 2: F i, W is in Hierarchy and L ( i ) ∈ W . In this case, we have the subfiles F i, { i } for i ∈ [3] .3) Case 3: F i, W is in Hierarchy and L ( i ) / ∈ W . In this case, we have the subfiles F i, { j } for i ∈ [3] and j ∈ [3] \ { i } . A. Case 1 If F ′ i, W is in Hierarchy , we have W ⊆ A . Since |A| − |W| = 1 , we assume that {A ( k ) } = A \ W . On the LHS of (66), F ′ i, W appears in W A , where from (58) we have W A = X j ∈ [ t +1] ( − j − ( x A ( j ) , F ′ , S\{A ( j ) } + . . . + x A ( j ) , |L| F ′|L| , S\{A ( j ) } ) . (70)Hence, the coefficient of F ′ i, W in W A is ( − k − x A ( k ) ,i .Let us then focus on the RHS of (66). F ′ i, W appears in W W∪{L ( i ) } . Since L ( i ) is the only leaderin W ∪ {L ( i ) } (i.e., Ind W∪{L ( i ) } = { i } ), the coefficient of F ′ i, W in W W∪{L ( i ) } is ( − − = 1 . Inaddition, by computing Ind W∪{L ( i ) } = { k } , we have β A , W∪{L ( i ) } = ( − k +0 x A ( k ) ,i = ( − k +1 x A ( k ) ,i = ( − k − x A ( k ) ,i . (71) Hence, the coefficient of F ′ i, W on the RHS of (66) (i.e., in β A , W∪{L ( i ) } W W∪{L ( i ) } ) is ( − k − x A ( k ) ,i × − k − x A ( k ) ,i , (72)which is the same as the coefficient of F ′ i, W on the LHS of (66). B. Case 2
Now we focus on one transformed subfile F ′ i, W in Hierarchy h > where L ( i ) ∈ W . Bydefinition, we have |W ∩ L| = h . On the RHS of (66), since L ( i ) ∈ W , F ′ i, W only appears in W W∪{A ( k ) } , where k ∈ Ind W . We define thatthe (cid:16) Ind − W ( k ) (cid:17) th smallest element in Ind W is k . (73)We focus on one k ∈ Ind W . A ( k ) is the k th element in A , and in A \ W there are Ind − W ( k ) − elements smaller than A ( k ) . Hence, in N W∪{A ( k ) } there are k − − (cid:16) Ind − W ( k ) − (cid:17) = k − Ind − W ( k ) elements smaller than A ( k ) . So from (58), it can be seen that the coefficient of F ′ i, W in W W∪{A ( k ) } is ( − k − Ind − W ( k ) x A ( k ) ,i . (74)In addition, we have β A , W∪{A ( k ) } = X u =( u ,...,u | Ind
W∪{A ( k ) }| ) ∈ Perm ( | Ind
W∪{A ( k ) } | ) ( − Tot ( Ind
W∪{A ( k ) } )+ P i ∈ [ | Ind
W∪{A ( k ) }| ] Card ([ | Ind
W∪{A ( k ) } | ] \{ u ,...,u i } ,u i ) Y i ∈ [ | Ind
W∪{A ( k ) } | ] x A (cid:0) Ind
W∪{A ( k ) } ( u i ) (cid:1) , Ind
W∪{A ( k ) } ( i ) (75a) = X u =( u ,...,u | Ind
W | ) ∈ Perm ( | Ind W | ) ( − ( Tot ( Ind W ) − k ) + P i ∈ [ | Ind
W | ] Card ([ | Ind W | ] \{ u ,...,u i } ,u i ) Y i ∈ [ | Ind W | ] x A (cid:0) Ind
W∪{A ( k ) } ( u i ) (cid:1) , Ind W ( i ) . (75b)From (74) and (75b), the coefficient of F ′ i, W in β A , W∪{A ( k ) } W W∪{A ( k ) } is ( − k − Ind − W ( k ) x A ( k ) ,i β A , W∪{A ( k ) } .In the following, we will prove X k ∈ Ind W ( − k − Ind − W ( k ) x A ( k ) ,i β A , W∪{A ( k ) } = 0 , (76)such that the coefficient of F ′ i, W on the RHS of (66) is . Let us focus on one k ∈ Ind W and one permutation u = ( u , . . . , u | Ind W | ) ∈ Perm ( | Ind W | ) .The term in (76) caused by k and u is ( − k − Ind − W ( k ) x A ( k ) ,i ( − ( Tot ( Ind W ) − k ) + P i ∈ [ | Ind
W | ] Card ([ | Ind W | ] \{ u ,...,u i } ,u i ) Y i ∈ [ | Ind W | ] x A (cid:0) Ind
W∪{A ( k ) } ( u i ) (cid:1) , Ind W ( i ) (77a) = ( − − Ind − W ( k )+1+ Tot ( Ind W )+ P i ∈ [ | Ind
W | ] Card ([ | Ind W | ] \{ u ,...,u i } ,u i ) x A ( k ) ,i Y i ∈ [ | Ind W | ] x A (cid:0) Ind
W∪{A ( k ) } ( u i ) (cid:1) , Ind W ( i ) . (77b)Notice that in the product x A ( k ) ,i Y i ∈ [ | Ind W | ] x A (cid:0) Ind
W∪{A ( k ) } ( u i ) (cid:1) , Ind W ( i ) , (78)there is one term whose second subscript is i ′ for each i ′ ∈ Ind W \ { i } , and there are two termswhose second subscript is i . We define thatthe (cid:0) Ind − W ( i ) (cid:1) th smallest element in Ind W is i . (79)Hence, the two terms in (78) whose second subscript is i are x A ( k ) ,i and x A ( k ′ ) ,i , where k ′ := Ind
W∪{A ( k ) } ( u Ind − W ( i ) ) .In addition, the combination k ′ and u ′ = ( u ′ , . . . , u ′| Ind W | ) also causes a term in (76) whichhas the product x A ( k ′ ) ,i Y i ∈ [ | Ind W | ] x A (cid:0) Ind
W∪{A ( k ′ ) } ( u ′ i ) (cid:1) , Ind W ( i ) . (80)The products in (78) and (80) are identical if u ′ is as follows, • for j ∈ [ | Ind W | ] \ { Ind − W ( i ) } , we have A (cid:0) Ind
W∪{A ( k ′ ) } ( u ′ j ) (cid:1) = A (cid:0) Ind
W∪{A ( k ) } ( u j ) (cid:1) ; (81)such that x A (cid:0) Ind
W∪{A ( k ′ ) } ( u ′ j ) (cid:1) , Ind W ( j ) = x A (cid:0) Ind
W∪{A ( k ) } ( u j ) (cid:1) , Ind W ( j ) ; (82) • for j = Ind − W ( i ) , we have A (cid:0) Ind
W∪{A ( k ′ ) } ( u ′ j ) (cid:1) = A ( k ); (83) such that x A (cid:0) Ind
W∪{A ( k ′ ) } ( u ′ j ) (cid:1) , Ind W ( j ) = x A ( k ) ,i . (84)It is obvious to check that there does not exist any other combination of k ′′ ∈ Ind W and u ′′ ∈ Perm ( | Ind W | ) , causing a term on the LHS of (76) which has the product in (78), exceptthe two above combinations.In Appendix C, we will prove that ( − − Ind − W ( k )+1+ Tot ( Ind W )+ P i ∈ [ | Ind W| ] Card ([ | Ind W | ] \{ u ,...,u i } ,u i ) +( − − Ind − W ( k ′ )+1+ Tot ( Ind W )+ P i ′ ∈ [ | Ind
W | ] Card ([ | Ind W | ] \{ u ′ ,...,u ′ i ′ } ,u ′ i ′ ) = 0 , (85)such that the coefficient of the product in (78) on the LHS of (76) is . In other words, for eachcombination of k and u on the LHS of (76), there is exactly one term caused by the combinationof k ′ and u ′ , such that the sum of these two caused terms is . Thus (76) is proved. C. Case 3
Lastly we focus on one transformed subfile F ′ i, W in Hierarchy h > where L ( i ) / ∈ W .By definition, we have |W ∩ L| = h . On the RHS of (66), since L ( i ) / ∈ W , F ′ i, W appears in W W∪{L ( i ) } . In addition, F ′ i, W also appears in W W∪{A ( k ) } , where k ∈ Ind W .Let us first focus on W W∪{L ( i ) } . Recall that the (cid:16) Ind − W∪{L ( i ) } ( i ) (cid:17) th element in Ind W∪{L ( i ) } is i . From (58), it can be seen that the coefficient of F ′ i, W in W W∪{L ( i ) } is ( − Ind − W∪{L ( i ) } ( i ) − . (86)In addition, we have β A , W∪{L ( i ) } = X u =( u ,...,u | Ind
W∪{L ( i ) }| ) ∈ Perm ( | Ind
W∪{L ( i ) } | ) ( − Tot ( Ind
W∪{L ( i ) } )+ P i ∈ [ | Ind
W∪{L ( i ) }| ] Card ([ | Ind
W∪{L ( i ) } | ] \{ u ,...,u i } ,u i ) Y i ∈ [ | Ind
W∪{L ( i ) } | ] x A (cid:0) Ind
W∪{L ( i ) } ( u i ) (cid:1) , Ind
W∪{L ( i ) } ( i ) (87a) = X u =( u ,...,u | Ind
W | +1 ) ∈ Perm ( | Ind W | +1) ( − Tot ( Ind W )+ P i ∈ [ | Ind W| +1] Card ([ | Ind W | +1] \{ u ,...,u i } ,u i ) Y i ∈ [ | Ind W | +1] x A (cid:0) Ind W ( u i ) (cid:1) , Ind
W∪{L ( i ) } ( i ) (87b) Let us then focus on W W∪{A ( k ) } , where k ∈ Ind W . It was proved in (74) that the coefficientof F ′ i, W in W W∪{A ( k ) } is ( − k − Ind − W ( k ) x A ( k ) ,i . (88)In addition, it was proved in (75b) that β A , W∪{A ( k ) } = X u =( u ,...,u | Ind W| ) ∈ Perm ( | Ind W | ) ( − ( Tot ( Ind W ) − k ) + P i ∈ [ | Ind
W | ] Card ([ | Ind W | ] \{ u ,...,u i } ,u i ) Y i ∈ [ | Ind W | ] x A (cid:0) Ind
W∪{A ( k ) } ( u i ) (cid:1) , Ind W ( i ) . (89)In the following, we will prove ( − Ind − W∪{L ( i ) } ( i ) − β A , W∪{L ( i ) } + X k ∈ Ind W ( − k − Ind − W ( k ) x A ( k ) ,i β A , W∪{A ( k ) } = 0 , (90)such that the coefficient of F ′ i, W on the RHS of (66) is . Notice that there are t − | Ind W | non-leaders in W . Since there are totally t + 1 non-leaders in A , we have | Ind W | = t + 1 − ( t − | Ind W | ) = | Ind W | + 1 . (91)Let us focus on one permutation u = ( u , . . . , u | Ind W | +1 ) ∈ Perm ( | Ind W | + 1) in β A , W∪{L ( i ) } .The term in (90) caused by u is ( − Ind − W∪{L ( i ) } ( i ) − ( − Tot ( Ind W )+ P i ∈ [ | Ind W| +1] Card ([ | Ind W | +1] \{ u ,...,u i } ,u i ) Y i ∈ [ | Ind W | +1] x A (cid:0) Ind W ( u i ) (cid:1) , Ind
W∪{L ( i ) } ( i ) (92a) = ( − Ind − W∪{L ( i ) } ( i )+ Tot ( Ind W )+ P i ∈ [ | Ind
W | +1]
Card ([ | Ind W | +1] \{ u ,...,u i } ,u i ) Y i ∈ [ | Ind W | +1] x A (cid:0) Ind W ( u i ) (cid:1) , Ind
W∪{L ( i ) } ( i ) . (92b)We can rewrite the product term in (92b) as follows (recall again that the (cid:16) Ind − W∪{L ( i ) } ( i ) (cid:17) th element in Ind W∪{L ( i ) } is i ), Y i ∈ [ | Ind W | +1] x A (cid:0) Ind W ( u i ) (cid:1) , Ind
W∪{L ( i ) } ( i ) = x A (cid:0) Ind W ( u Ind − W∪{L ( i ) } ( i ) ) (cid:1) ,i Y i ∈ [ | Ind W | +1] \{ Ind − W∪{L ( i ) } ( i ) } x A (cid:0) Ind W ( u i ) (cid:1) , Ind
W∪{L ( i ) } ( i ) (93a) = x A ( e k ) ,i Y i ∈ [ | Ind W | +1] \{ Ind − W∪{L ( i ) } ( i ) } x A (cid:0) Ind W ( u i ) (cid:1) , Ind
W∪{L ( i ) } ( i ) , (93b) where we define e k := Ind W ( u Ind − W∪{L ( i ) } ( i ) ) .Hence, on the LHS of (90), besides ( − Ind − W∪{L ( i ) } ( i ) − β A , W∪{L ( i ) } , only the caused term bythe combination of e k and e u = ( e u , . . . , e u | Ind W | ) has the product in (93b), where e u = ( g ( u ) , . . . , g ( u Ind − W∪{L ( i ) } ( i ) − ) , g ( u Ind − W∪{L ( i ) } ( i )+1 ) , . . . , g ( u | Ind W | +1 )) , (94a) g ( u j ) := u j , if u j < u Ind − W∪{L ( i ) } ( i ) u j − if u j > u Ind − W∪{L ( i ) } ( i ) . (94b)In Appendix D, we will prove that ( − Ind − W∪{L ( i ) } ( i )+ Tot ( Ind W )+ P i ∈ [ | Ind W| +1] Card ([ | Ind W | +1] \{ u ,...,u i } ,u i ) +( − − Ind − W ( e k )+1+ Tot ( Ind W )+ P e i ∈ [ | Ind W| ] Card ([ | Ind W | ] \{ e u ,..., e u e i } , e u e i ) = 0 , (95)such that the coefficient of the product in (93b) on the LHS of (90) is . Hence, for eachpermutation u ∈ Perm ( | Ind W | + 1) , there is exactly one term caused by the combination of e k ∈ Ind W and e u ∈ Perm ( | Ind W | ) , such that the sum of these two caused terms are .In addition, on the LHS of (90), there are ( | Ind W | +1)! terms in ( − Ind − W∪{L ( i ) } ( i ) − β A , W∪{L ( i ) } .Recall that in (91), we proved | Ind W | = | Ind W | + 1 . Hence, on the LHS of (90), there are | Ind W | ! × ( | Ind W | + 1) = ( | Ind W | + 1)! terms in P k ∈ Ind W ( − k − Ind − W ( k ) x A ( k ) ,i β A , W∪{A ( k ) } . Inconclusion, we prove (90). A PPENDIX CP ROOF OF (85)To prove (85), it is equivalent to prove ( − − Ind − W ( k ) − Ind − W ( k ′ )+ P i ∈ [ | Ind W| ] Card ([ | Ind W | ] \{ u ,...,u i } ,u i )+ P i ′ ∈ [ | Ind
W | ] Card ([ | Ind W | ] \{ u ′ ,...,u ′ i ′ } ,u ′ i ′ ) = − . (96)Let us focus on P i ∈ [ | Ind W | ] Card ([ | Ind W | ] \ { u , . . . , u i } , u i ) . By the definition of the functionCard ( · ) in (65), we have X i ∈ [ | Ind W | ] Card ([ | Ind W | ] \ { u , . . . , u i } , u i )= X i ∈ [ | Ind W | ]: i = Ind − W ( i ) Card (cid:0) ([ | Ind W | ] \ { u Ind − W ( i ) } ) \ { u , . . . , u Ind − W ( i ) − , u Ind − W ( i )+1 , . . . , u i } , u i (cid:1) + |{ i ∈ [ Ind − W ( i ) −
1] : u Ind − W ( i ) < u i }| + |{ i ∈ [ Ind − W ( i ) + 1 : | Ind W | ] : u i < u Ind − W ( i ) }| (97a) = X i ∈ [ | Ind W | ]: i = Ind − W ( i ) Card (cid:0) ([ | Ind W | ] \ { u Ind − W ( i ) } ) \ { u , . . . , u Ind − W ( i ) − , u Ind − W ( i )+1 , . . . , u i } , u i (cid:1) + ( Ind − W ( i ) − − |{ i ∈ [ Ind − W ( i ) −
1] : u i < u Ind − W ( i ) }| ) + |{ i ∈ [ Ind − W ( i ) + 1 : | Ind W | ] : u i < u Ind − W ( i ) }| (97b) = X i ∈ [ | Ind W | ]: i = Ind − W ( i ) Card (cid:0) ([ | Ind W | ] \ { u Ind − W ( i ) } ) \ { u , . . . , u Ind − W ( i ) − , u Ind − W ( i )+1 , . . . , u i } , u i (cid:1) + ( Ind − W ( i ) − − |{ i ∈ [ Ind − W ( i ) −
1] : u i < u Ind − W ( i ) }| )+ Card ([ | Ind W | ] , u Ind − W ( i ) ) − |{ i ∈ [ Ind − W ( i ) −
1] : u i < u Ind − W ( i ) }| . (97c)Similarly, for P i ′ ∈ [ | Ind W | ] Card ([ | Ind W | ] \ { u ′ , . . . , u ′ i ′ } , u ′ i ′ ) , from the same derivation as (97c), wehave X i ′ ∈ [ | Ind W | ] Card ([ | Ind W | ] \ { u ′ , . . . , u ′ i ′ } , u ′ i ′ )= X i ′ ∈ [ | Ind W | ]: i ′ = Ind − W ( i ) Card (cid:0) ([ | Ind W | ] \ { u ′ Ind − W ( i ) } ) \ { u ′ , . . . , u ′ Ind − W ( i ) − , u ′ Ind − W ( i )+1 , . . . , u ′ i } , u ′ i (cid:1) + ( Ind − W ( i ) − − |{ i ′ ∈ [ Ind − W ( i ) −
1] : u ′ i ′ < u ′ Ind − W ( i ) }| )+ Card ([ | Ind W | ] , u ′ Ind − W ( i ) ) − |{ i ′ ∈ [ Ind − W ( i ) −
1] : u ′ i ′ < u ′ Ind − W ( i ) }| . (98)In addition, from (81), it can be seen that X i ∈ [ | Ind W | ]: i = Ind − W ( i ) Card (cid:0) ([ | Ind W | ] \ { u Ind − W ( i ) } ) \ { u , . . . , u Ind − W ( i ) − , u Ind − W ( i )+1 , . . . , u i } , u i (cid:1) = X i ′ ∈ [ | Ind W | ]: i ′ = Ind − W ( i ) Card (cid:0) ([ | Ind W | ] \ { u ′ Ind − W ( i ) } ) \ { u ′ , . . . , u ′ Ind − W ( i ) − , u ′ Ind − W ( i )+1 , . . . , u ′ i } , u ′ i (cid:1) . (99)From (97c)-(99), and the fact that ( − a = ( − for any integer a , we have ( − P i ∈ [ | Ind W| ] Card ([ | Ind W | ] \{ u ,...,u i } ,u i )+ P i ′ ∈ [ | Ind W| ] Card ([ | Ind W | ] \{ u ′ ,...,u ′ i ′ } ,u ′ i ′ ) = ( − Card ([ | Ind W | ] ,u Ind − W ( i ) )+ Card ([ | Ind W | ] ,u ′ Ind − W ( i ) ) . (100)Without loss of generality, we assume k < k ′ . Recall that Ind W∪{A ( k ) } ( u Ind − W ( i ) ) = k ′ . By thedefinition in (73), we can see that in Ind W , there are Ind − W ( k ′ ) − elements smaller than k ′ . Bythe assumption, k < k ′ . Hence, in Ind W∪{A ( k ) } , there are Ind − W ( k ′ ) − elements smaller than k ′ .In other words, u Ind − W ( i ) = Ind − W ( k ′ ) − , (101) which leads to Card ( { u , . . . , u | Ind W | } , u Ind − W ( i ) ) = Ind − W ( k ′ ) − . (102)Similarly, recall that Ind W∪{A ( k ′ ) } ( u ′ Ind − W ( i ) ) = k . In Ind W , there are Ind − W ( k ) − elements smallerthan k . By the assumption, k < k ′ . Hence, in Ind W∪{A ( k ′ ) } , there are Ind − W ( k ) − elements smallerthan k . In other words, u ′ Ind − W ( i ) = Ind − W ( k ) , (103)which leads to Card ( { u ′ , . . . , u ′| Ind W | } , u ′ Ind − W ( i ) ) = Ind − W ( k ) − . (104)We take (102) and (104) into (100) to obtain, ( − P i ∈ [ | Ind W| ] Card ([ | Ind W | ] \{ u ,...,u i } ,u i )+ P i ′ ∈ [ | Ind W| ] Card ([ | Ind W | ] \{ u ′ ,...,u ′ i ′ } ,u ′ i ′ ) = ( − Ind − W ( k ′ ) − Ind − W ( k ) − (105a) = ( − Ind − W ( k ′ )+ Ind − W ( k )+1 . (105b)Finally, we have ( − − Ind − W ( k ) − Ind − W ( k ′ )+ P i ∈ [ | Ind W| ] Card ([ | Ind W | ] \{ u ,...,u i } ,u i )+ P i ′ ∈ [ | Ind W| ] Card ([ | Ind W | ] \{ u ′ ,...,u ′ i ′ } ,u ′ i ′ ) = ( − − Ind − W ( k ) − Ind − W ( k ′ )+ Ind − W ( k ′ )+ Ind − W ( k )+1 (106a) = − , (106b)which proves (96). A PPENDIX DP ROOF OF (95)To prove (95), it is equivalent to prove ( − Ind − W∪{L ( i ) } ( i ) − Ind − W ( e k )+ P i ∈ [ | Ind
W | +1]
Card ([ | Ind W | +1] \{ u ,...,u i } ,u i )+ P e i ∈ [ | Ind W| ] Card ([ | Ind W | ] \{ e u ,..., e u e i } , e u e i ) = − . (107)Let us focus on P i ∈ [ | Ind W | +1] Card ([ | Ind W | + 1] \ { u , . . . , u i } , u i ) . By the definition of the func-tion Card ( · ) in (65), we have X i ∈ [ | Ind W | +1] Card ([ | Ind W | + 1] \ { u , . . . , u i } , u i ) = X i ∈ [ | Ind W | +1]: i = Ind − W∪{L ( i ) } ( i ) Card (cid:0) ([ | Ind W | + 1] \ { u Ind − W∪{L ( i ) } ( i ) } ) \ { u , . . . , u Ind − W∪{L ( i ) } ( i ) − , u Ind − W∪{L ( i ) } ( i )+1 , . . . , u i } , u i (cid:1) + |{ i ∈ [ Ind − W∪{L ( i ) } ( i ) −
1] : u Ind − W∪{L ( i ) } ( i ) < u i }| + |{ i ∈ [ Ind − W∪{L ( i ) } ( i ) + 1 : | Ind W | + 1] : u i < u | Ind W | +1 }| (108a) = X i ∈ [ | Ind W | +1]: i = Ind − W∪{L ( i ) } ( i ) Card (cid:0) ([ | Ind W | + 1] \ { u Ind − W∪{L ( i ) } ( i ) } ) \ { u , . . . , u Ind − W∪{L ( i ) } ( i ) − , u Ind − W∪{L ( i ) } ( i )+1 , . . . , u i } , u i (cid:1) + Ind − W∪{L ( i ) } ( i ) − − |{ i ∈ [ Ind − W∪{L ( i ) } ( i ) −
1] : u i < u Ind − W∪{L ( i ) } ( i ) }| + Card ([ | Ind W | + 1] , u Ind − W∪{L ( i ) } ( i ) ) − |{ i ∈ [ Ind − W∪{L ( i ) } ( i ) −
1] : u i < u Ind − W∪{L ( i ) } ( i ) }| (108b)From the construction of e u in (94a), we have X i ∈ [ | Ind W | +1]: i = Ind − W∪{L ( i ) } ( i ) Card (cid:0) ([ | Ind W | + 1] \ { u Ind − W∪{L ( i ) } ( i ) } ) \ { u , . . . , u Ind − W∪{L ( i ) } ( i ) − , u Ind − W∪{L ( i ) } ( i )+1 , . . . , u i } , u i (cid:1) = X e i ∈ [ | Ind W | ] Card ([ | Ind W | ] \ { e u , . . . , e u e i } , e u e i ) . (109)From (108b) and (109), and the fact that ( − a = ( − for any integer a , we have ( − Ind − W∪{L ( i ) } ( i ) − Ind − W ( e k )+ P i ∈ [ | Ind W| +1] Card ([ | Ind W | +1] \{ u ,...,u i } ,u i )+ P e i ∈ [ | Ind W| ] Card ([ | Ind W | ] \{ e u ,..., e u e i } , e u e i ) = ( − − Ind − W ( e k )+ Card ([ | Ind W | +1] ,u Ind − W∪{L ( i ) } ( i ) ) (110a) = − , (110b)where (110b) comes from that e k := Ind W ( u Ind − W∪{L ( i ) } ( i ) ) , and thus u Ind − W∪{L ( i ) } ( i ) = Ind − W ( e k ) .R EFERENCES [1] M. A. Maddah-Ali and U. Niesen, “Fundamental limits of caching,”
IEEE Trans. Infor. Theory , vol. 60, no. 5, pp. 2856–2867, May 2014.[2] K. Wan, D. Tuninetti, and P. Piantanida, “On the optimality of uncoded cache placement,” in IEEE Infor. Theory Workshop ,Sep. 2016.[3] Q. Yu, M. A. Maddah-Ali, and S. Avestimehr, “The exact rate-memory tradeoff for caching with uncoded prefetching,”
IEEE Trans. Infor. Theory , vol. 64, pp. 1281 – 1296, Feb. 2018.[4] Q. Yu, M. A. Maddah-Ali, and A. S. Avestimehr, “Characterizing the rate-memory tradeoff in cache networks within afactor of 2,”
IEEE Trans. Infor. Theory , vol. 65, no. 1, pp. 647–663, Jan. 2019. [5] M. Ji, G. Caire, and A. Molisch, “Fundamental limits of caching in wireless d2d networks,” IEEE Trans. Inf. Theory ,vol. 62, no. 1, pp. 849–869, 2016.[6] K. Wan and G. Caire, “On coded caching with private demands,” arXiv:1908.10821 , Aug. 2019.[7] K. Wan, H. Sun, M. Ji, D. Tuninetti, and G. Caire, “Device-to-device private caching with trusted server,” arXiv:1909.12748 ,Sep. 2019.[8] S. Li, M. A. Maddah-Ali, Q. Yu, and A. S. Avestimehr, “A fundamental tradeoff between computation and communicationin distributed computing,”
IEEE Trans. Inf. Theory , vol. 64, no. 1, pp. 109–128, Jan. 2018.[9] M. A. Attia and R. Tandon, “Near optimal coded data shuffling for distributed learning,” vol. 65, no. 11, pp. 7325–7349,Nov. 2019.[10] A. Elmahdy and S. Mohajer, “On the fundamental limits of coded data shuffling for distributed learning systems,” arXiv:1807.04255, a preliminary version is in IEEE Int. Symp. Inf. Theory 2018 , Jul. 2018.[11] K. Wan, D. Tuninetti, M. Ji, G. Caire, and P. Piantanida, “Fundamental limits of decentralized data shuffling,” arXiv:1807.00056 , Jun. 2018.[12] M. Ji, A. Tulino, J. Llorca, and G. Caire, “Caching-aided coded multicasting with multiple random requests,” in Proc.IEEE Inf. Theory Workshop (ITW) , May. 2015.[13] A. Sengupta and R. Tandon, “Improved approximation of storage-rate tradeoff for caching with multiple demands,”
IEEETrans. Commun. , vol. 65, no. 5, pp. 1940–1955, May. 2017.[14] J. So, B. Guler, A. S. Avestimehr, and P. Mohassel, “Codedprivateml: A fast and privacy-preserving framework fordistributed machine learning,” arXiv:1902.00641 , Feb. 2019.[15] M. Aliasgari, O. Simeone, and J. Kliewer, “Private and secure distributed matrix multiplication with flexible communicationload,” arXiv:1909.00407 , Sep. 2019.[16] H. Sun and S. A. Jafar, “The capacity of private computation,”
IEEE Trans. Inf. Theory , vol. 65, no. 5, pp. 3880–3897,Jun. 2019.[17] C. Yapar, K. Wan, R. F. Schaefer, and G. Caire, “On the optimality of d2d coded caching with uncoded cache placementand one-shot delivery,” in IEEE Int. Symp. Inf. Theoryin IEEE Int. Symp. Inf. Theory