A General Coded Caching Scheme for Scalar Linear Function Retrieval
AA General Coded Caching Schemefor Scalar Linear Function Retrieval
Yinbin Ma and Daniela Tuninetti,
University of Illinois Chicago, Chicago, IL 60607, USAEmail:{yma52, danielat}@uic.edu
Abstract —Coded caching aims to minimize the network’s peak-time communication load by leveraging the information pre-stored in the local caches at the users. The original single fileretrieval setting by Maddah-Ali and Niesen has been recentlyextended to general Scalar Linear Function Retrieval (SLFR)by Wan et al. , who proposed a linear scheme that surprisinglyachieves the same optimal load (under the constraint of uncodedcache placement) as in single file retrieval. This paper’s goalis to characterize the conditions under which a general SLFRlinear scheme is optimal and gain practical insights into why thespecific choices made by Wan et al. work. This paper shows thatthe optimal decoding coefficients are necessarily the product oftwo terms, one only involving the encoding coefficients and theother only the demands. In addition, the relationships among theencoding coefficients are shown to be captured by the cycles ofcertain graphs. Thus, a general linear scheme for SLFR can befound by solving a spanning tree problem.
I. I
NTRODUCTION
Coded caching, originally introduced by Maddah-Ali andNiesen (MAN) in [1], has been the focus of much researchefforts recently as it predicts, for networks with a serverdelivering a single file to each cache-aided user, that it ispossible to achieve a communication load that does not scalewith the number of users. Yu et al. in [2] improved on thedelivery phase of the MAN scheme by removing the MANmulticast message transmissions that are redundant when afile is requested by multiple users, and thus showed thatthe converse bound under the constraint of uncoded cacheplacement by Wan et al. in [3] is tight. Wan et al. in [4]recently extended the MAN setup so as to allow users torequest general scalar linear combinations of the files storedat the server. Despite the fact that the number of possibledemands increases exponentially in the number of files, [4]surprisingly showed that the optimal communication load isthe same as for the single file retrieval setting, at least underuncoded cache placement.The scheme proposed in [4] is linear. As in [2], theserver selects of set of leaders (whose demand vectors area linearly independent spanning set of the set of all possibledemands) and creates multicast messages by performing linearcombinations of demanded subfiles that were not cached;the coefficients for such linear combinations are referred toas encoding coefficients and can be optimized. As in [2],multicast messages that would only be useful for non-leaderusers are not sent and have to be locally reconstructed as linearcombinations of sent multicast messages; the coefficients for such linear combinations are referred to as decoding coeffi-cients and must guarantee that each user correctly decodes itsdemanded linear combination of files. The choice of encodingand decoding coefficients in [4] is rather non trivial and nota simple extension of [2], which actually fails to guaranteesuccessful decoding on finite fields of characteristics strictlylarger than two. The encoding coefficients chosen in [4],inspired by private function retrieval in [5], all have unitmodulo but alternate in sign among leaders and among non-leaders. Such a choice works (with corresponding decodingcoefficients given, up to a sign, by determinants of certainmatrices derived from the demand matrix) but the reason whyit is so could not be explained.This paper aims to gain insights into why the choicesin [4] work by analyzing the most general linear scheme(i.e., general encoding and decoding coefficients). Our maincontribution is to show that the optimal decoding coefficientsare necessarily the product of two terms, one only involvingthe encoding coefficients and the other only the determinantsof certain matrices derived from the demands. In addition, wecharacterize the relationships the encoding coefficients needto satisfy in order to guaranteed successful decoding as cycleson certain graphs.
Thus, we show that a general SLFR linearscheme can be found by solving a spanning tree problem.
The rest of the paper is organized as follow. Section IIintroduces the cache-aided scalar linear function retrieval(SLFR) problem and summarizes related work. Section IIIpresents our main result, which is proved in Section IV.Section V concludes the paper. Some examples can be foundin Appendix.In this paper we use the following notation convention. • Calligraphic symbols denote sets, bold symbols vectors,and sans-serif symbols system parameters. • | · | is the cardinality of a set or the length of a vector. • det( M ) is the determinant of the matrix M . • {E} is the indicator function of the event E . • M [ Q , S ] is the submatrix of M obtained by selecting therows indexed by Q and the columns indexed by S . • For an integer b , we let [ b ] := { , . . . , b } . • For a ground set G and an integer t , we let Ω t G := {T ⊆G : |T | = t } . Moreover, S \ Q := { k : k ∈ S , k / ∈ Q} . • Ind S ,k returns the position of the element k ∈ S ,where the element of the integer set S are consideredin increasing order. For example, Ind { , } , = 1 and Ind { , } , = 2 . By convention Ind S ,k = 0 if k (cid:54)∈ S . a r X i v : . [ c s . I T ] F e b I. P
ROBLEM F ORMULATION AND K NOWN R ESULTS
A. Problem Formulation A ( K , N , q , M , R ) SLFR problem has one central server thathas access to a library of N files (denoted as F , . . . , F N ),each of B independent and uniformly distributed symbolsover the finite field F q , for some prime-power q . The servercommunicates through an error-free shared link at load R to K users, where each has a local memory to store up to M files. The worst-case load R (cid:63) ( M ) , M ∈ [0 , N ] , for the SLFRproblem is defined as in [4], which is not explicitly writtenhere for same of space (as it also appears next). B. Known Results
It was shown in [4] that requesting arbitrary scalar linearfunctions of the files from the server does not incur any loadpenalty compared to the case of requesting a single file, that is,the lower convex envelope of the following points is achievable ( M , R ) = (cid:32) 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) (cid:33) , ∀ t ∈ [0 : K ] . (1)Moreover, the tradeoff in (1) is optimal among all schemeswith uncoded cache placement [2], [4] and to within a factortwo otherwise [6]. The scheme in [4] is as follows. a) Cache Placement: Partition the position indices as [ B ] = (cid:26) I T : I T ⊆ [ B ] , T ∈ Ω t [ K ] , |I T | = B / (cid:18) K t (cid:19)(cid:27) , (2)and define (with a Matlab-like notation) the sub-files as F i, T := F i ( I T ) ∈ F B / ( K t ) q , ∀T ∈ Ω t [ K ] , ∀ i ∈ [ N ] . (3)The cache of user k ∈ [ K ] is populated as Z k = { F i, T : T ∈ Ω t [ K ] , k ∈ T , i ∈ [ N ] } ∈ F BN ( K − t − ) / ( K t ) q . (4)The memory size is thus M = N (cid:0) K − t − (cid:1) / (cid:0) K t (cid:1) = N t/ K as in (1). b) Delivery: The demand of user k ∈ [ K ] is representedby the row vector d k = ( d k, , . . . , d k, N ) ∈ F Nq , meaning thathe needs to successfully retrieve the scalar linear function (i.e.,operations are element-wise across files) B k := d k, F + . . . + d k, N F N ∈ F Bq , ∀ k ∈ [ K ] . (5)As for the sub-files, define the demand-blocks as B k, T = B k ( I T ) ∈ F B / ( K t ) q , ∀T ∈ Ω t [ K ] , ∀ k ∈ [ K ] . (6)Some demand-blocks can be computed based on the cachecontent available locally at the users in (4), while the re-maining ones need to be delivered by the server. Let D :=[ d ; . . . ; d K ] ∈ F K × Nq be the demand matrix . Let L ⊆ [ K ] such that rank q ( D ) = rank q ( D [ L , :]) = |L| =: r be the leader set , which is not unique but its size is (as every finite-dimensional vector space has a basis). Let D (cid:48) ∈ F K ×|L| q denotethe transformed demand matrix defined as [ D (cid:48) ] k,(cid:96) = (cid:40) { k = (cid:96) } if k ∈ L x k,(cid:96) if k (cid:54)∈ L , ∀ k ∈ [ K ] , ∀ (cid:96) ∈ L , (7) i.e., the demand-blocks of non-leaders in (5) are expressed asa linear combination of the demand-blocks of the leaders as B k, T = (cid:88) (cid:96) ∈L x k,(cid:96) B (cid:96), T , ∀T ∈ Ω t [ K ] , ∀ k ∈ [ K ] \L , (8)where the existence of the coefficients { x u,(cid:96) ∈ F q : u ∈ L , (cid:96) ∈L} in (8) follows from linear algebra. The server forms thefollowing multicast messages W S = (cid:88) k ∈S α k, S\{ k } B k, S\{ k } ∈ F B / ( K t ) q , ∀S ∈ Ω t +1[ K ] , (9)for some encoding coefficients { α k, S\{ k } ∈ F q \ { } : k ∈ [ K ] , S ∈ Ω t +1[ K ] } . (10)The server sends all multicast messages in (9) that are usefulfor the leaders, that is, X ∈ F ∆+ B ( ( K t +1 ) − ( K −|L| t +1 ) ) / ( K t ) q for X = { W S : S ∈ Ω t +1[ K ] , |S ∩ L| > } ∪ {L , D (cid:48) } . (11)Note that sending the chosen leader set and the transformeddemand matrix requires ∆ = |L|(cid:100) log q ( K ) (cid:101) + K + |L| symbols,where ∆ does not scale with the file length B . The worst-caseload is for r = |L| = min( K , N ) and equals R in (1).For a given S ∈ Ω t +1[ K ] , user k ∈ S can decode the missingdemand-block B k, S\{ k } from W S . The multicast messages { W A : A ∈ Ω t +1[ K ] \L } must be locally reconstructed from thetransmitted ones in (11) so that each user can recover all itsmissing demand-blocks. For K − r ≥ t + 1 , we seek to express W A = (cid:88) S∈ Ω t +1[ K ] , |S∩L| > β ( A ) S W S , ∀A ∈ Ω t +1[ K ] \L , (12)by an appropriate choice of the decoding coefficients { β ( A ) S ∈ F q : S ∈ Ω t +1[ K ] , |S ∩ L| > , A ∈ Ω t +1[ K ] \L } . (13)The choice of decoding coefficients must work for all realiza-tions of the demand-blocks .In [4] it was proposed that in (9) one alternates between ± the encoding coefficients as α k, S\{ k } = ( − Ind
S∩L ,k + Ind
S\L ,k , ∀ k ∈ S . (14)which results in decoding coefficients that are equal, up toa sign, to determinants of certain sub-matrices of D (cid:48) in (7).A reason for the choice of alternating signs in (14) (and theresulting decoding coefficients) was not given in [4]. The openquestion is whether such a choice is fundamental.We answer this open question by analyzing a general linearscheme in the form of (9) and (12). We show that: (1) thesigns of the encoding coefficients must follow a pattern wherethey alternate, but not necessarily as in (14), and their moduloneed not be one; (2) the decoding coefficients are proportionalto the determinants of certain matrices obtained from thetransformed demand matrix, but the proportionality coefficientneed not have modulo one; and, finally and importantly, (3)the encoding and decoding coefficients must satisfy certainrelationships that are captured by the cycles of a graph. The leader set L , the encoding coefficients in (10) and the decodingcoefficients in (13) are a function of D in general; such a dependency isnot made explicit here in order not to clutter the notation. II. M
AIN R ESULT
Our main result is to show that the linear scheme in (9)and (12) is correct if and only if the following holds.The local reconstruction of non-sent multicast messagesin (12) simplifies to solving (cid:88) S∈ Ω t +1 A∪L β ( A ) S W S : β ( A ) A = − , ∀A ∈ Ω t +1[ K ] \L , (15)where in (15) the summation is over subsets of A ∪ L (in total (cid:0) |L| + t +1 t +1 (cid:1) terms in (15)) rather than over some subsets of [ K ] (in total (cid:0) K t +1 (cid:1) − (cid:0) K −|L| t +1 (cid:1) terms in (12)). Eq(15) is solved, forany realization of the files, by using decoding coefficients β ( A ) S = (cid:101) β ( A ) S · det ( D (cid:48) [ A \ S , S \ A ]) , (16a) ∀S ∈ Ω t +1 A∪L , ∀A ∈ Ω t +1[ K ] \L , (16b)where the part of the decoding coefficients that does notdepend on the demands (denoted as (cid:101) β ( A ) { k }∪T next) and theencoding coefficients (denoted as α k, T next) must satisfy (cid:101) β ( A ) { k }∪T · α k, T = ( − φ ( A ) k, T · c ( A ) T , (17a) φ ( A ) k, T = (cid:40) Ind ( { k }∪T ) \A ,k k ∈ L \ T Ind
A\T ,k k ∈ A \ T , (17b) ∀T ∈ Ω t A∪L , ∀ k ∈ ( A ∪ L ) \ T , (17c)for some constants { c ( A ) T ∈ F q : T ∈ Ω t A∪L , ∀ k ∈ ( A ∪L ) \ T } . Finally, the relationships in (17) can be representedon an undirected graph that has the (cid:101) β ( A ) S ’s and the c ( A ) T ’sas vertices and whose edges are labeled by the encodingcoefficients according to the constraints in (17a). A spanningtree on such a graph identifies all the encoding coefficientsthat are free to vary , in other words, cycles on such a graphidentify constraints that the encoding coefficients must satisfy.
Remark.
The reason why the signs of the encoding coeffi-cients (and the resulting decoding coefficients) must alternatein [4] is because of the condition in (17b), which is satisfiedby the choice in (14); however the alternating patten in (14)in just one possible feasible linear scheme. The choice ofcoefficients in (14) (and the resulting decoding coefficients)has the following advantages: (a) the scheme does not involvedivisions other than by elements of unit modulo, which inturns allows one to extend the scheme to monomial retrievalas well [4]; and (b) the scheme works irrespective of thecharacteristics of the finite field. (cid:3)
IV. P
ROOF OF M AIN R ESULT
We shall start to prove the result in Section III from thecase K − |L| = t + 1 in Section IV-A (i.e., only the multicastmessage indexed by A = L must be reconstructed in (12)),then in Section IV-B we shall argue that the case K −|L| > t +1 can be solved by analyzing several systems with only |L| + t +1 users each. Moreover, we provide a complete characterizationof all feasible linear schemes via graph theoretic properties.The proof holds for all r = |L| ∈ [min( K , N )] and t ∈ [0 : K ] . A. Case K − |L| = t + 1 We consider here a system with K users, r = |L| leaders,and memory size parameterized by t , where ( t, r ) are fixedand satisfy K = r + t + 1 . For a subset T of [ K ] , we let T := [ K ] \ T . In particular, L is the set of non-leader users.Define the transformed demand matrix as in (7). Onlythe multicast message indexed by A = L needs to bereconstructed, thus for notation convenience we drop A from β ( A ) S in (12). We re-write (12) with β L = − (but actuallyany non-zero value will do), as follow F B / ( K t ) q (cid:51) (cid:88) S∈ Ω t +1[ K ] β S W S (18a) = (cid:88) S∈ Ω t +1[ K ] β S (cid:88) k ∈S α k, S\{ k } (cid:88) (cid:96) ∈L [ D (cid:48) ] k,(cid:96) B (cid:96), S\{ k } (18b) = (cid:88) T ∈ Ω t [ K ] (cid:88) (cid:96) ∈L (cid:88) k ∈T β { k }∪T α k, T [ D (cid:48) ] k,(cid:96) B (cid:96), T . (18c)Since (18) must hold for all { B (cid:96), T ∈ F B / ( K t ) q : (cid:96) ∈ L , T ∈ Ω t [ K ] } , we equivalently rewrite it, ∀ (cid:96) ∈ L , ∀T ∈ Ω t [ K ] , as F q (cid:51) (cid:88) k ∈T β { k }∪T α k, T [ D (cid:48) ] k,(cid:96) (19a) = (cid:88) k ∈T ∩L β { k }∪T α k, T { k = (cid:96) } (19b) + (cid:88) k ∈T ∩L β { k }∪T α k, T x k,(cid:96) , (19c)by the dentition of transformed demand matrix in (7). Wefinally rewrite (19) by separating it into two cases (cid:88) k ∈T ∩L β { k }∪T α k, T x k,(cid:96) = (cid:40) (cid:96) ∈ L ∩ T , − β { (cid:96) }∪T α (cid:96), T (cid:96) ∈ L ∩ T , ∀T ∈ Ω t [ K ] . (20)Next, we say that a set T ⊆ [ K ] is in ‘hierarchy h ’ if |T ∩L| = h for some h ∈ [0 : min( |T | , |L| )] . We also say that β S is in hierarchy h if S is in hierarchy h . We next seek to showthat in general the decoding coefficients in hierarchy h +1 canbe expressed as a linear combination of those in hierarchy h .Initialization / hierarchy h = 1 : β L = − is the onlydecoding coefficient in hierarchy . By picking T = L \ { u } , u ∈ L , and (cid:96) ∈ L in (20) (and thus T ∩ L = { u } ), we expressthe decoding coefficients in hierarchy as follows β { (cid:96) }∪L\{ u } = α u, L\{ u } α (cid:96), L\{ u } x u,(cid:96) , ∀ u ∈ L , ∀ (cid:96) ∈ L . (21) Hierarchy h : For any
T ∈ Ω t [ K ] , from (20) with (cid:96) ∈ T , (cid:88) k ∈T ∩L β { k }∪T α k, T x k,(cid:96) = 0 , ∀ (cid:96) ∈ L ∩ T . (22)In particular, for a T in hierarchy h > , we indicate WLOG(recall that here |L| = K − r = t + 1 = |T | + 1 and thus |T ∩L| = h , |T ∩L| = t − h , |T ∩L| = r − h , |T ∩L| = h +1 ) T ∩ L = { (cid:96) , . . . , (cid:96) h } : (cid:96) < . . . < (cid:96) h , (leaders) , (23) ∩ L = { j , . . . , j h , j h +1 } : j < . . . < j h +1 , (24)and collect the h constraints in (22) in matrix form as indicatedin (25) and (26), at the top of the next page, for all T ∈ Ω t [ K ] .By Cramer’s rule, the solution of (26) can be written as ( − h +1 − i det (cid:0) D (cid:48) [ T ∩ L \ { j i } , L ∩ T ] (cid:1) det (cid:0) D (cid:48) [ T ∩ L \ { j h +1 } , L ∩ T ] (cid:1) (27a) = β { j i }∪T α j i , T β { j h +1 }∪T α j h +1 , T , ∀ i ∈ [ h ] , ∀ j i ∈ T ∩ L , (27b)or equivalently (27) can be written as (recall j ∈ T ∩ L ) ( − β { j }∪T α j , T det (cid:0) D (cid:48) [ T ∩ L \ { j } , L ∩ T ] (cid:1) = . . . (28a) = ( − h +1 β { j h +1 }∪T α j h +1 , T det (cid:0) D (cid:48) [ T ∩ L \ { j h +1 } , L ∩ T ] (cid:1) . (28b)Notice that all the decoding coefficients in (28) are in hierarchy h if the set T is hierarchy h . Hierarchy h + 1 : We plug the decoding coefficients inhierarchy h from (28) into (20) with (cid:96) ∈ T and, by definitionof determinant (i.e., Laplace expansion along a column), weobtain that for all T ∈ Ω t [ K ] − β { (cid:96) }∪T α (cid:96), T = (cid:88) k ∈T ∩L β { k }∪T α k, T x k,(cid:96) (29a) = β { j h +1 }∪T α j h +1 , T det (cid:0) D (cid:48) [ T ∩ L \ { j h +1 } , L ∩ T ] (cid:1) (29b) · (cid:88) i ∈ [ h +1] ( − h +1 − i det (cid:0) D (cid:48) [ T ∩ L \ { j i } , L ∩ T ] (cid:1) x j i ,(cid:96) = ( − h +1 β { j h +1 }∪T α j h +1 , T det (cid:0) D (cid:48) [ T ∩ L \ { j h +1 } , L ∩ T ] (cid:1) (29c) · ( − − Ind
L∩T ∪{ (cid:96) } ,(cid:96) det (cid:0) D (cid:48) [ T ∩ L , L ∩ T ∪ { (cid:96) } ] (cid:1) , (29d)or equivalently, ∀T ∈ Ω t [ K ] , ∀ (cid:96) ∈ T ∩ L , we have ( − Ind
L∩T ∪{ (cid:96) } ,(cid:96) β { (cid:96) }∪T α (cid:96), T det (cid:0) D (cid:48) [ T ∩ L , L ∩ T ∪ { (cid:96) } ] (cid:1) = eq(28) , (30)Notice that all the decoding coefficients in (30) are in hierarchy h + 1 if the set T is hierarchy h . Combing everything together:
We can interpret (28)and (30) as follows: for a set
T ∈ Ω t [ K ] and an element k ∈ T , we create a set S = T ∪ { k } ∈ Ω t +1[ K ] that satisfies thefollowing: add a non-leader k = j ∈ T ∩ L : T ∩ L \ { j } = L \ ( { j } ∪ T ) , (31a) L ∩ T = ( { j } ∪ T ) \ L , (31b)or add a leader k = (cid:96) ∈ T ∩ L : T ∩ L = L \ ( { (cid:96) } ∪ T ) , (31c) L ∩ T ∪ { (cid:96) } = ( { (cid:96) } ∪ T ) \ L , (31d)thus (recall T ∩ L = L \ T , T ∩ L = L \ T and T = [ K ] \ T ) c ( L ) T = ( − φ ( L ) k, T α k, T · (cid:101) β ( L ) { k }∪T , ∀T ∈ Ω t [ K ] , ∀ k ∈ T , (32a) (cid:101) β ( L ) { k }∪T := β { k }∪T det (cid:0) D (cid:48) [ L \ ( { k } ∪ T ) , ( { k } ∪ T ) \ L ] (cid:1) , (32b) φ ( L ) k, T := (cid:40) Ind
L\T ,k k ∈ L \ T , Ind ( { k }∪T ) \L ,k k ∈ L \ T , , (32c)for some constatns { c ( L ) T : T ∈ Ω t [ K ] } .The term in (32b) (that only depends on { k }∪T as opposedto on both k and T ) can be further expressed as a function ofthe encoding coefficients as follows. For a set S ∈ Ω t +1[ K ] , S (cid:54) = L , in hierarchy h and by setting WLOG S ∩ L = { (cid:96) , . . . , (cid:96) h } : (cid:96) < . . . < (cid:96) h , (leaders) (33) S ∩ L = { j , . . . , j h } : j < . . . < j h , (non leaders) (34) S ∩ L = J , L = { j , . . . , j h } ∪ J , (35)we iteratively use (29) to express β S with S = { (cid:96) . . . (cid:96) h } ∪ J as in (36) at the top of the next page and where the lastequality follows since by definition β { j h ...j }∪J = β L = − and by convention det ( D (cid:48) [ ∅ , ∅ ]) = 1 . Eq (36) shows that eachdecoding coefficient is proportional to the determinant of asub-matrix of the transformed demand matrix and that theproportionality coefficient (denoted as (cid:101) β ( L ) { (cid:96) ...(cid:96) h }∪J ) dependsonly on the encoding coefficients; the encoding coefficientshowever are not all free to vary, as they need to satisfy therelationships imposed by (32b). Graph representation:
The relationships among V := { c ( L ) T : T ∈ Ω t [ K ] } and V := { (cid:101) β ( L ) S : S ∈ Ω t +1[ K ] } imposedby (32) can be represented by a graph. We create an undirectedgraph G ( V , E ) , where V := V ∪ V is the vertex set and E := { ( (cid:101) β ( L ) { k }∪T , c ( L ) T ) : T ∈ Ω t [ K ] , k ∈ T } is the edge set.We assign label ( − φ ( L ) k, T α k, T to edge ( (cid:101) β ( L ) { k }∪T , c ( L ) T ) ∈ E tocapture the relationship in (32). We elect (cid:101) β L to be the rootnode and assign to it the value − (but we could start fromany other vertex with any non-zero value). We then create aspanning tree from that root . By doing so, we find valuesfor all the vertices by using (32). One can easily see, by theproperties of spanning trees, that the encoding coefficients onthe edges of the spanning tree are free to vary (i.e., they canbe be any non-zero value), while the encoding coefficients onedges that are not part of the spanning tree are determinedthrough the following relationship: every path from the rootto a node determines the value of the node by using (32) andall those values must be equal; in other words, every cyclein the graph, obtained by adding a edge that is not on thespanning tree to the spanning tree, is a constraint.This concludes the proof for the case K − r = t + 1 . Example:
Fig. 1 shows the described graph for the caseof K = 4 users, r = 2 leaders, and memory size t = 1 (i.e.,each user can cache one file); the edges of a possible spanningtree are marked by a solid red line; the edges that are not in A spanning tree is a subset of the graph, which has all the vertices of thegraph covered with minimum possible number of edges . Hence, a spanningtree does not have cycles and it cannot be disconnected. Moreover, everyconnected and undirected graph has at least one spanning tree. β { j }∪T α j , T . . . β { j h }∪T α j h , T β { j h +1 }∪T α j h +1 , T (cid:3) x j ,(cid:96) · · · x j ,(cid:96) h ... . . . ... x j h ,(cid:96) · · · x j h ,(cid:96) h x j h +1 ,(cid:96) · · · x j h +1 ,(cid:96) h (cid:124) (cid:123)(cid:122) (cid:125) = D (cid:48) [ T ∩L , L∩T ] ∈ F h +1 × h q = 0 ∈ F × h q , (25) (cid:104) β { j }∪T α j , T β { jh +1 }∪T α jh +1 , T . . . β { jh }∪T α jh, T β { jh +1 }∪T α jh +1 , T (cid:105) x j ,(cid:96) · · · x j ,(cid:96) h ... . . . ... x j h ,(cid:96) · · · x j h ,(cid:96) h (cid:124) (cid:123)(cid:122) (cid:125) = D (cid:48) [ T ∩L\{ j h +1 } , L∩T ] ∈ F h × h q = − (cid:2) x j h +1 ,(cid:96) . . . x j h +1 ,(cid:96) h (cid:3)(cid:124) (cid:123)(cid:122) (cid:125) = D (cid:48) [ { j h +1 } , L∩T ] ∈ F × h q , (26) (cid:101) β ( L ) { (cid:96) ...(cid:96) h }∪J = β { (cid:96) ...(cid:96) h }∪J det (cid:0) D (cid:48) [ S ∩ L , S ∩ L ] (cid:1) = − α j h , { (cid:96) ...(cid:96) h − }∪J α (cid:96) h , { (cid:96) ...(cid:96) h − }∪J β { j h }∪{ (cid:96) ...(cid:96) h − }∪J det (cid:0) D (cid:48) [ S ∩ L \ { j h } , S ∩ L \ { (cid:96) h } ] (cid:1) (36a) = ( − h α j h , { (cid:96) ...(cid:96) h − }∪J α (cid:96) h , { (cid:96) ...(cid:96) h − }∪J α j h − , { j h }∪{ (cid:96) ...(cid:96) h − }∪J α (cid:96) h − , { j h }∪{ (cid:96) ...(cid:96) h − }∪J . . . α j , { j h ...j }∪J α (cid:96) , { j h ...j }∪J β { j h ...j }∪J det ( D (cid:48) [ ∅ , ∅ ]) (36b) = ( − h +1 h (cid:89) i =1 α j i , { j h ...j i +1 }∪{ (cid:96) ...(cid:96) i − }∪J α (cid:96) i , { j h ...j i +1 }∪{ (cid:96) ...(cid:96) i − }∪J , (36c) the spanning tree (doted blue line edges) correspond to thefollowing constraintsvertex c : α , { } = − α , { } α , { } α , { } α , { } α , { } , (37a)vertex c : α , { } = − α , { } α , { } α , { } α , { } α , { } , (37b)vertex (cid:101) β { , } : α , { } = − α , { } α , { } α , { } α , { } α , { } . (37c)The relationships in (37) can arrived at by directly solv-ing (12) as shown in Appendix A. B. Case K − |L| > t + 1 It is easy to see that in order to locally reconstruct all non-sent multicast messages as in (12) we need not sum over allsent multicast messages indexed by {S ∈ Ω t +1[ K ] : |S ∩ L| > } but only on those indexed by {S ∈ Ω t +1 A∪L : S (cid:54) = A} . Bydoing so, we can equivalently re-write (12) as in (15). In otherwords, for reconstructing multicast messages W A we considera “reduced system” with users in A ∪ L for which W A isonly multicast message to be reconstructed. The analysis wedid in Section IV-A applies to this “reduced system” with |A ∪ L| = t + 1 + r users. After the substitutions A ∪ L instead of [ K ] , and A instead of L , the conditions in (32)reads as stated in (17). Graph representation:
The relationships in (17) can berepresented on a graph as we did in Section III. The resultinggraph now has as many disconnects components as there aremulticast messages to reconstruct. The edges of the variouscomponents are labeled by the encoding coefficients. Asan example, Fig. 2 shows the graph and a set of possiblespanning trees (one per disconnected component) for the case K = 5 , r = 2 , t = 1 , by using the same convention as in α { } - α { } - α { } α { } - α { } α { } α { } α { } - α { } α { } α { } - α { } β ˜ { } ← α { } α { } α { } α { } c { } ← α { } α { } α { } c { } ← α { } α { } α { } β ˜ { } ← α { } α { } c { } ←α { } β ˜ { } ← α { } α { } c { } ←α { } β ˜ { } ← α { } α { } β ˜ { } ← α { } α { } β ˜ { } ← - Fig. 1:
The graph and a possible spanning tree for the case K = 4 , r = 2 , t = 1 . For legibility, we removed the superscript L = { , } from the vertices. The edges are labeled by an encodingcoefficient with an appropriate sign. Solid edges form a spanningtree; the encoding coefficients on dotted edges are determined byusing (32b). The (cid:101) β -vertexes are in a yellow box and the c -vertexesin a cyan box; the expression on the RHS of the symbol ← in abox is the value assigned to the vertex when we travel the graphfrom the root (i.e., (cid:101) β { , } = − ) along the spanning tree. Fig. 1. Unlike for the case K − r = t + 1 , here some encodingcoefficients appear more than once in the graph, meaningthat finding a spanning tree independently for each connectcomponent may result in some encoding coefficients being partof one spanning tree (and thus being free to vary) while notbeing part of other spanning trees (and thus being determinedby the corresponding ‘cycle’ constraint). Since our goal hereis to determined all encoding coefficients that are free to vary,we propose the following greed algorithm.1) We assign the “priority score” { k ∈L} + 2 |T ∩ L| toncoding coefficient α k, T , and sort all encoding coeffi-cients in decreasing order of priority score.2) We check each group of coefficients with the samepriority score, and mark an encoding coefficient as“free” if the corresponding edges do not form a cyclewith prior free coefficients in any of the components.3) We end after all coefficients have been checked.The edges/encoding coefficients marked as “free” by thisgreedy algorithm are free to vary, as they are part of thespanning tree for each of the components in which they appear.The priority score aims to find the edges that are in thelargest number of components at each step, and cycles aresimultaneously broken in all components in order to buildthe spanning trees. This greedy algorithm guarantees thatthe edges that are marked as “not free” (and are markedas such in every component they appear in) are in a cyclewith the same set of “free” edges in all components theyappear in, that is, although the same encoding coefficientappears to be constrained by multiple cycles, all those cyclesinvolve edges with the same label and thus do not conflict.Appendix B explains the details of the greedy algorithm bydirectly solving (12). V. C ONCLUSION
In this paper, we investigated the constraints that a linearscheme for cache-aided scalar linear function retrieval mustsatisfy in order to be feasible. We showed that the constraintsamong the parameters of a feasible linear scheme are capturedby the cycles of a certain graph. Equivalently, we showed thata spanning tree for the graph identifies the parameters of thescheme that are free to vary. The structure of our generalscheme sheds light into a scheme that had been previouslyproposed in the literature. Ongoing work includes using similarideas to explain the scheme in [5].This work was supported in part by NSF Award 1910309.A
PPENDIX AE XAMPLE : K = 4 , r = 2 , t = 1 WLOG, let L = { , } and thus L = { , } . The multicastmessages sent by the server are W { , } = α , { } B , { } + α , { } B , { } , (all leaders) , (38) W { , } = α , { } B , { } + α , { } B , { } , (mixed) , (39) W { , } = α , { } B , { } + α , { } B , { } , (mixed) , (40) W { , } = α , { } B , { } + α , { } B , { } , (mixed) , (41) W { , } = α , { } B , { } + α , { } B , { } , (mixed) (42)and the multicast messages that is not sent is W { , } = α , { } B , { } + α , { } B , { } , (all non leaders) . (43)In order to reconstruct W { , } at users, we seek the decodingcoefficients { β { , }S : S ∈ Ω , S (cid:54) = { , }} such that W { , } = β { , }{ , } W { , } (44) + β { , }{ , } W { , } + β { , }{ , } W { , } (45) + β { , }{ , } W { , } + β { , }{ , } W { , } , (46)that is, we aim to solve the following α , { } [ x , B , { } + x , B , { } ] + α , { } [ x , B , { } + x , B , { } ] (47) = β { , }{ , } (cid:0) α , { } B , { } + α , { } B , { } (cid:1) (48) + β { , }{ , } (cid:0) α , { } B , { } + α , { } [ x , B , { } + x , B , { } ] (cid:1) (49) + β { , }{ , } (cid:0) α , { } B , { } + α , { } [ x , B , { } + x , B , { } ] (cid:1) (50) + β { , }{ , } (cid:0) α , { } B , { } + α , { } [ x , B , { } + x , B , { } ] (cid:1) (51) + β { , }{ , } (cid:0) α , { } B , { } + α , { } [ x , B , { } + x , B , { } ] (cid:1) (52)for any realization of the demand-blocks. We thus equate thecoefficients on the RRS and on the LRS of the above equation,as follows.We start with the hierarchy 1 decoding coefficientsfor B , { } : α , { } x , = β { , }{ , } α , { } (53) ⇐⇒ β { , }{ , } x , = α , { } α , { } = (cid:101) β { , }{ , } , (54)for B , { } : α , { } x , = β { , }{ , } α , { } (55) ⇐⇒ β { , }{ , } x , = α , { } α , { } = (cid:101) β { , }{ , } , (56)for B , { } : α , { } x , = β { , }{ , } α , { } (57) ⇐⇒ β { , }{ , } x , = α , { } α , { } = (cid:101) β { , }{ , } , (58)for B , { } : α , { } x , = β { , }{ , } α , { } (59) ⇐⇒ β { , }{ , } x , = α , { } α , { } = (cid:101) β { , }{ , } . (60)In Fig. 1 (recall we did not write the superscript { , } ), start-ing with (cid:101) β { , }{ , } = β { , }{ , } = − , we arrive at the hierarchy 1 (cid:101) β { , }{ (cid:96),j } , (cid:96) ∈ { , } , j ∈ { , } through c { , }{ } = α , { } and c { , }{ } = α , { } .Next we havefor B , { } : 0 = β { , }{ , } α , { } x , + β { , }{ , } α , { } x , (61) ⇐⇒ α , { } α , { } α , { } + α , { } α , { } α , { } = 0 , (62)for B , { } : 0 = β { , }{ , } α , { } x , + β { , }{ , } α , { } x , (63) ⇐⇒ α , { } α , { } α , { } + α , { } α , { } α , { } = 0 , (64)where (62) and (64) are two cycles in Fig. 1 (starting from (cid:101) β { , }{ , } along the edge with label − α , { } , one is clockwise and { } - α { } - α { } α { } - α { } α { } α { } α { } - α { } α { } α { } - α { } β ˜ { } c { } c { } β ˜ { } c { } β ˜ { } c { } β ˜ { } β ˜ { } β ˜ { } (a) Component for W { , } α { } - α { } - α { } α { } - α { } α { } α { } α { } - α { } α { } α { } - α { } β ˜ { } c { } c { } β ˜ { } c { } β ˜ { } c { } β ˜ { } β ˜ { } β ˜ { } (b) Component for W { , } α { } - α { } - α { } α { } - α { } α { } α { } α { } - α { } α { } α { } - α { } β ˜ { } c { } c { } β ˜ { } c { } β ˜ { } c { } β ˜ { } β ˜ { } β ˜ { } (c) Component for W { , } Fig. 2:
The graph and possible spanning trees for the case K = 5 , r = 2 , t = 1 . The convention is as in Fig. 1. For sake of legibility, weomitted the superscripts in the various sub-figures, which should be the index of the multicast message listed in the sub-caption. the other is counterclockwise) and impose constraints amongthe involved encoding coefficients. Furthermore, (62) and (64)cover another two vertexes c { , }{ } and c { , }{ } . Indeed in Fig. 1,by proceeding from (cid:101) β { , } along the edge with label α , { } ,we get c { , }{ } = α , { } α , { } α , { } (65)and, from (cid:101) β { , } along the edge with label − α , { } , we get c { , }{ } = − α , { } α , { } α , { } . (66)Similarly for c { , }{ } . By breaking these cycles we obtain (37a)and (37b).Finally, with the condition in (62) and (64), we get thehierarchy 2 decoding coefficientsfor B , { } : − β { , }{ , } α , { } (67) = β { , }{ , } α , { } x , + β { , }{ , } α , { } x , (68) = ( − x , x , + x , x , ) α , { } α , { } α , { } (69) ⇐⇒ (cid:101) β { , }{ , } = − β { , }{ , } ( x , x , − x , x , ) (70) = − c { , }{ } α , { } = − α , { } α , { } α , { } α , { } ; (71)for B , { } : − β { , }{ , } α , { } (72) = β { , }{ , } α , { } x , + β { , }{ , } α , { } x , (73) = − ( x , x , − x , x , ) α , { } α , { } α , { } (74) ⇐⇒ (cid:101) β { , }{ , } = β { , }{ , } ( x , x , − x , x , ) (75) = c { , }{ } α , { } = α , { } α , { } α , { } α , { } . (76)Indeed in Fig. 1, we have two paths that lead to (cid:101) β { , }{ , } : (i) byproceeding from c { } along the edge with label α , { } we getto (cid:101) β { , }{ , } as in (76), while (ii) from c { } along the edge withlabel − α , { } we get (cid:101) β { , }{ , } as in (71); but the two must beequal, thus we get the condition in (37c).By combining the conditions in (62), (64), (71), and (76),we have − α , { } α , { } = α , { } α , { } α , { } α , { } (77) = α , { } α , { } α , { } α , { } (78) = − α , { } α , { } α , { } α , { } α , { } α , { } , (79)which is the same as the relationships (37) we obtained fromthe spanning tree in Fig. 1.A PPENDIX BE XAMPLE : K = 5 , r = 2 , t = 1 WLOG, let L = { , } and thus L = { , , } . Themulticast messages sent by the server are W { , } = α , { } B , { } + α , { } B , { } , (all leaders) , (80) W { , } = α , { } B , { } + α , { } B , { } , (mixed) , (81) W { , } = α , { } B , { } + α , { } B , { } , (mixed) , (82) W { , } = α , { } B , { } + α , { } B , { } , (mixed) , (83) W { , } = α , { } B , { } + α , { } B , { } , (mixed) , (84) W { , } = α , { } B , { } + α , { } B , { } , (mixed) , (85) W { , } = α , { } B , { } + α , { } B , { } , (mixed) , (86)and those that we not sent are W { , } = α , { } B , { } + α , { } B , { } , (all non leaders) , (87) { , } = α , { } B , { } + α , { } B , { } , (all non leaders) , (88) W { , } = α , { } B , { } + α , { } B , { } , (all non leaders) . (89)For every A ∈ Ω { , , } , the non-send multicast message W A can be reconstruct from { W S : S ∈ Ω { , }∪A , S (cid:54) = A} ,by a procedure equivalent to (80)-(86) after appropriate rela-beling of the indices of the non-leader users. To locally recon-struct all the non-sent multicast messages we thus proceed asfor “reduced systems” with parameters K (cid:48) = 4 , r = 2 , t = 1 .By symmetry and from (79), the relationships among theencoding coefficients are A = { , } : − α , { } α , { } = α , { } α , { } α , { } α , { } (90) = α , { } α , { } α , { } α , { } (91) = − α , { } α , { } α , { } α , { } α , { } α , { } , (92) A = { , } : − α , { } α , { } = α , { } α , { } α , { } α , { } (93) = α , { } α , { } α , { } α , { } (94) = − α , { } α , { } α , { } α , { } α , { } α , { } , (95) A = { , } : − α , { } α , { } = α , { } α , { } α , { } α , { } (96) = α , { } α , { } α , { } α , { } (97) = − α , { } α , { } α , { } α , { } α , { } α , { } , (98)where the coefficients highlighted in cyan are assigned todotted edges in Fig. 2. For example, for A = { , } (andsimilarly for A = { , } and A = { , } ), the relationshipsrevealed in Fig. 2a are α , { } = − α , { } α , { } α , { } α , { } α , { } , (99) α , { } = − α , { } α , { } α , { } α , { } α , { } , (100) α , { } = − α , { } α , { } α , { } α , { } α , { } . (101)By substituting the fixed coefficients in (99)-(101) into (92),we will eliminate other free coefficients, that is, (92) areequivalent to (99)-(101), which we obtained from the spanningtrees in Fig. 2 by using the greedy algorithm in Section IV-B.R EFERENCES[1] M. A. Maddah-Ali and U. Niesen, “Fundamental limits of caching,”
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