Non-linear index coding outperforming the linear optimum

Non-linear index co ding outp erfor ming the linear opti m um Ey al Lub etzky ∗ Uri Sta v † Abstract The following source coding pro blem w as int ro duced b y Birk and Kol: a sender holds a word x ∈ { 0 , 1 } n , and wishes to br oadcast a codeword to n rece iv e r s, R 1 , . . . , R n . The rece iver R i is int erested in x i , and has prior side information comprising some subset o f the n bits. This corres p onds to a directed g raph G on n vertices, wher e ij is a n edge iff R i knows the bit x j . An index c o de for G is an enco ding sc heme which enables each R i to a lw ays r econstruct x i , given his side information. The minimal word length of an index co de was s tudied by Ba r-Y ossef, Birk, Jayram and Kol [4]. They in tro duced a graph par ameter, minrk 2 ( G ), whic h completely characterizes the length of an optimal line ar index co de for G . The authors of [4] show ed that in v arious cases linear co des attain the o ptimal word length, and conjectured that linear index co ding is in fact always optimal. In this work, we disprov e the main conjecture of [4] in the following strong sense: for any ε > 0 and sufficiently large n , there is an n -vertex graph G so that every linear index co de f or G re q uires co dewords of length at lea s t n 1 − ε , and y et a non-linear index co de for G has a word length of n ε . This is ac hieved by an explicit co nstruction, which extends Alon’s v a riant o f the celebrated Ramsey co nstruction o f F rankl and Wilson. In addition, we study optimal index co des in v a rious, less restricted, natura l models, a nd prov e sev eral r elated pro perties of the gra ph parameter minrk( G ). 1 In tro duc tion Source co ding deals with a scenario in whic h a sender h as s ome data string x he wish es to transmit through a br oadcast channel to r e c eive rs . Th e fi rst and classica l result in this area is Sh annon’s Source Co ding Theorem. T h is has b een follo w ed b y v arious scenarios whic h differ in the nature of th e data to b e transmitted, the broadcast c hannel and some assu mptions on t he computational abilities of the users. Another family of source co d ing pr oblems, wh ic h attracted a considerable amoun t of atten tion o v er the yea rs, d eals with the assu mption that the r eceiv ers p ossess some prior kno w ledge on the data s tr ing x . It was sho wn that in some cases eve n some restricted assumptions on this kno wledge ma y drastica lly affe ct the nature of the cod ing pr ob lem. In this pap er we consider a v ariant of source co ding whic h w as first prop osed b y Bi rk and Kol [5]. In this v arian t, called Informed Source Co d ing On Demand (ISCOD), eac h r ecei v er h as some prior sid e information, comprisin g some sub set of the in put wo rd x . The send er is aw are of the ∗ Microsof t Researc h, One Microsoft W ay , Red m on d , W A 98052-6399 , USA. Email: eyal@microso ft.com. † School of Computer Science, Ra ymond and Beverly S ac kler F aculty of Exact Sciences, T el Aviv Universit y , T el Aviv, 69978, Israel. Email: u ristav@tau.ac.il. 1 p ortion of x kno w n to eac h rece iv er. Moreo ve r, eac h receiv er is inte rested in just part of the data. F ollo win g [4], w e restrict ourselve s to the pr oblem whic h is formaliz ed as follo ws. Definition 1 ( index code ) . A sender wishes to send a wo r d x ∈ { 0 , 1 } n to n r e c eivers R 1 , . . . , R n . Each R i knows some of th e bits of x and i s inter este d solely in the bit x i . An index code of length ℓ for this se tting i s a binary c o de of wor d-length ℓ , which enables R i to r e c over x i for any x and i . Using a graph mo del for th e side-information, this problem can be restat ed a s a graph parameter. F or a directed graph G and a ve rtex v , let N + G ( v ) b e the s et of out-neighbors of v in G , and f or x ∈ { 0 , 1 } n and S ⊂ [ n ] = { 1 , . . . , n } , le t x | S b e the r estrictio n of x to the co ordinates of S . Definition 2 ( ℓ ( G )) . The setting of Definition 1 is c har acterize d by the dir e cte d side inf ormation graph G on the vertex set [ n ] , wher e ( i, j ) is an e dge iff R i knows the value of x j . An index co de of length ℓ for G is a fu nction E : { 0 , 1 } n → { 0 , 1 } ℓ and functions D 1 , . . . , D n , so that for al l i ∈ [ n ] and x ∈ { 0 , 1 } n , D i ( E ( x ) , x | N + G ( i ) ) = x i . D enote the minimal length of an index c o de for G by ℓ ( G ) . Example: Supp ose that every receiv er R i kno w s in adv ance the whole w ord x , except for the single bit x i he wishes to r eco ver. T he corresp ond ing side information graph G is the complete graph K n (that is, ( i, j ) is an edge for all i 6 = j ). By br oadcasti ng the X OR of all the bits of x , eac h receiv er can easily compute its missin g bit: E ( x ) = n L i =1 x i , D i ( E ( x ) , x | { j : j 6 = i } ) = E ( x ) ⊕ ( L j 6 = i x j ) = x i . In this case the co de has length ℓ = 1 and E is a linear fu nction of x ov er GF (2). The problem of Inf ormed Sour ce Codin g On De mand (ISCO D) w as pr esented b y Birk and K ol [5]. Th ey were m otiv ated by v arious app licat ions of distributed comm unication such as satellit e comm un icati on n et w orks with cac hing cli en ts. In suc h applications, the clien ts h av e limited s torage and maintai n part of the transmitted in formation. Subs equen tly , the clien ts receiv e requests for arbitrary information blo c ks, an d ma y u se a slow backw ard channel to advise the serv er of their status. Th e serve r, pla ying the r ole of the sender in De finition 1, then b roadcasts a single transm is- sion to all client s (the receiv ers). As observed by Birk and Kol [5 ], when the sender h as only partial kno w ledge of the side information (e.g., the numb er of m issing blo c k s for eac h user), an erasure correcting co de suc h as Reed-Solomon Co de p erforms well. Th is is also the case if ev ery user is exp ected to b e able to d ecode the whole information. The authors of [5] pr esen t some b ound s and h eu ristics for obtaining efficien t encod ing sc h emes, as well as proto cols for implementing the ab o ve scenario. See [5] and [4] for more details on the relation b et ween the source co ding problem, as formulat ed abov e, and the I SCOD problem, as w ell as the communicati on complexity of the indexing fu nction, random access cod es and n etw ork coding. Bar-Y o ssef, B irk, Jayram and K ol [4] further inv estigated index co ding. They sho wed that this problem is different in nature fr om the wel l-kno wn source co ding p roblems previously studied by Witsenhausen in [12]. Their main con tribution is an upp er b ound on ℓ ( G ), the optimal length of an index co de (Definition 2) . The upp er b ound is a graph p arameter denoted by minrk 2 ( G ), which is also sh o wn to b e the length of the optimal line ar ind ex co de. It is sho wn in [4] that in sev eral cases 2 linear co des are in fact optimal, e.g., for directed acyclic graphs, p erfect graphs, o dd cycles and o dd an ti-holes. An information theoretic lo wer b oun d on ℓ ( G ) is obtained: it is at least the size o f a maximal acyclic induced subgraph of G . Th is low er b oun d holds ev en for the relaxed problem of r andomize d index co des, where the sender is allo w ed to u se (p ublic) random coins d uring enco ding, and the receiv ers are expected to d ecode their information correctly with high probabilit y o ver these coin flips. Nevertheless, they sho w that in some cases the lo wer bou n d is not tight. Ha ving pro ved th at the up p er b ound ℓ ( G ) ≤ minrk 2 ( G ) is tigh t for sev eral natural graph families and under some relaxed restrictions on the co de (“semi-linearly-decod able”) , th e authors of [4] conjectured that the length of th e optimal index cod e is in fact equal to minr k 2 ( G ). That is, they conjectured that linear index co ding is alw a ys o ptimal, and concluded that this was the main op en pr ob lem to b e in vestiga ted. Before stating the main results of this pap er, w e review the definition of minrk 2 ( G ) an d other related graph theoretic parameters. 1.1 Definitions, notations and background Let G = ( V , E ) b e a directed grap h on the verte x set V = [ n ]. The adjacency matrix of G , denoted b y A G = ( a ij ), is the n × n binary matrix where a ij = 1 iff ( i, j ) ∈ E . An indep endent set of G is a set of v ertices whic h ha ve no edges b et wee n them, and the indep endenc e nu mb er of G , α ( G ), is the cardin ality of a maxim um indep enden t set. Th e c hr omatic numb er of G , χ ( G ), is the minim um n um b er of indep en d en t sets wh ose union is all of V . Let G denote th e gr aph c omplement of G . A clique of G is an ind ep enden t set of G (i.e., a set of v ertices s u c h that all edges b et ween them b elong to G ), and the clique numb er of G , ω ( G ), is the cardinalit y of a maximum clique. Without b eing formal, a graph G is called “Ramsey” if b oth α ( G ) and ω ( G ) are “small”. In [4], a binary n × n matrix A = ( a ij ) w as said to “fit” G if A has 1-s on its diago nal, and 0 in all the indices i, j where i 6 = j and ( i, j ) / ∈ E . T he paramete r minrk 2 ( G ) w as d efined to b e the minimal possible rank o v er GF (2) of a matrix whic h fits G . T o extend t his definition to a g eneral field, let A = ( a ij ) b e an n × n matrix o ver some field F . W e sa y that A r epr e se nts the graph G o ver F if a ii 6 = 0 for all i , and a ij = 0 whenev er i 6 = j and ( i, j ) / ∈ E . The minr ank of a directed graph G with resp ect to the field F is defin ed b y minrk F ( G ) = min { rank F ( A ) : A repr esen ts G o v er F } . F or the co mmon case where F is a fin ite field, we abbreviate: minrk p k ( G ) = minrk GF ( p k ) ( G ) . The notion of minrk( G ) for an u ndirected graph G was fir st considered in the cont ext of graph capacitie s by Hae mers [8],[9]. T he Shannon capacit y of the graph G , d enoted b y c ( G ), is a noto- riously c h allengi ng parameter, wh ic h was defined b y S hannon in [11], and remains unkn o wn ev en for simple graphs, such as C 7 , the cycle on 7 ve rtices. Lo wer b ounds for c ( G ) are giv en in terms of indep endence num b ers of certain graphs, and in particular, α ( G ) ≤ c ( G ). Haemers show ed that for all F , minrk F ( G ) is s andwic hed b et w een c ( G ) and χ ( G ), the chromatic num b er of the complemen t of G , alto gether givi ng α ( G ) ≤ c ( G ) ≤ minrk F ( G ) ≤ χ ( G ) . (1) 3 While minrk F ( G ) can prov e to b e difficult to compute, the m ost useful upp er b ound f or c ( G ) is ϑ ( G ), the Lo v´ asz ϑ -function, wh ic h w as introdu ced in the seminal pap er [10] to compute c ( C 5 ). The matrix-rank argumen t was thereafter in tro duced by Haemers to answ er some questions of [10], and has since b een used (und er some v arian ts) in add itional settings to obtain b etter b ounds than those p ro vided b y the ϑ -function (cf., e. g., [1]). 1.2 New results The main result of this pap er is an improv ed index cod ing scheme, whic h is shown to strictly impro ve up on th e minrk 2 ( G ) b ound. This dispro ves the main conjecture of [4] regarding the optimalit y of linear in dex co ding, as sta ted by the follo win g theorem. Theorem 1.1. F or any ε > 0 and any sufficiently lar ge n , th er e is an n -vertex gr aph G so that: 1. Any line ar index c o de for G r e quir es n 1 − ε bits, that is, minrk 2 ( G ) ≥ n 1 − ε . 2. Ther e exists a non-line ar index c o de f or G using n ε bits, that is, ℓ ( G ) ≤ n ε . Mor e over, the gr aph G is undir e cte d and c an b e c onstructe d explicitly. Note that this in fact disprov es the conjecture of Bar-Y ossef et al. in the follo win g strong sense: the ratio b et w een an optimal co de and an optimal linear co de o ver GF (2) can b e n 1 − o (1) . The essence of the pro of lies in the fact that, in some cases, linear co des ov er higher-order fields ma y yield significan tly b etter index co ding sc hemes. The term “linear co des o v er GF ( p )” is u sed here to describ e a co ding scheme, in w hic h the inp ut w ord is encoded in to a sequence of linear functionals of its sym b ols o ve r GF ( p ), which are su bsequen tly u s ed for the decoding (t he protocol for transmitting these fu nctionals need not b e linear). How ev er, as the n ext theorem demonstrates, ev en this extended family of in d ex co des ma y b e sub optimal. Theorem 1.2. F or any ε > 0 and any sufficiently lar ge n , th er e is an n -vertex gr aph G so that: 1. Any line ar ind ex c o de for G over some field F r e quir es √ n symb ols, tha t is, m inrk F ( G ) ≥ √ n . 2. Ther e exists a non-line ar index c o de f or G using n ε bits, that is, ℓ ( G ) ≤ n ε . Mor e over, the gr aph G is undir e cte d and c an b e c onstructe d explicitly. In ord er to p ro ve the ab o v e t wo theorems, w e in tro duce the f ollo wing upp er b oun d on ℓ ( G ), whic h is a simple ext ension o f a result of [4] (the sp ecial case F = GF (2 )), and is a sp ecial case of Prop osition 2.1 (Secti on 2). ℓ ( G ) ≤ min F : | F | < ∞ ⌈ min r k F ( G ) log 2 | F | ⌉ . (2) The proof of Theorem 1.1 relies on the fact that for some graphs, the min im um of (2) is attained when F 6 = GF (2), in w h ic h case th e linear co de o v er GF (2) is sub optimal. Pr op ositio n 2.2 (Section 2) pr o vides a construction of su c h g raphs, and is the main ingredien t in the pro of of Theorem 1.1. This p r op osition, whic h ma y b e of indep enden t in terest, s tates that for any pair of finite fields w ith 4 distinct c haracteristics, F and K , the gap b et ween m inrk F and minrk K can b e n 1 − o (1) . Theorem 1.1 is then obtained a s a corol lary of (2) and a sp ecial case of Prop osition 2. 2. Moreo v er, as Theorem 1.2 sh o ws, the up p er b ound of (2) is not alw ays tigh t. T o see this, w e com bine the constru ction in the ab ov e men tioned Proposition 2 .2 with some additional ideas. As an a dditional corollary , Pr op ositio n 2.2 yiel ds that minrk F ( G ) /ϑ ( G ) (wh ere ϑ is the Lo v´ asz ϑ -function and | V ( G ) | = n ) is in some cases (roughly) at lea st √ n , wher eas in other cases it is (roughly) at most 1 / √ n . This addresses anot her question of [4] on the relati on betw een these t wo parameters. The relat ion b et w een ℓ ( G ) and the Sh annon capacit y of G , c ( G ), is addressed as well, as a b y-pro duct o f the pro of of Theorem 1.2. W e also extend the main construction of Prop osition 2.2 and giv e, for any prescrib ed set of finite fields { F i } and an additional fin ite field K of a distinct c haracteristic, a construction of a graph G so that minr k F i ( G ) is “la rge” for all i , whereas minrk K ( G ) is “small” . Prop osition 1.3. F or any fixe d t , let F 1 , . . . , F t denote finite fields, and let K denote a finite field of a distinct char acteristic. F or any ε > 0 an d a sufficie ntly lar ge n , ther e is an explicit c onstruction of a gr aph G on n vertic es, so that minrk K ( G ) ≤ n ε , wher e as for al l i ∈ [ t ] , minrk F i ( G ) ≥ n 1 − ε . In the second part of this pap er, we revisit the problem definition. It is shown that the restricted problem give n in Definition 1 captures many other case s arising from the original d istributed applications, which motiv ated the stu dy of Informed S ou r ce Co ding On Demand. In p articular, w e suggest app ropriate m od els and reductions for cases in w hic h multiple us ers are interested in the same bit, there are m u ltiple rounds of transmission and the transmitted w ords are o v er a la rge alphab et. Th ese mo dels are obtained as natural extensions of the original problem, and exhibit in teresting relations to the parameters ℓ ( G ) and min r k( G ). 1.3 T ec hniques A ke y elemen t in the pr oof of the main result is an extended v ersion of the Ramsey grap h constructed b y Alon [1], whic h is a v ariant of the well- kno w n Ramsey construction of F rankl and Wilson [7]. This graph, G p,q for some large p rimes p, q , w as us ed by Alon in ord er to d ispro ve an old conject ure of S hannon [11] on the Shannon capacit y of a un ion of graphs. Using some prop erties of the m inrk p arameter, one can sho w that the graph G p,q has a “small” minrk p and a “large” minrk q , implyin g that the optimal linear index cod e o v er GF ( p ) ma y b e significan tly b etter than the o ne o ver GF ( q ). Ho w ev er, it is imp erativ e in the ab o ve co nstruction that b oth p and q will b e large, whereas w e are interested in the case q = 2, corresp onding to minrk 2 . T o this end, w e extend the ab o ve co nstruction of [1] to prime-p o w ers , u sing some classic al results on congru en cies of binomial coefficients. Th is allo ws omitting the requiremen t that p, q should b e la rge, b y taking sufficien tly large p o w ers of arbitr ary distinct primes p an d q . Using v ariants of th e ab ov e construction, w e extend the results to m ultiple fields, to obtain Theorem 1.2 and Prop osition 1.3. E n route, we d eriv e seve ral pr op erties of the min rank parameter, whic h ma y b e of indep end en t inte rest. The pr oofs of the results throughout the pap er com bin e argument s from Linear Algebra and Num b er Theory along with some additional ideas, inspired b y the theory of graph capacitie s under 5 the s tr ong graph pr od uct definition. 1.4 Organization The rest of the pap er is organized as follo w s . Section 2 con tains a description of the basic constru c- tion, and the p roof of Theorem 1.1 . Th e extension of this resu lt to multiple fields, including the pro of of T heorem 1.2, app ears in Section 3. In Section 4, we study the v arious extensions of the original problem. Section 5 con tains some concluding remarks and op en p roblems. 2 Linear index co des o v er higher-order fields In this section, w e pro ve Theorem 1.1, b y constructing graph s for whic h a giv en linear in d ex co de o ve r a higher-order field outp erforms all linear index co des o v er GF (2). 2.1 Pro of of T heorem 1.1 The first ingredien t in the pro of is a linear index coding scheme, whic h is an extension of the ideas in [4] for larger fields. This n otion is formulated in the next prop osition, whose p ro of app ears in Subsection 2.2. Prop osition 2.1. L et G b e a gr aph, and let A b e a matrix which r epr e se nts G over some field F (not ne c essarily finite). Then ℓ ( G ) ≤ ⌈ log 2 |{ Ax : x ∈ { 0 , 1 } n }| ⌉ . In p articular, the fol lowing holds: ℓ ( G ) ≤ min F : | F | < ∞ ⌈ minrk F ( G ) log 2 | F | ⌉ . The second and m ain in gredien t in the pro of of Theorem 1.1 is Prop osition 2.2, whose p roof app ears in Sub sectio n 2. 3. Here and in what follo ws, all loga rithms are in th e natural base unless stated otherwise. Prop osition 2.2. L et F and K denote two finite fields with distinct char acteristics. Ther e is an explicit c onstruction of a family of gr aphs G = G ( n ) on n vertic es, so that minrk F ( G ) ≤ exp  p (2 + o (1)) log n log log n  = n o (1) , (3) and yet: minrk K ( G ) ≥ n / exp  p (2 + o (1)) log n log log n  = n 1 − o (1) , (4) wher e the o (1) -terms tend to 0 as n → ∞ . In ord er to deriv e Th eorem 1.1 from P rop ositions 2.1 and 2.2, apply P rop osition 2.2, setting F = GF ( p ) and K = GF (2), wh er e p > 2 is any fi xed (odd) prime. Let ε > 0; for an y sufficien tly large n , the graph obtained ab o ve satisfies minrk 2 ( G ) ≥ n/ exp( O ( p log n log log n )) ≥ n 1 − ε , 6 and hence, by the r esults of [4], any linear index co de ov er GF (2) requ ires a w ord length of at least n 1 − ε bits. O n the other hand, Pr op osition 2 .1 implies that ℓ ( G ) ≤ ⌈ minrk p ( G ) log 2 ( p ) ⌉ ≤ exp( O ( p log n log log n )) ≤ n ε .  2.2 Pro of of P rop osition 2.1 Let V = [ n ] denote the vertex set of G , A = ( a ij ) denote a matrix which represen ts G ov er some field F (not necessarily fin ite), and S = { Ax : x ∈ { 0 , 1 } n } ⊂ F n . F or some arbitrary ord ering of the elemen ts of S , the enco ding of x ∈ { 0 , 1 } n is the lab el of Ax , requiring a w ord -length of ⌈ log 2 | S |⌉ bits. F or deco ding, the i -th r ecei v er R i examines ( Ax ) i , and since th e diagonal of A d o es not conta in ze ro entries by definition, w e h a v e: a − 1 ii ( Ax ) i = a − 1 ii X j a ij x j = x i + a − 1 ii X j ∈ N + G ( i ) a ij x j , (5) where the last equalit y is by the fact that A repr esen ts G . As R i kno w s { x j : j ∈ N + G ( i ) } , this allo ws R i to reco v er x i . Th erefore, indeed ℓ ( G ) ≤ ⌈ log 2 | S |⌉ . T o conclude th e pro of, note that in case F is finite, we ha v e | S | ≤ | F | rank F ( A ) , as required. F ur th ermore, in this case it is p ossible to us e a lin ear co de u tilizing the same word-length. Th e sender tran s mits a bin ary -enco ding of the inner-pro du cts ( u 1 · x, . . . , u r · x ) ∈ F r , where { u 1 , . . . , u r } is a basis for th e ro ws of A ov er F .  Remark 2.3: As prov ed for the case F = GF (2) in [4], it is p ossible to sh o w that th e ab o v e b ound is tight for th e case of linear co d es o v er F . That is, the length of an optimal linear index co de o v er a fin ite field F is ⌈ minrk F log 2 | F |⌉ . 2.3 Pro of of P rop osition 2.2 W e first co nsider the case F = GF ( p ) and K = GF ( q ) for distinct primes p and q . Let ε > 0, and let k d enote a (large) in teger sat isfying 1 q l < p k < (1 + ε ) q l , wh ere l = ⌊ k log q p ⌋ . (6) Define: s = p k q l − 1 and r = p 3 k . (7) 1 It is easy to v erify that there are infinitely man y suc h integers k , as p, q are distinct primes, and hence the set { k log q p (mo d 1) } k ∈ N is dense in [0 , 1]. 7 The graph G on n =  r s  v ertices 2 is defin ed as follo ws. Its v ertices a re a ll s -element sub sets of [ r ], and t w o v ertices are adjacen t iff their corresp ond ing sets ha ve an in tersection whose cardin alit y is congruen t to − 1 mo dulo p k : V ( G ) =  [ r ] s  , (8) ( X, Y ) ∈ E ( G ) ⇐ ⇒ ( X 6 = Y , | X ∩ Y | ≡ − 1 (mo d p k ) . F or some in teger d to b e determined later, define th e inclusi on matrix M d to b e th e  r s  ×  r d  binary matrix, ind exed b y all s -elemen t and d -elemen t subsets of [ r ], where ( M d ) A,B = 1 iff B ⊂ A , for all A ∈  [ r ] s  and B ∈  [ r ] d  . Noti ce that the n × n matrix M d ( M d ) T satisfies the follo wing for all A, B ∈ V (not n ecessarily distinct): ( M d ( M d ) T ) A,B =      X ∈  [ r ] d  : X ⊂ ( A ∩ B )      =  | A ∩ B | d  . (9) Define P = M p k − 1 ( M p k − 1 ) T and Q = M q l − 1 ( M q l − 1 ) T . W e claim that P rep resen ts G o v er GF ( p ) whereas Q represents G o v er GF ( q ). T o see this, w e need the follo wing simple observ ation, whic h is a sp ecial case of Lucas’s Theorem (cf., e.g., [6]) on congruencies of binomial co efficien ts. It w as used, for instance, in [3] for constructing lo w-degree representa tions of OR fun ctions mo dulo comp osite num b ers, as w ell a s in [7]. Observ ation 2.4. F or every prime p and inte gers i, j, e with i < p e ,  j + p e i  ≡  j i  (mo d p ) . Consider some A ∈ V ; since s satisfies b oth s ≡ ( p k − 1) (mo d p k ) and s ≡ ( q l − 1) (mo d q l ), com binin g (9) with Ob s erv ation 2.4 gi v es P A,A =  s p k − 1  ≡ 1 (mod p ) , and Q A,A =  s q l − 1  ≡ 1 (mo d q ) . Th us, indeed the diagonal en tr ies of P and Q are non-zero; it remains to show that their ( A, B )- en tries are 0 wherever A, B are distinct n on-adjacen t v er tices. T o this end , tak e A, B ∈ V so that A 6 = B and AB / ∈ E ( G ); b y (8), | A ∩ B | 6≡ − 1 (mod p k ), h en ce P A,B =  | A ∩ B | p k − 1  ≡ 0 (mod p ) , the last equiv alence again follo wing from O bserv ation 2.4, as  x p k − 1  = 0 for all x ∈ { 0 , . . . , p k − 2 } . Finally , supp ose that A, B ∈ V satisfy A 6 = B and AB / ∈ E ( G ). That is, AB ∈ E ( G ), h ence 2 By well know n prop erties of t he densit y of prime num b ers, and standard graph theoretic arguments, proving th e assertion of the prop osition for th ese v alues of n in fact implies the result for any n . 8 | A ∩ B | ≡ − 1 (mod p k ). The Chinese Remainder Th eorem now implies | A ∩ B | 6≡ − 1 (mod q l ), otherwise we w ould get | A ∩ B | = s and A = B . Since  x q l − 1  = 0 for all x ∈ { 0 , . . . , q l − 2 } , we get Q A,B =  | A ∩ B | q l − 1  ≡ 0 (mod q ) . Altoget her, P represent s G o ve r GF ( p ), and Q represen ts G ov er GF ( q ). Therefore, minrk p ( G ) is at most rank p ( P ) ≤ rank p ( M p k − 1 ), and similarly , minrk q ( G ) is at most rank q ( Q ) ≤ rank q ( M q l − 1 ). As M d has  r d  columns, n =  p 3 k p k q l  and q l < p k < (1 + ε ) q l , a straigh tforw ard ca lculation no w gives: minrk p ( G ) ≤  r p k − 1  < exp  p (1 + ε + o (1)) 2 log n log log n  , minrk q ( G ) ≤  r q l − 1  < exp  p (1 + ε + o (1)) 2 log n log log n  . The n ext simple claim r elat es minrk q ( G ) and minrk q ( G ): Claim 2.5. F or any gr aph G on n vertic es and any field F , minrk F ( G ) · minrk F ( G ) ≥ n . Pr o of. W e use the follo wing definition of graph pr od uct due to Shannon [11]: G 1 × G 2 , th e str ong gr aph pr o duct of G 1 and G 2 , is the graph whose v ertex set is V ( G 1 ) × V ( G 2 ), wh ere t w o d istin ct v ertices ( u 1 , u 2 ) 6 = ( v 1 , v 2 ) are adjacen t iff for all i ∈ { 1 , 2 } , either u i = v i or ( u i , v i ) ∈ E ( G i ). As obs erv ed by Haemers [8], if A 1 and A 2 represent G 1 and G 2 resp ectiv ely ov er F , then the tensor pro duct A 1 ⊗ A 2 represent s G 1 × G 2 o ve r F . T o see th is, notice that the diagonal of A 1 ⊗ A 2 do es not co n tain zero ent ries, and that if ( u 1 , u 2 ) 6 = ( v 1 , v 2 ) are disconnected v er tices in G 1 × G 2 , then by d efinition ( A 1 ) ( u 1 ,v 1 ) ( A 2 ) ( u 2 ,v 2 ) = 0, sin ce in this case for some i ∈ { 1 , 2 } we ha v e u i 6 = v i and u i v i / ∈ E ( G i ). Letting A 1 and A 2 denote matrices wh ic h attain min rk F ( G ) and minr k F ( G ) resp ectiv ely , the ab o ve discu s sion implies that: minrk F ( G × G ) ≤ rank( A 1 ⊗ A 2 ) = minrk F ( G ) · minr k F ( G ) . Ho wev er, the set { ( u, u ) : u ∈ V ( G ) } is an indep endent -set of G × G , since for u 6 = v , either uv ∈ E ( G ) and uv / ∈ E ( G ) or vice versa. Therefore, (1) give s m in rk F ( G × G ) ≥ α ( G × G ) ≥ n , completing t he pr oof of the claim.  This concludes the pro of of the prop osition for the case F = GF ( p ), K = GF ( q ), w here p, q are t wo distinct primes. The ge neralization to the case of prime-p ow ers is an immediate co nsequence of th e next claim: Claim 2.6. L e t G b e a gr aph, p b e a prime and k b e an inte ger. The fol lowing holds: 1 k minrk p ( G ) ≤ m inrk p k ( G ) ≤ m inrk p ( G ) . (10) 9 Pr o of. The statemen t minr k p k ( G ) ≤ minr k p ( G ) follo ws immediately from the fact that an y ma- trix A whic h repr esen ts G o ver GF ( p ) also rep resen ts G o ver GF ( p k ), and in addition s atisfies rank p k ( A ) ≤ rank p ( A ). T o sho w that minrk p ( G ) ≤ k minrk p k ( G ), let V = [ n ] denote the v ertex s et of G , and let A = ( a ij ) denote a matrix whic h repr esen ts G o ver GF ( p k ) with rank r = minrk p k ( G ). As usual, w e rep r esen t the elemen ts of GF ( p k ) as p olynomials of degree at most k − 1 o ver GF ( p ) in the v ariable x . Sin ce the result of multiplying eac h row of A b y a non-zero elemen t of GF ( p k ) is a matrix of r ank r which also represents G ov er GF ( p k ), assume withou t loss of generalit y that a ii = 1 for all i ∈ [ n ]. By this assumption, the n × n matrix B = ( b ij ), whic h conta ins the free co efficien ts of the p olynomials in A , rep resen ts G o ver GF ( p ). T o complete the p ro of, we claim that rank p ( B ) ≤ k r . T his follo ws from the simple fact th at, if { u 1 , . . . , u r } is a b asis for the rows of A o v er GF ( p k ), then the set S r i =1 { u i , x · u i , . . . , x k − 1 · u i } spans the ro ws of A when view ed as k n -dimensional vecto rs o v er GF ( p ).  This conclud es the p ro of of Prop osition 2.2 .  Remark 2.7: Alon’s Ramsey construction [1] is the graph on the vertex set V =  [ r ] s  , wh ere r = p 3 and s = pq − 1 for some large primes p ∼ q , and t w o distinct vertic es A, B are adjacen t iff | A ∩ B | ≡ − 1 (mo d p ). Ou r construction allo ws p and q to b e large prime-p o w er s p k ∼ q l . Note that the original co nstruction b y F rankl and Wilson [7] had the parameters r = q 3 and s = q 2 − 1 for some prime-p ow er q , and tw o distinct v ertices A and B are adjacen t iff | A ∩ B | ≡ − 1 (mo d q ). Remark 2.8: Another corollary of Pr op ositio n 2.2 is that th e ratio b etw een minrk F ( G ) and ϑ ( G ) can b e arbitrarily large. T o see this, consider the n -v ertex graph G constructed in Prop osition 2.2 for F = GF ( p ) and K = GF ( q ), where p and q are t w o distinct primes: it satisfies minrk p ( G ) ≤ n o (1) and minrk q ( G ) ≤ n o (1) . Clearly , G is verte x transitiv e (that is, its automorph ism group is closed under all vertex s u bstitutions), as we can alw a ys r elabel the elemen ts of the groun d set [ r ]. By [10] (Theorem 9), ev ery v ertex tr ansitiv e graph G o n n v ertices satisfies ϑ ( G ) ϑ ( G ) = n . Assume without loss of ge neralit y that ϑ ( G ) ≥ √ n ≥ ϑ ( G ) (ot herwise, switc h the roles of p and q and of G and G ). As minrk p ( G ) ≤ n o (1) and min rk p ( G ) ≥ n 1 − o (1) , we dedu ce that ϑ ( G ) ≥ n 1 2 − o (1) · minrk p ( G ) , and yet minrk p ( G ) ≥ n 1 2 − o (1) · ϑ ( G ) . 3 Outp erforming linear index co des o ve r mult iple fields In th is section we use v arian ts of the graphs constructed in Prop osition 2.2 in order to prov e Theorem 1.2 and Prop osition 1.3. 10 3.1 Pro of of T heorem 1.2 Let ε > 0, and let G b e the graph constructed by Prop osition 2.2 for F = GF (2), K = GF (3), and a sufficien tly large n su c h that minrk 2 ( G ) ≤ n ε/ 2 and minrk 3 ( G ) ≤ n ε/ 2 . Let H den ote the graph G + G , that is, the d isj oin t union of G and its complement . W e clai m that ℓ ( H ) < 3 n ε/ 2 , and y et c ( H ) ≥ √ 2 n = p | V ( H ) | . T o see this, obs er ve that in order to obtain an index co de for a give n graph , one ma y alw ays arbitrarily partiti on the graph int o su bgraphs and co ncatenate their individual ind ex co des: Observ ation 3.1. F or any gr aph G and any p artition of G to sub gr aphs G 1 , . . . , G r (that is, G i is an induc e d su b gr aph of G on some V i , and V = ∪ i V i ), we have ℓ ( G ) ≤ P r i =1 ℓ ( G i ) . In p articular, in our case, b y com bining the ab o ve with Prop osition 2.1, we ha ve ℓ ( H ) ≤ ℓ ( G ) + ℓ ( G ) ≤ n ε/ 2 + ⌈ log 2 3 n ε/ 2 ⌉ < 3 n ε/ 2 . Finally , lab el th e v ertices of G as { v 1 , . . . , v n } and the corresp onding v ertices of G as { v ′ 1 , . . . , v ′ n } . F ollo win g the arguments of the pro of of Claim 2.5, it is easy to v erify that the set of vertices { ( v i , v ′ i ) : i ∈ [ n ] } ∪ { ( v ′ i , v i ) : i ∈ [ n ] } is an indep endent set of s ize 2 n in G × G + G × G , whic h is an indu ced subgraph of H × H . T herefore, c ( H ) ≥ √ 2 n .  Remark 3.2: A stand ard argum en t giv es a slight imp ro ve men t in the ab o v e lo wer b ound on c ( H ), to c ( H ) ≥ 2 √ n . See, e.g., [1 ] (pro of of Theorem 2.1) for further details. 3.2 Pro of of P rop osition 1.3 Notice that m inrk p e ( G ) ≤ m inrk p d ( G ) for an y prime p and integ ers e > d . Therefore, we can assume without loss o f generalit y th at all the F i -s are fields with p airwise d istinct c haracteristics. Let G i denote the graph obtained by applying Proposition 2.2 on K and F i , so that: minrk K ( G i ) ≤ n ε/ 2 and min r k F i ( G i ) ≥ n 1 − ε/ 2 , and let G = P t i =1 G i b e the disjoin t union of these graphs. S ince the adjacency matrix of G is a diagonal blo c k-matrix of th e adj acency matrices corresp ondin g to th e individu al G i -s, w e ob tain that minrk K ( G ) = t X i =1 minrk K ( G i ) ≤ tn ε/ 2 < n ε , Clearly , for eve ry i , minrk F i ( G ) ≥ m inrk F i ( G i ), completing the pro of.  4 The problem definition revisite d Call the problem of finding the optimal in dex co de, as defined in Defin ition 1, Problem 1 . A t first glance, Problem 1 seems to ca pture only v ery r estricted instances of the source co ding problem for ISCOD, and its motiv ating applications in comm unication. Namely , the ma in restrictions are: 11 (1) Each r e c eiver r e qu ests exactly one dat a blo ck. (2) Each data blo ck is r e qu e ste d only onc e. (3) Every data blo ck c onsists of a single bit. In [5], wh ere D efinition 1 was stated, it is p r o v ed that the sour ce coding p roblem for ISCOD can b e reduced to a similar one which satisfies restriction (1 ). This is ac hiev ed by replacing a u ser that requ ests k > 1 bloc ks b y k users, all ha ving th e same side information, and e ac h requesting a differen t blo c k. On the other hand, restriction (2) app eared in [5] to simplify the problem and to enable the side-inform ation to b e mo deled b y a directed graph . 3 Restriction (3) is stated assuming a larger blo c k siz e do es not dramatically effect the n ature of the problem. In what follo ws , we aim to reconsider the last tw o restrictions. 4.1 Larger alphabet and mul tiple rounds Supp ose the data string x is o ver a p ossibly larger alphab et, e.g ., { 0 , 1 } t for some t ≥ 1: Problem 2: The generalization of Problem 1, where eac h input symb ol x i ∈ { 0 , 1 } t comprises a blo ck of t bits. Ev ery user is interested in a single blo c k , and kn ows a su bset of the other blo c ks. By co nsidering eac h of th e t bits o f the symb ol as one in dep enden t round of transmiss ion, one can verify that the follo wing form u lation is equiv alen t: Problem 2’: The generalizatio n of Problem 1 to t ≥ 1 round s o v er the same side information graph G . The sender wishes to transmit t words x 1 , . . . , x t ∈ { 0 , 1 } n , with the same side information setting. Receiv er R i is alwa ys in terested in the i -th bit of the input words, x 1 i , . . . , x t i . The abov e problem can b e redu ced to Problem 1 by considering the graph G [ t ], d efined as follo ws. F or some int eger t , let G [ t ] denote the t - blow-up of G (with indep end en t sets), that is, the graph on the vertex set V ( G ) × [ t ], w here ( u, i ) and ( v , j ) are adjace n t iff uv ∈ E ( G ). In deed, Problem 2 reduces to Problem 1 with the side information graph G [ t ], by assigning a receiv er to eac h of the data bits. Therefore, this extension is in fact a sp ecia l case of the original seemingly restricted p roblem. Clearly , one ma y choose to treat eac h roun d of transmission indep endently , at a total cost of t · ℓ ( G ) transmitted bits, thus ℓ ( G [ t ]) ≤ t · ℓ ( G ). Th e next remark sho ws that this b ound is sometimes tigh t: Remark 4.1: If an un directed graph G satisfies ℓ ( G ) = α ( G ) (this holds, e.g., for all grap h s satisfying α ( G ) = χ ( G ), a nd namely for p erfect graphs), then ℓ ( G [ t ]) = t · ℓ ( G ), as t · ℓ ( G ) ≥ ℓ ( G [ t ]) ≥ α ( G [ t ]) = t · α ( G ) = t · ℓ ( G ) . Ho wev er, as th e n ext r emark states, one ma y indeed sa v e on comm unication when sending a unified trans m ission for the en tire set of roun ds (or b lo ck of symbols): 3 It fol lo wed the observ ation that if the same block is requested by seve ral receiv ers, then mos t of th e communication sa v ing comes from transmitting t h is blo ck once ( duplic ate elimination ). 12 Remark 4.2: In a su bsequen t w ork [2 ], we show that there are graphs for whic h ℓ ( G [ t ]) < t · ℓ ( G ). That is, trans m ission of t rounds ma y strictly improv e up on th e p er f ormance of t indep endent transmissions. This jus tifi es the study of the index c o ding r ate defi ned b y lim t →∞ ℓ ( G [ t ]) t (the limit exists b y sub-additivit y). Th is corresp onds to the av erage length of a codewo rd p er round, wh en the num b er of rounds tends t o infi nit y . A natural extension of Problem 2’ is the case where the underlying side information graph c hanges b et we en rounds: Problem 3: The generalizat ion of Problem 1 to t ≥ 1 rounds: the sender wish es to tran s mit t w ord s x 1 , . . . , x t ∈ { 0 , 1 } n , with resp ectiv e side information graphs G 1 , . . . , G t . Receiv er R i is alw a ys in terested in the i - th bit of the in put words, x 1 i , . . . , x t i . Ev en in this more general setting, a r eduction to Problem 1 is p ossible: let G = G 1 ◦ · · · ◦ G t denote the directed graph on the ve rtex set V ( G ) = [ n ] × [ t ], w here for all i 1 , i 2 ∈ [ n ] and k 1 , k 2 ∈ [ t ], (( i 1 , k 1 ) , ( i 2 , k 2 )) is an edge of G iff ( i 1 , i 2 ) ∈ E ( G k 2 ). Again, it is straigh tforward to see that ℓ ( G ) is precisely the solution for Problem 3. In the general setting of Problem 3, it is eve n simpler to s ee that ind ep enden t transm iss ions ma y consume significantl y m ore comm un ication. F or ins tance, consider the follo w ing case. W e hav e t w o receiv ers, R 1 and R 2 , and tw o rounds for transmitting the b inary wo rds x = x 1 x 2 and y = y 1 y 2 . Supp ose that in the first r ound receiv er R 1 kno w s x 2 and in the second trans m ission receiv er R 2 kno w s y 1 . In this case, eac h round - if transmitted separately - requires 2 bits to b e transmitted. Y et, if the serv er transm its the 3 bits x 1 ⊕ y 1 , x 2 ⊕ y 2 , x 1 ⊕ y 2 , then b oth recei v ers can r econstruct their m issing bits (a nd moreo ver, reconstru ct all of x and y ). This in fact is a sp ecial case of the follo w ing construction. W e defin e a pair of graphs G 1 , G 2 suc h that ℓ ( G 1 ) = ℓ ( G 2 ) = n and ye t only ℓ ( G 1 ◦ G 2 ) = n + 1 b its n eed to transm itted for consecutive transmissions. Th is is stated in the next claim, w here the tr ansitive tournament gr aph on n v ertices is isomorphic to the dir ecte d graph on the v ertex set [ n ], w here ( i, j ) is an e dge iff i < j . Claim 4.3. L et G 1 denote the tr ansitive tournament gr aph on n vertic e s, and let G 2 denote the gr aph obtaine d fr om G 1 by r eversing al l e dges. Then ℓ ( G 1 ) + ℓ ( G 2 ) = 2 n , and yet ℓ ( G 1 ◦ G 2 ) = n + 1 . Pr o of. Without loss of generalit y , assume th at E ( G 1 ) = { ( i, j ) : i < j } and E ( G 2 ) = { ( i, j ) : i > j } . Since G 1 and G 2 are b oth acyclic, the fact th at ℓ ( G 1 ) = ℓ ( G 2 ) = n follo ws from the lo w er b ound of [4] ( ℓ ( G ) is alwa ys at least the size a m axim um induced acyclic su bgraph of G ). Recall that by definition, G 1 ◦ G 2 is the disjoint u nion of G 1 and G 2 , with the additional edges { (( i, 1) , ( j, 2)) : j < i } and { (( i, 2) , ( j, 1)) : j > i } . Therefore, G 1 ◦ G 2 has an ind uced acyclic graph of size n + 1: for instance, the set { ( i, 1) : i ∈ [ n ] } ∪ { ( n, 2) } induces suc h a graph. W e deduce that ℓ ( G 1 ◦ G 2 ) ≥ n + 1. 13 T o co mplete the pr oof of the claim, we giv e an enco ding sc heme for G 1 ◦ G 2 whic h requires the transmission of n + 1 bits, hence ℓ ( G 1 ◦ G 2 ) ≤ n + 1. Denote the t w o words to b e transmitted by x = x 1 . . . x n and y = y 1 . . . y n . Th e co ding sc h eme is linear: b y transmitting x i ⊕ y i for i ∈ [ n ] and ⊕ i ∈ [ n ] x i , it is not difficult to see that eac h r ecei v er is able to d ecode its miss ing b its (in fact, eac h receiv er can r econstru ct all the bits of x and y ).  4.2 Shared requests Problem 4: The generalization of Problem 1 to m ≥ n receiv ers, eac h in terested in a single b it (i.e., we allo w sev eral users to ask for the same bit). In this case, the one-to-one corresp ondence b etw een message bits and receiv ers no lo nger holds, th u s the directed side information graph seems un suitable. Ho w ev er, it is still p ossible to obtain b ounds on th e optimal linear and non-linear co des using sligh tly different mo dels. Let P 4 denote an instance of Problem 4, and let ℓ ( P 4 ) denote the length of an optimal index co de in this setting. It is con venien t to model the side-information of P 2 using a b inary m × n matrix, where the ij entry is 1 iff the i -th user knows the j -th bit (if m = n , this m atrix is the adjacency matrix of the side information graph ). With this in mind, we extend the notion of represent ing the side-information graph as follo ws: an m × n ma trix B r e pr esents P 4 o ve r F i ff for all i and j : • If the i -th rece iv er is interested in th e bit x j , then B ij 6 = 0. • If the i -th rece iv er is n either in terested in n or kno ws the bit x j , th en B ij = 0. Notice that in th e sp ecial case m = n , the ab o ve defin ition coincides w ith the usual definition of represent ing the s id e-information graph. L et minrk F ( P 4 ) denote the minim um rank of a matrix B that rep resen ts P 4 o ve r F . It is straigh tforw ard to verify that results analogous to Proposition 2.1 and Remark 2.3 hold f or this extended notion of mat rix represen tation: Theorem 4 .4. L et P 4 denote an instanc e of Pr oblem 4. Then the length of an optima l line ar c o de is minrk 2 ( P 4 ) , and the upp er b ounds of The or em 2.1 on arbitr ary index c o des hold for P 4 as wel l. Next, giv en P 4 , define the follo wing t w o dir ecte d m -v ertex graphs G ind and G cl . Both v ertex sets corresp ond to the m u sers, where eac h set of u sers intereste d in the same bit f orm s an indep endent set in G ind and a clique in G cl . In the remaining cases, in b oth graph s ( v i , v j ) is an edge iff the i -th u s er kno w s the b it in w hic h the j -th user is in terested (for m = n , b oth graphs are equal to the usu al side-information graph defined in Defin ition 2). T he follo wing simple claim pr o vides additional b oun ds on ℓ ( P 4 ); we omit th e details of its pro of. Claim 4.5. If P 4 denotes an insta nc e of Pr oblem 4, and G ind and G cl ar e define d as ab ove, then: 1. ℓ ( G cl ) ≤ ℓ ( P 4 ) , and in add ition, minrk F ( G cl ) ≤ min rk F ( P 4 ) for al l F . 2. ℓ ( P 4 ) ≤ ℓ ( G ind ) , and in add ition, minrk F ( P 4 ) ≤ minrk F ( G ind ) for al l F . 14 5 Concluding remarks and op en problems • In this pap er we ha ve introdu ced constructions of graphs for w hic h linear index co ding is sub optimal (Theorem 1.1), thus d ispro ving the main conjecture of [4]. It is in fact sho wn that an y linear index code for these n -v ertex graphs r equires a w ord length of n 1 − o (1) bits (barely impro ving the na ¨ ıv e proto col whic h requires n bits), y et a giv en index co de for th ese graph s utilizes w ords w hic h a re o nly n o (1) bits lo ng. • Th e graph s constructed extend Alon’s v ariant of the Ramsey construction giv en by F rankl and Wilson. F or these graphs, linear index codes ov er higher-order fields outp erform the linear co des o v er GF (2). F urthermore, a v ariant of this construction (g iv en in T heorem 1.2) sho w s that th er e are graphs where linear co des ov er any field are sub optimal. • Th e main qu estion for further work is trying to obtain tight b oun d s on ℓ ( G ) for a general graph G . In addition, it w ou ld b e in teresting to determine the exp ected v alue of ℓ ( G ) for the random graph G ∼ G ( n, 1 2 ). • In Theorem 1.1, we hav e constructed n -v ertex graphs, w here the ratio b etw een the parameters minrk 2 ( G ) and ℓ ( G ) w as n/ exp( O ( √ log n log log n )). It can b e in teresting to obtain an ev en larger gap b et w een these t wo parameters, and n amely , to sho w n -v ertex graphs G where minrk 2 ( G ) /ℓ ( G ) = e Θ( n ). Th is ma y require either a different approac h to the problem, or significan tly improving the giv en Ramsey constr u ctions. • In addition, we show ed that more general s cenarios of ind ex co ding, as presen ted in [5], can b e r educed to the main problem, wh ic h recen tly attracted att en tion. In this cont ext, w e ha v e demonstrated that one ma y sav e on comm u nicatio n when transmitting t binary w ords at once, rather than tran s mitting these w ords in dep enden tly . W e h av e shown this for the case where the underlying side-information graph is allo wed to c hange d y n amically . • Th e most interesting scenario is that of large d ata blo c ks o ver a fixed side-information graph. As in [4], w e ha ve confi n ed ourselv es in th is pap er mainly to the case in w h ic h eac h of the data blo c ks consists of a single bit. Ho w ever, this analysis of in dex co ding is relev ant to the motiv ating application only if th e communicatio n which is required to co ordinate the side in formation graph is negligi ble with resp ect to the size of the data blo c ks themselv es. Therefore, w e should , in fact, consider a scenario in which an n -w ord of b -bits blo c ks is transmitted, where b ≫ n . In this case, it is clearly p ossible to use an optimal index co de for eac h b it in the block indep endently , transmitting b · ℓ ( G ) bits altogether. Nev ertheless, this proto col is not guaran teed to b e optimal, which yields th e follo wing natural qu estion: Is there a side information graph G on n v ertices and intege r b , for which tran s mitting an n -w ord wh ic h consists of b -bits blocks requires less than b · ℓ ( G ) bits? Remark 5.1: After th e completion of this w ork, with Noga Alon, we w ere able to answ er the last question in the affirm ative . Th is will app ear i n a subsequen t w ork [2] . Ac kno wledgemen t: W e are grateful to Noga Alon and Od ed Reg ev for helpful discussions. W e w ou ld also lik e to thank the F OCS 2 007 p rogram committee for helpful suggestions. 15 References [1] N. Alon, The S hannon capacit y of a union, Com binatorica 1 8 (1998), 301 -310. [2] N. Alon, E. Lub etzky and U. S ta v, The broadcast rate of a graph , to app ear. [3] D.A.M. Barrington, R. Beigel , and S. 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