Separating NOF communication complexity classes RP and NP

Separatin g NOF commun ication complexity classes RP and NP Matei Da vid Computer Science Depart ment Uni vers ity of T oronto matei at cs toronto edu T oniann Pitassi ∗ Computer Science Department Uni vers ity of T oronto toni at cs toronto edu November 9, 2018 Abstract W e p rovide a non-explicit separation of the num ber-on-fore head commun ication complexity classes RP and NP w hen the number o f players is up to δ · log n for any δ < 1. Recent lo wer bounds o n Set- Disjointness [1 0, 7] p rovide an exp licit separation between th ese classes whe n the num ber of player s is only up to o ( log log n ) . 1 Introd uction In the number -on-forehea d (NOF ) model of communicati on c omple xity , k players are trying to e v aluate a functi on F defined on kn bits. T he input of F is part itioned into k pieces of n bit s ea ch, call them x 1 , . . . , x k , and x i is plac ed, metapho rically , on the fore head of player i . Thus, eac h player sees ( k − 1 ) n of the kn input bits. The player s communica te by writing bits on a shared blackboard in order to compute F . This model was introdu ced by [5] and it has m any application s, including circuit lower bound s [9, 11], time/s pace tradeo ff s for T uring M achine s, pse udo-ra ndom number generators for sp ace-bo unded T uring Mac hines [2], and proof system lo wer bound s [4]. In this mode l, a protoco l is said to be “efficien t” if it ha s c omple xity ( log n ) O ( 1 ) . Correspo ndingl y , P cc k , R P cc k , BPP cc k and NP cc k are the classes of function s ha ving efficie nt deterministic , one-s ided-er ror randomized, (two-s ided-er ror) rando mized and nonde terministi c protocol s, respecti vely . The usual inclusions between these classe s apply , so P cc k ⊆ RP cc k ⊆ NP cc k and R P cc k ⊆ BPP cc k . One of the most fundament al questio ns in NO F communication complexit y is to provide separat ions between these classes. In [3], Beame et al. sho w that RP cc k 6 = P cc k for k ≤ n O ( 1 ) player s. Recently , [7, 10] sho w tha t NP cc k 6⊂ BPP cc k (and thus, that NP cc k 6 = R P cc k ) for k ≤ o ( log log n ) players. Our main result in this paper is the follo w ing. Theor em 1.1 (Main Theorem) . NP cc k 6⊂ B PP cc k (and thus, NP cc k 6 = R P cc k ) for all δ < 1 and all k ≤ δ · log n. Until ver y recently , it was far from clear how to obtain communication complexity lower bounds in the number -on-forehea d model for any function that could separa te nondete rministic from randomized com- ple xity . The dif fi culty can be desc ribed as follo ws. The only metho d currently kno wn for obtain ing multi- party NO F lo wer b ounds is the discrep ancy met hod [ 2, 13, 8]. Lo wer bounds usi ng discrepanc y are obtained ∗ Research supported by NSERC. 1 by sh o wing that the functio n in ques tion has smal l discr epanc y w ith respect to some distrib ution. Unfortu - nately , it is not hard to see t hat e ve ry function with small non determini stic comple xity has high dis crepanc y with respect to e v ery distrib ution (see, for exampl e, Lemm a 3.1 in [7].) Thus, the discrepa ncy method seemed doomed to fa ilure and new tec hnique s seemed to be require d. Ho wev er , in v ery rec ent work , these d if ficulties were o v ercome to ob tain a surprisingly elegan t lo wer bou nd for the Set-Disjointn ess fu nction [7, 10]. The idea behind their proofs as well as ours is as follo ws. In a recent paper , Sherstov [15] (and implicitl y also in Razboro v [14]) applied the discrepanc y m ethod in a m ore genera l way for the 2-player model in order to overc ome the above dif ficulties . The gene raliz ed discre panc y meth od was ad apted to t he number -on-forehea d model in [7, 10] and can be described at a h igh le vel as follo ws. Start with some candid ate function F , wher e F has s mall non determin istic comple xity , and we want to prov e that F has high randomiz ed communication comple xity . N o w come up w ith a function G and a distr ib ution λ such tha t: (1) F and G are highl y correlated with respect to λ ; and (2) G has small discre panc y with respect to λ . It is not hard to see that if such a G can be found, then since G has small discre panc y , it requires lar ge randomized compl exity , and more ov er since F and G are very correlated, this in turn implies lo wer bounds on the rando mized co mplexi ty of F as well. Thus, to u se the genera lized discrepanc y meth od, the prob lem is to come up with the functions F and G . T o accompli sh this, we will use ano ther wonderful idea due to Sher stov [16], and substa ntially generalize d to apply to the nu mber -on-fo rehead setting by Chatto padhy ay [6]. W e cons ider special functions of the form F φ . This will be a functio n on ( k + 1 ) n bits, computed by k + 1 players. P layer 0 recei ves an n -bit vecto r x . Player i , for 1 ≤ i ≤ k gets an n -bit vector y i . The function φ tak es as input y 1 , . . . , y k and outputs an n -bit string z , where z has exactly m 1’ s. W e will view φ a selecti ng m bits/indic es of Player 0’ s input, x . The functio n F φ will be the OR function applie d to the m bits of x as specified by φ ( y 1 , . . . y k ) . (In earlie r terminol ogy , the k + 1 players will apply the O R funct ion to Play er 0’ s unmasked input.) Note that regard less of what function φ is chosen, F φ will hav e a small nondetermin istic protocol. Player 0 simply guesses an index j that is one of the indices chosen by φ , and then any of the other players can easily verify whether or not x j is 1 in that positio n. When φ is the bitwise AND function, then F φ is the Set-Disjoin tness function . W e will sho w that for almost all φ , the randomized communication complexit y of F φ is large as long as k is at mos t a constant times lo g n . Becaus e we will be work ing with a random φ , as a bonus, our ar gument is substanti ally simple r tha t the prev ious bounds obtained for Set-Disjointne ss. 2 Definitions and Notation 2.1 Communication Complexity In the numbe r- on-fore head (NO F) multi party communicatio n complexi ty game [5] there are k player s that are try ing to collaborate to co mpute a fu nction F : X 1 × . . . × X k → { 0 , 1 } where each X i = { 0 , 1 } n . The kn input b its are p artitio ned into k sets, each o f size n . For ( x 1 , . . . , x k ) ∈ { 0 , 1 } kn , an d for ea ch i , pla yer i kn o ws the v alues of all of the inputs exce pt for x i (which concep tually is though t of as being placed on player i ’ s forehe ad). The players exchan ge bits accordin g to an agreed-upo n protocol, by writing them on a public blackbo ard. A pr otoc ol spe cifies, for eve ry possibl e blackboard contents, whether or not the communication is ove r , the output if ov er and the next player to speak if not. A protoco l also specifies w hat each play er writes as 2 a function of the blackboa rd contents and of the inputs seen by that player . The cost of a protocol is the maximum number of bits written on the black board. In a determinist ic pr otocol , the blackb oard is initially empty . A r ando mized pr otocol of cost c is simply a probab ility distrib ution ov er determin istic protoc ols of cost c , which can be vie wed as a protocol in which the players hav e access t o a sh ar ed ra ndom string. A non-deter ministic pr otocol is one where an initi al g uess string appears on the blackboa rd at the begin ning of the protocol, and the players are trying to verify that the outp ut of the functio n is 1 in the usual sense: there exists a gue ss string where the outp ut of the protoco l is 1 if and only if the outpu t of the func tion is 1. The determin istic communication comple xity of F , written D k ( F ) , is the minimum cost of a determinis tic protoc ol for F that always output s the correct an swer . For 0 ≤ ε < 1 / 2, le t R k , ε ( F ) denote th e min imum c ost of a randomized protoc ol for F which, for e ve ry input, mak es an er ror with probabilit y at most ε (ov er the choice of the determin istic protocols ). The ( two-sided -err or) ran domized communic ation comple xity of F is R k ( F ) = R k , 1 / 3 ( F ) . Let R 1 k , ε ( F ) deno te the minimum cost of a randomiz ed protocol for F w hich is correct on all 0-inputs, and for ev ery 1-input, it makes an error with probabil ity at most ε . The one-sid ed-err or ran domized communicati on comple xity of F is R 1 k ( F ) = R 1 k , 1 / 3 ( F ) . The non-determini stic communication comple xity of F , written N k ( F ) , is the minimum cost of a non-dete rministic p rotocol for F . W e usual ly drop the subscr ipt k when the number of players is clear from the conte xt. Since any function F n on k n bits can be computed using only n bits of communicati on, follo wing [1 ], for sequen ces of functio ns F = ( F n ) n ∈ N , protocol s are consider ed “ef ficient” or “polynomia l” if only polylog - arithmica lly many bits are excha nged. Accor dingly , let P cc k , RP cc k , BPP cc k and NP cc k denote the classes of functi on f amilies F for w hich D k ( F n ) , R 1 k ( F n ) , R k ( F n ) and N k ( F n ) are ( log n ) O ( 1 ) , respe cti vel y . Even thoug h the standard communication complex ity definition s abov e are gi ven for functions with range { 0 , 1 } , we find it more con venien t to wo rk with the range {− 1 , 1 } . W e transf orm the former into the latter by mapping 0 → 1 (representin g false ) and 1 → − 1 (represen ting true ). Thus, for e xample, when the range of F is {− 1 , 1 } , in a non-determin istic protocol the players are trying to veri fy that the output of F is -1. The most important method to prov e lower bounds for randomized commun ication complexit y uses the concep t of discrepa ncy . An i-cyl inder Γ i in X 1 × . . . × X k is a set such that for all x 1 ∈ X 1 , . . . , x k ∈ X k , x ′ i ∈ X i we ha ve ( x 1 , . . . , x i , . . . , x k ) ∈ Γ i if and only if ( x 1 , . . . , x ′ i , . . . , x k ) ∈ Γ i . A cyl inder inter sectio n is a set of the form T k i = 1 Γ i where each Γ i is an i -cylin der in X 1 × · · · × X k . For a set S , let 1 S be its character istic func tion, which is 1 if the input is in S and 0 otherwise. Let λ be a distrib ution on the inputs of F . The discr epancy of F on Γ under λ is disc Γ k , λ ( F ) = | E x ∼ λ [ F ( x ) 1 Γ ( x )] | . The disc r epan cy of F und er λ is disc k , λ ( F ) = max Γ disc Γ k , λ ( F ) . The standar d discr epanc y method [2] connec ts the discrepan cy of a function F with its randomiz ed commun ication complex ity as fo llo ws: for ev ery distrib ution λ , R k , ε ( F ) ≥ log  1 − 2 ε disc k , λ ( F )  . 2.2 Notation Through out this paper , the functio ns whose communicatio n comple xity w e are analyzing are denoted by capita l letters such as F . As mentioned in the introd uction, we will be restricting our attentio n to certain functi ons which are construc ted from a base functi on, usually denot ed by lower case f , and a maski ng functi on, usually denoted by φ . In genera l, m denotes the size of the input to the base function f , and the range of this function is {− 1 , 1 } . A specific base function w e will work with is the OR function, which tak es on th e valu e -1 if and on ly if any of its input bits is 1. The masking function φ tak es as inp ut k string s 3 of n b its each, usu ally denot ed by y 1 , . . . , y k , an d it’ s outpu t is an m -element subs et of [ 1 , n ] . W e al ways ha v e m ≤ n . Starting with a base function f and a masking functio n φ , we constru ct a function Lift ( f , φ ) on ( k + 1 ) n inp ut bits as follo w s. Gi ve n n -bit inpu ts x , y 1 , . . . , y k , φ is e valuat ed on the latt er k inputs to select a set of m bits in x on which we apply f . Formally , Lift ( f , φ )( x , y 1 , . . . , y k ) = f ( x | φ ( y 1 , . . . , y k )) , where for a set S ⊆ [ 1 , n ] , x | S denotes the substrin g of x index ed by the elements in S . W e are interested in the communica tion comple xity of Lift ( f , φ ) in the NOF model with k + 1 players , where player 0 gets x and player s 1 th rough k get y 1 throug h y k , respe cti vel y . 2.3 Corr elation, Fourier Rep re sentation a nd Degr ee Let f , g : { 0 , 1 } m → R . Let µ be a di strib utio n on the set { 0 , 1 } m . W e define th e corr elation betwee n f and g under µ to be corr µ ( f , g ) = E x ∼ µ [ f ( x ) g ( x )] . Whene ve r we omit to mention a spec ific distrib ution when computin g the correlati on, an ex pected v alue or a probabilit y , it is to be assumed that we are talking abou t the uniform distr ib ution. For S ⊆ [ 1 , m ] , let χ S ( x ) = ( − 1 ) ∑ i ∈ S x i be the Fourier character of the set S . Let f : { 0 , 1 } m → R and let f S = corr ( f , χ S ) . Then f ( x ) = ∑ S ⊆ [ 1 , m ] f S χ S ( x ) is th e Fourier repre sentati on of f . The exact de gr ee of f is the size o f the lar gest S su ch that f S is non-zero . T he ε -appr oximate de gr ee of f , denoted by de g ε ( f ) is the smallest d for which there exists a func tion g o f exact de gree d such that m ax x | f ( x ) − g ( x ) | ≤ ε . 2.4 Set Families Let S = ( S 1 , . . . , S z ) be a multi -set of m -elemen t su bsets of [ 1 , n ] . Let the rang e of S , denote d by S S , be the set o f indices from [ 1 , n ] that appear in at least one s et in S . Let t he boun dary of S , denoted by ∂ S , be the set of indic es from [ 1 , n ] that app ear in exactl y on e set in the collectio n S . 3 Statement of Results Our main technic al resu lt is the follo wing. Theor em 3.1. Let δ < 1 be a consta nt. Let ε = ( 1 − δ ) / 4 . L et m = n ε and let k ≤ δ · log n. Ther e e xists a functi on φ suc h that R k + 1 ( Lift ( OR , φ )) ≥ n Ω ( 1 ) . Pr oof of Main T heor em 1.1 fr om Theor em 3.1. Conside r the function φ whose existe nce is guaranteed by Theorem 3.1. On the one hand, the Theorem implies that Lift ( OR , φ ) / ∈ BPP cc k + 1 . On the other hand, the follo wing is a nonde terministi c protocol for Lift ( OR , φ ) : gu ess an index i ∈ [ 1 , n ] using log n bits; player 0 (the one holding x on its forehead) locally computes φ ( y 1 , . . . , y k ) and communi- cates a 1 if i belongs to that set; playe r 1 communic ates a 1 if x i = 1. The cost of this proto col is O ( log n ) . Easily , Lift ( OR , φ )( x , y 1 , . . . , y k ) = − 1 iff there exists a guess i such that both player s communicate a 1. Thus, Lift ( OR , φ ) ∈ NP cc k + 1 . 4 4 Pr oof of Main Result W e obtain our lower bo unds on the b ounded -error communication complexi ty of Lift ( OR , φ ) using an a nal- ysis that follo w s [7]. In their paper , Chattopa dhyay and Ada analyze the Set-Disjointne ss function, and for that reason, their masking functio n φ must be the AND function. In our case, intuiti vel y , we allo w φ to be a random function. While our results no longer apply to Set-Disjoint ness, we still obtain a separatio n between BPP cc k and N P cc k becaus e, no m atter what masking fun ction is used, L ift ( OR , φ ) a lway s has a cheap nonde terministi c protocol. At a more technical lev el, the results of [7] become tri vial when k ≥ log log n because of the relations hip between n (the size of the in put to F ) and m (the number of bits the ba se fun ction OR gets app lied to .) For their analy sis to go through , the y need n = 2 2 k m O ( 1 ) . In our case, n = m O ( 1 ) is su ffici ent, and this allo ws our results to be non-tr iv ial fo r k ≤ δ log n for any δ < 1. 4.1 Overview of Proof As ment ioned earlie r , we will sta rt with the base func tion f = OR on m input bits, m < n . W e lif t the base functi on f in order to obtain the lifted functio n F φ = Lift ( f , φ ) . Recall that F φ is a function on ( k + 1 ) n inputs with small nondetermini stic comple xity , and is obtained by applying the base function (in this case the OR function) to the unmas ked bits of Player 0’ s input, x . W e wa nt to prov e that for a rand om φ , F φ has high rando mized co mmunicatio n comple xity . Paturi [12] pro v ed that no func tion that is a sum of lo w-deg ree Fourier characters can w ell-app roximate th e OR func tion. This implies that there exi sts a fun ction g (als o on m bits ) and a distrib ution µ ov er al l m -bit inputs s uch that the funct ions g and f = OR are highly corr elated ov er µ and furt hermore, g is orthogo nal to all small Fo urier characters . This is our Lemma 4.1, and it was origin ally pro ve d usin g dual ity by Shersto v [15] in the conte xt of 2-player lower bou nds for quantum communication complexi ty . No w we li ft the f unction g in o rder to get the functio n G φ = Lift ( g , φ ) . Define λ to be a distrib utio n ov er all ( k + 1 ) n -bit inp uts that is the natural extens ion of µ . Since g and f = OR are highly co rrelated over µ , it is not hard to se e (using the definit ions and the fa ct that λ is the nat ural extens ion of µ to the lif ted space) that the lifted ve rsions, F φ and G φ are also highly correla ted o ve r λ . By the gen eralize d discrep ancy method (Lemma 4.2), in order to prov e that th e randomize d comple xity of F φ is high, it suf fices to pro ve that G φ has small discrepanc y . This final step is accomplis hed b y Lemmas 4.4, 4.5, and 4.6, using two important proper ties of g and φ . T he cru cial propert y of g that w e ex ploit is tha t it is orthog onal to the space of all small Four ier characters . This pr operty will be used to prov e Lemma 4.4. Secondly , we want φ to beh a ve lik e a random func tion with resp ect to all sub-cube s. This second pr operty is ex ploited in order to prove Lemma 4.6. W e no w proceed with the formal proof. 4.2 Pr oof of Main Theor em The follo wing lemma is from [15]. Intuiti vely it sho ws the foll o wing. Let f be a base fun ction on m bits, and w ith the propert y that no function in the low-de gree Fourier subspace can approximate f . (W e will be interested in f = OR.) The lemma states that this implies the existe nce of another function g and a distrib ution µ such that g is in the orthogon al subspace of low-de gree Fourier characte rs and g well- 5 approx imates f . Lemma 4.1 (Orthogonali ty Lemma, Lemma 5.1 in [7]) . If f : { 0 , 1 } m → {− 1 , 1 } is a functio n with δ ′ - appr oximate de gre e d , ther e ex ist a functio n g : { 0 , 1 } m → {− 1 , 1 } and a distrib ution µ on { 0 , 1 } m suc h that: (i) corr µ ( g , f ) ≥ δ ′ ; and (ii) for every T ⊆ [ 1 , m ] w ith | T | ≤ d and every functio n h : { 0 , 1 } | T | → R , E x ∼ µ [ g ( x ) · h ( x | T )] = 0 . The next lemma is the generalize d disc repanc y lemma from [7]. It states that if two functions F and G are highly corr elated, and if G has small discrepa ncy (a nd hence high communic ation comp lexit y), then the communica tion co mplex ity of F is also high. Lemma 4.2 (Generalized Discrepan cy L emma, L emma 3.2 in [7]) . Let Z = Z 1 × · · · × Z k . Let F , G : Z → {− 1 , 1 } and let λ be a distrib ution on Z such tha t co rr λ ( G , F ) ≥ δ ′ . T hen, for ev ery ε ′ < δ ′ / 2 , R k , ε ′ ( F ) ≥ log  δ ′ − 2 · ε ′ disc k , λ ( G )  . The follo w ing lemma is standard and used in ev ery discre panc y ar gument. See [2, 13, 8] for details . Lemma 4.3 (The st andard BNS ar gument) . Let Z = X × Y 1 × · · · × Y k and let F : Z → {− 1 , 1 } . Let Γ ⊆ Z be a cylinde r in ters ection. W e write y for ( y 1 , . . . , y k ) . Then,  E x , y [ F ( x , y ) 1 Γ ( x , y )]  2 k ≤ E y 0 , y 1 "      E x " ∏ u ∈{ 0 , 1 } k F ( x , y u 1 1 , . . . , y u k k ) #      # . Using the ab ov e lemmas, W e will now prov e Theorem 3.1. B y [12], deg 5 / 6 ( OR ) ≥ c √ m for some cons tant c . By Lemm a 4.1, applie d w ith f = O R, ther e e xist a func tion g a nd a distrib ution µ such that: (i) corr µ ( g , OR ) ≥ 5 / 6; and (ii) for ev ery T ⊆ [ 1 , m ] with T ≤ c √ m and ev ery funct ion h : { 0 , 1 } | T | → R , E x ∼ µ [ g ( x ) h ( x | T )] = 0. For ev ery masking functio n φ , let F φ = Lift ( OR , φ ) and let G φ = Lift ( g , φ ) . As in [7], w e define the distrib ution λ on { 0 , 1 } ( k + 1 ) n as follo ws. For x ∈ { 0 , 1 } n and y = ( y 1 , . . . , y k ) ∈ { 0 , 1 } kn , let λ ( x , y ) = µ ( x | φ ( y )) 2 ( k + 1 ) n − m . It can be easily verified tha t corr λ ( G φ , F φ ) = corr µ ( g , OR ) ≥ 5 / 6. Thus, by Lemma 4.2, R ( F φ ) ≥ log  5 / 6 − 2 ( 1 / 3 ) disc λ ( G φ )  = log  1 disc λ ( G φ )  − Θ ( 1 ) . Let Γ be the cyli nder intersecti on that witnesse s th e discrepan cy of G φ under λ . T hen, disc λ ( G φ ) = disc Γ λ ( G φ ) =   E ( x , y ) ∼ λ [ G φ ( x , y ) 1 Γ ( x , y )]   = 2 m | E x , y [ µ ( x | φ ( y )) g ( x | φ ( y )) 1 Γ ( x , y )] | 6 where the last equality follo ws from the connection between λ and the uniform distrib ution . F inally , by Lemma 4.3, we obtai n ∀ φ ,  disc λ ( G φ )  2 k ≤ 2 m 2 k E y 0 , y 1 "      E x " ∏ u ∈{ 0 , 1 } k µ ( x | φ ( y u 1 1 , . . . , y u k k )) g ( x | φ ( y u 1 1 , . . . , y u k k )) #      # . It is at this point that we di ver ge fro m the analysis in [7]. Let A = A ( y 0 , y 1 ) be the ev ent “ ∃ i such that y 0 i = y 1 i ”. Clearly , this e ve nt depen ds onl y on the choice of y 0 and y 1 . B y a simpl e union bound, Pr y 0 , y 1 [ A ] ≤ k / 2 n = 2 − n + l ogk . F urthermo re, Pr y 0 , y 1 [ A ] ≤ 1, and since | µ g | ≤ 1, E y 0 , y 1 [ . . . | A ] ≤ 1. T hus, ∀ φ ,  disc λ ( G φ )  2 k ≤ 2 − n + m 2 k + log k + 2 m 2 k E y 0 , y 1 "      E x " ∏ u ∈{ 0 , 1 } k µ ( x | φ ( y u 1 1 , . . . , y u k k )) g ( x | φ ( y u 1 1 , . . . , y u k k )) #        A # . For the remaining part of the analysis , we fix the choice s of y 0 and y 1 in such a way that the e ve nt A does not occur . For u ∈ { 0 , 1 } k , define S u = S u ( y 0 , y 1 , φ ) = φ ( y u 1 1 , . . . , y u k k ) . Let S = S ( y 0 , y 1 , φ ) be the multi-set ( S u : u ∈ { 0 , 1 } k ) . Even though th e se ts S u and the mult i-set S depend on y 0 , y 1 and φ , we will usually omit exp licitly indicating this depend ence in our proofs in order to reduce the clutter . W e define the number of conflic ts in S to be q ( S ) = m 2 k − | S S | . Intuiti vely , | S S | measures the range o f S , while m 2 k is the maximum possib le v alue for this rang e. W e use the follo w ing three L emmas to comple te our p roof. Lemma 4.4. F or ev ery y 0 , y 1 and φ , if A ( y 0 , y 1 ) and q ( S ( y 0 , y 1 , φ )) < c · √ m · 2 k / 2 , then E x " ∏ u ∈{ 0 , 1 } k µ ( x | S u ( y 0 , y 1 , φ )) g ( x | S u ( y 0 , y 1 , φ )) # = 0 . Lemma 4.5. F or ev ery y 0 , y 1 and φ , if A ( y 0 , y 1 ) , E x " ∏ u ∈{ 0 , 1 } k µ ( x | S u ( y 0 , y 1 , φ )) # ≤ 2 q ( S ( y 0 , y 1 , φ )) 2 m · 2 k . Lemma 4.6. F or ev ery y 0 , y 1 , if A ( y 0 , y 1 ) , when φ is ch osen at rand om, Pr φ [ q ( S ( y 0 , y 1 , φ )) = q | A ( y 0 , y 1 )] ≤  m · 2 k n  q . Before pro ving these Lemmas, we compl ete the proof of our main Theore m. Since the bound on dis c λ ( G φ ) holds for e ve ry φ , we can write E φ   disc λ ( G φ )  2 k  ≤ 2 − n + m 2 k + log k + 2 m 2 k E y 0 , y 1 , φ "      E x " ∏ u ∈{ 0 , 1 } k µ ( x | S u ) g ( x | S u ) #        A # . 7 Moreo ver , E y 0 , y 1 , φ "      E x " ∏ u ∈{ 0 , 1 } k µ ( x | S u ) g ( x | S u ) #        A # ≤ ∑ q ≥ 0 Pr φ [ q ( S ) = q | A ] E y 0 , y 1 , φ "      E x " ∏ u ∈{ 0 , 1 } k µ ( x | S u ) g ( x | S u ) #        A , q ( S ) = q # (by Lemma 4.4) ≤ ∑ q ≥ c √ m 2 k / 2 Pr φ [ q ( S ) = q | A ] E y 0 , y 1 , φ "      E x " ∏ u ∈{ 0 , 1 } k µ ( x | S u ) g ( x | S u ) #        A , q ( S ) = q # (becau se | g | = 1) ≤ ∑ q ≥ c √ m 2 k / 2 Pr φ [ q ( S ) = q | A ] E y 0 , y 1 , φ "      E x " ∏ u ∈{ 0 , 1 } k µ ( x | S u ) #        A , q ( S ) = q # (by Lemma 4.5) ≤ ∑ q ≥ c √ m 2 k / 2 Pr φ [ q ( S ) = q | A ] 2 q 2 m 2 k (by Lemma 4.6) ≤ ∑ q ≥ c √ m 2 k / 2  m 2 k n  q 2 q 2 m 2 k = 1 2 m 2 k ∑ q ≥ c √ m 2 k / 2  2 m 2 k n  q . W e ha ve chosen ε = ( 1 − δ ) / 4, so 1 − ε − δ = 3 ε . Furthe rmore, m = n ε and k ≤ δ log n , so m 2 k / n ≤ n − 1 + ε + δ = n − 3 ε < 1 / 4 when n is lar ge en ough. Thus, 2 m 2 k / n < 1 / 2. Using ∑ q ≥ q 0 w q = w q 0 / ( 1 − w ) ≤ 2 w q 0 for w < 1 / 2, we obtain E y 0 , y 1 , φ "      E x " ∏ u ∈{ 0 , 1 } k µ ( x | S u ) g ( x | S u ) #        A # ≤ 2 1 − c √ m 2 k / 2 2 m 2 k . Putting e ve rything together , E φ   disc λ ( G φ )  2 k  ≤ 2 − n + m 2 k + log k + 2 m 2 k 2 − m 2 k 2 1 − c √ m 2 k / 2 . For the e xpon ent of the first term, note that log k ≤ m 2 k and n ≥ 4 m 2 k , so − n + m 2 k + log k ≤ − 2 m 2 k . When m is large enough, − 2 m 2 k ≤ − c √ m 2 k / 4. For the expo nent of the s econd term, note that 1 ≤ c √ m 2 k / 4 when m is large enough, so 1 − c √ m 2 k / 2 ≤ − c √ m 2 k / 4. Thus, the sum of the two terms is at most 2 1 − c √ m 2 k / 4 . When m is lar ge enough , 1 ≤ c √ m 2 k / 8, so E φ   disc λ ( G φ )  2 k  ≤ 2 − c √ m 2 k / 8 . Therefore , the re exists some φ such that disc λ ( G φ ) ≤ 2 − c √ m / 8 . For this φ , R ( F φ ) ≥ log  1 disc λ ( G φ )  − Θ ( 1 ) ≥ Θ ( 1 ) √ m = Θ ( 1 ) n ε ≥ n Ω ( 1 ) . 8 5 Pr oofs of Lemm as Pr oof of Lemm a 4.4. W e write S u for S u ( y 0 , y 1 , φ ) and S for S ( y 0 , y 1 , φ ) . Assume q ( S ) < c √ m 2 k / 2. Let r ( S ) = | S S | be the size of the rang e of S , and let b ( S ) = | ∂ S | be the size of the boundary of S . Note that r ( S ) − b ( S ) ≤ q ( S ) because ev ery j ∈ ∪ S \ ∂ S occurs in at least 2 sets in S , thus contrib utes at least 1 to q ( S ) . Furthermore, r ( S ) + q ( S ) = m 2 k . T hen, b ( S ) ≥ r ( S ) − q ( S ) = m 2 k − 2 q ( S ) > ( m − c √ m ) 2 k . There are 2 k sets in the multi-set S so by the pi geonho le principle, there exist s v such th at | S v ∩ ∂ S | > m − c √ m . W e can write E x " ∏ u ∈{ 0 , 1 } k µ ( x | S u ) g ( x | S u ) # = E x | S v " µ ( x | S v ) g ( x | S v ) E x | [ 1 , n ] \ S v " ∏ u ∈{ 0 , 1 } k , u 6 = v µ ( x | S u ) g ( x | S u ) ## . Let T = S v \ ∂ S . So | T | ≤ c √ m . Let h = E x | [ 1 , n ] \ S v  ∏ u 6 = v µ ( x | S u ) g ( x | S u )  . Note that h is a function that depen ds only on x | T . T hen, by the proper ty (ii) of g an d µ , E x | S v [ µ ( x | S v ) g ( x | S v ) h ( x | T )] = 0. Pr oof of Lemm a 4.5. W e write S u for S u ( y 0 , y 1 , φ ) and S for S ( y 0 , y 1 , φ ) . W e see that E x " ∏ u ∈{ 0 , 1 } k µ ( x | S u ) # = E x | [ 1 , n ] \ S S " E x | S S " ∏ u ∈{ 0 , 1 } k µ ( x | S u ) ## = E x | S S " ∏ u ∈{ 0 , 1 } k µ ( x | S u ) # . Every u ∈ { 0 , 1 } k can be interpreted as an integer in the range [ 0 , 2 k − 1 ] . W ith this in mind, for 0 ≤ j ≤ 2 k − 1, let S j be the sub-mult i-set of S consisting of the sets up to and includin g S j , S j = ( S 0 , . . . , S j ) . So, S = S 2 k − 1 . Define S − 1 = / 0. For 0 ≤ j ≤ 2 k − 1, let G j = E x | S S j [ ∏ j i = 0 µ ( x | S i )] and let H j ( x | S j \ ∂ S j ) = E x | S j ∩ ∂ S j [ µ ( x | S j )] . Letting G − 1 = 1, observ e that, fo r 0 ≤ j ≤ 2 k − 1, G j = E x | S S j − 1 " j − 1 ∏ i = 0 µ ( x | S i ) ! H j ( x | S j \ ∂ S j ) # ≤ ( max ( H j )) · G j − 1 . T o obtain a bound on max ( H j ) , consider an arbitrary partiti on of [ 1 , m ] into two sets E , F . Let ν be a dis- trib utio n on [ 1 , m ] , and let ρ ( x | E ) = E x | F [ ν ( x )] . Then, ρ ( x | E ) = ∑ x | F 2 −| F | ν ( x ) = 2 −| F | ∑ x | F ν ( x ) ≤ 2 −| F | = 2 | E |− m , simply using the fact that ν is a prob ability di strib utio n. Thus, max ( H j ) ≤ 2 | S j \ ∂ S j |− m . Inducti vel y , E x " 2 k − 1 ∏ i = 0 µ ( x | S i ) # = G 2 k − 1 ≤ 2 ∑ 2 k − 1 j = 0 | S j \ ∂ S j | 2 m 2 k . Consider some index z ∈ S S . Suppose this index appears in l sets S j 1 , . . . , S j l from S , with j 1 < · · · < j l . Then, this index contrib utes exa ctly l − 1 to the expressi on ∑ 2 k − 1 j = 0 | S j \ ∂ S j | , once for ev ery j = j 2 , . . . , j l (for j = j 1 , z ∈ ∂ S j becaus e no set before S j contai ns z .) Since this holds for ev ery index z , we see that ∑ 2 k − 1 j = 0 | S j \ ∂ S j | = q ( S ) and therefor e E x [ ∏ u ∈{ 0 , 1 } k µ ( x | S u )] ≤ 2 q ( S ) − m 2 k . Pr oof of Lemm a 4.6. Fix y 0 , y 1 such th at A . The multi- set S is co nstruc ted fro m the sets S u = φ ( y u 1 1 , . . . , y u k k ) for u ∈ { 0 , 1 } k . Since A did not occur , the 2 k points w here φ gets e va luated are distinct. Furthermo re, φ is chosen at random, which is equi v alent to choos ing 2 k random m -element subsets of [ 1 , n ] . W e can ov eresti mate the number of conflicts in S as follo ws. Instead of choosing, for each subset, m elements 9 from [ 1 , n ] witho ut rep lacement, suppose we chose them with replacemen t. 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