Upper and Lower Bounds on the Power of Advice

Transcription

1 Upper and Lower Bounds on the Power o Advice Aradev Chattopadhyay Je Edmonds Faith Ellen Toniann Pitassi March 1, 2016 Abstract Proving superpolylogarithmic lower bounds or dynamic data structures has remained an open problem despite years o research. Pǎtraşcu proposed an exciting approach or breaing this barrier via a two player communication model in which one player gets private advice at the beginning o the protocol. He gave reductions rom the problem o solving an asymmetric version o set-disjointness in his model to a diverse collection o natural dynamic data structure problems in the cell probe model. He also conjectured that, or any hard problem in the standard two-party communication model, the asymmetric version o the problem is hard in his model, provided not too much advice is given. In this paper, we prove several surprising results about his model. We show that there exist Boolean unctions requiring linear randomized communication complexity in the two-party model, or which the asymmetric versions in his model have deterministic protocols with exponentially smaller complexity. For set-disjointness, which also requires linear randomized communication complexity in the two-party model, we give a deterministic protocol or the asymmetric version in his model with a quadratic improvement in complexity. These results demonstrate that Pǎtraşcu s conjecture, as stated, is alse. In addition, we show that the randomized and deterministic communication complexities o problems in his model dier by no more than a logarithmic multiplicative actor. We also prove lower bounds in some restricted versions o this model or natural unctions such as set-disjointness and inner product. All o our upper bounds conorm to these restrictions. Moreover, a special case o one o these lower bounds implies a new proo o a strong lower bound on the tradeo between the query time and the amortized update time o dynamic datastructures with non-adaptive query algorithms. 1 Introduction In the cell probe model [25, 9, 34], the complexity o an algorithm is measured by the number o (ixed size) memory cells it accesses. Lower bounds in the cell probe model have been obtained or numerous static data structure problems, yielding lower bounds or these problems in the unit cost random access machine. Obtaining lower bounds or dynamic data structures in the cell probe model has been a challenge. In 1989, Fredman and Sas [10] introduced the chronogram method and used it to prove an Ω(log n/ log log n) lower bound on the worst case time per operation or the partial sums problem. Tata Institute o Fundamental Research, aradev.c@tir.res.in Yor University, je@cs.yoru.ca University o Toronto, aith@cs.toronto.edu University o Toronto, toni@cs.toronto.edu

2 In 1998, Alstrup, Huseldt and Rauhe [1] got the same bound or the dynamic mared ancestor problem. Reductions rom these problems to a variety o other dynamic data structure problems have also been obtained [1, 12, 13, 14]. In 2004, Pǎtraşcu and Demaine [30] introduced a beautiul inormation theoretic technique to prove Ω(log n) lower bounds or the partial sums problem and dynamic connectivity in undirected graphs. Later, Pǎtraşcu [29] used a reduction rom setdisjointness in an asymmetric two-party communication model to prove an Ω(log n/(log log n) 2 ) lower bound or the dynamic mared ancestor problem. More recently, Larsen proved the irst Ω(log 2 n) lower bounds, or the dynamic range counting problem [21] and dynamic polynomial evaluation (over a large ield) [22]. Despite these advances, it remains a longstanding open problem to prove polynomial (or even super-polylogarithmic) lower bounds or any dynamic data structure problem. 1.1 Pǎtraşcu s Conjecture Pǎtraşcu [28] listed a diverse collection o natural dynamic data structure problems that are conjectured to require superpolylogarithmic time per operation, including determining the existence o paths in dynamic directed graphs and inding the length o shortest paths in dynamic undirected graphs. He proposed an exciting new approach or obtaining polynomial lower bounds or all o these problems using a new communication model that we call the A B (B C) model. It augments the standard two-party communication model between two players Bob and Charlie, by providing advice (given by Alice) to one o the players (Bob). For any Boolean unction : X Y {0, 1}, Pǎtraşcu deined an asymmetric communication problem SEL 1 : {1,..., } X Y {0, 1}, where SEL 1 (i, x 1,..., x, y) = (x i, y). In the A B (B C) model, there are two players, Bob and Charlie, who, with advice rom Alice, compute SEL 1 (i, x 1,..., x, y) as ollows: Alice receives x 1,..., x and y, Bob receives y and i, and Charlie receives x 1,..., x and i. Alice irst sends some advice privately to Bob and then remains silent. Thereater, Bob and Charlie can communicate bac and orth, alternating arbitrarily, until they have computed the output o the unction. The last bit that is sent is the output o the protocol, which is supposed to be the value o the unction. This can also be viewed as a restricted version o a three-player number-on-the-orehead problem, in which Alice has i on her orehead, Bob has x 1,..., x on his orehead, and Charlie has y on his orehead. Pǎtraşcu presented simple reductions rom the problem o computing SEL 1 DISJ in the A B (B C) model, where DISJ denotes the set-disjointness problem, to many dynamic problems in the cell probe model. These reductions prove that, i SEL 1 DISJ cannot be solved by a protocol in which Alice gives o(ntw) bits o advice and Bob and Charlie communicate a total o o(tw) bits, then the worst case time per operation o the dynamic problems is Ω(t) in the cell probe model with w bit words. He conjectured that there exist positive constants δ < 1 and γ > 1+δ such that SEL 1 DISJ cannot be solved or Θ(n γ ) i Alice gives o(n 1+δ ) bits o advice and Bob and Charlie communicate a total o o(n δ ) bits. I his conjecture is true, then all o the dynamic problems presented in [28] require n Ω(1) time per operation in the cell probe model with O(log n) bit words. More generally, he stated the ollowing conjecture, which does not speciy whether the communication protocols involved are deterministic or randomized. Conjecture 1 (Pǎtraşcu) Let : {0, 1} n {0, 1} n {0, 1} be any unction. Consider a protocol 2

3 π or computing SEL 1 in the A B (B C) model. I Alice sends o() bits, then the cost o communication between Bob and Charlie is Ω(c), where c is the 2-party communication complexity o. The intuition is that, i Alice sends o() bits o advice, then, or many o the instances (x 1, y),..., (x, y), she is providing very little inormation. This suggests that, in the worst case, solving one o the these instances should be essentially as hard as computing in the standard two-party model. Furthermore, the generality o this conjecture, namely that it maes no assumptions about the structure o, invites the possibility o an inormation theoretic round elimination argument. 1.2 Reutations To our surprise, this intuition is not correct. While it is true that Alice cannot provide much inormation about the x i s, it turns out that she can provide a succinct message that will help Charlie learn y. This is the main intuition behind all o our upper bounds. For example, it is easy to disprove Pǎtraşcu s conjecture or deterministic protocols by considering the equality unction, EQ, where EQ(x, y) = 1 i and only i x = y. It has a very simple deterministic protocol in which Alice sends Bob the minimum j {1,..., } such that y = x j. I there is no such j, she sends him 0. Bob orwards this message to Charlie, who can determine that the output should be 1 i and only i he receives j 0 and x j = x i. Here, Alice teaches y to Charlie (via Bob) using a very short message. We exploit this intuition to prove a much stronger result, using notions rom learning theory and recent results about sign matrices. Speciically, we show that, even i a Boolean unction has large randomized complexity in the two-party model, SEL 1 can have small deterministic complexity in the A B (B C) model. Theorem 2 There exists a Boolean unction with two-party randomized communication complexity Ω(n) such that SEL 1 has a deterministic protocol in the A B (B C) model in which the total number o bits communicated is O(log 2 ). Note that when n O(1), the total amount o communication is O(log 2 n). Interestingly, we prove the upper bound using the harder side o Yao s min-max principle. Although it is standard to use the min-max principle or proving lower bounds, we are not aware o its application to prove upper bounds, especially or communication protocols. A natural hope would be that Pǎtraşcu s conjecture is still true or certain speciic Boolean unctions with Ω(n) two-party randomized complexity, such as set-disjointness. Our next result shows that this is not the case or set-disjointness. We directly design a protocol or set-disjointness, in which Alice reveals a careully chosen subset o y s bits so that, on the remaining bits, determining DISJ(x i, y) is easy, or each i {1,..., }, because either x i has ew 1 s or a large raction o the positions o 1 s in x i are also positions o 1 s in y. Theorem 3 There is a deterministic protocol or SEL 1 DISJ in the A B (B C) model, in which Alice sends at most n log bits, Bob sends at most 1 + n log bits, and Charlie sends at most n log n bits. 3

4 Note that when n O(1), the total amount o communication is O ( n log n ). It is worth remaring that Theorem 3 does not eliminate the possibility o proving strong lower bounds or dynamic data structure problems via the A B (B C) model. To obtain polynomial lower bounds or the dynamic problems listed above, it suices to prove that, or every protocol in which Alice sends o() bits o advice, Bob and Charlie must communicate Ω(n δ ) bits to compute, or some constant 0 < δ 1/2. Theorem 2 and Theorem 3 show that such a lower bound argument has to crucially use the structure o the set-disjointness unction. We also show that the randomized and deterministic communication complexities o computing SEL 1 in the A B (B C) model do not dier by much. Furthermore, or any Boolean SEL 1 DISJ unction with two-party randomized communication complexity R, we show that SEL 1 deterministic communication complexity O((R + log n + log ) log ) in the A B (B C) model. This immediately shows that problems which, in the 2-party model, have eicient randomized protocols, but are hard deterministically, give rise to easy asymmetric problems in the A B (B C) model. 1.3 Restricted Lower Bounds Finally, we provide lower bounds in the A B (B C) model or some restricted classes o protocols, which include those protocols used or our upper bounds in Theorem 2 and Theorem 3. In those protocols, Alice sends ar ewer bits o advice than she is allowed to. Moreover, ater Alice s message is sent, Bob and Charlie engage in a very limited orm o interaction. Our lower bounds show that each o these restrictions, by itsel, does not allow improvements in our upper bounds. For analyzing protocols where Alice s advice is less than n bits, we convert the problem into a direct product problem with n instances. Then we obtain our lower bounds using recent strong direct product theorems. For analyzing restricted interactions between Bob and Charlie, we present an inormation theoretic argument. We show that, i there is a limited interaction protocol in which Bob and Charlie communicate ew bits, then x 1,..., x can be compressed to substantially ewer than n bits, which is impossible, in general. While limited interaction protocols are a natural restriction o general ones, there is an additional motivation to study them. In particular, they are very related to restricted dynamic data structure algorithms. Consider Pǎtraşcu s three phase dynamic problem [28]. In the irst phase, data gets inserted and the algorithm pre-processes this eiciently. The second phase consists o a series o updates which are each processed in amortized time t u. The third phase gets a query and the algorithm outputs the answer in time t q. Suppose that the third phase is non-adaptive in the sense that, or each query, there is a ixed set o cells that are probed, which does not depend on the results o the queries. All other phases have no restrictions. Can one prove lower bounds or such algorithms? Ater the preliminary version o our wor [7] appeared, Brody and Larsen [6] proved strong lower bounds or such algorithms, which are independent o the pre-processing time allowed in the irst phase o the algorithm. We show that these lower bounds ollow rom our lower bounds on restricted protocols. More precisely, we observe that every non-adaptive query algorithm yields a very restrictive, non-interactive protocol in the A B (B C) model: In these protocols, Bob sends no messages at all. Our lower bounds or less restricted protocols then imply their non-adaptive lower bounds. has 4

5 1.4 Organization In Section 2, we introduce notation and deine necessary concepts rom two-party communication complexity. In Section 3.1, we prove that randomization does not help or solving SEL 1 in the A B (B C) model. Then, in Section 3.2, we present a unction with Ω(n) randomized twoparty complexity such that SEL 1 can be computed deterministically with only log O(1) bits o communication. In Section 3.3, we prove our upper bound or set-disjointness. In Section 4, we prove lower bounds in restricted settings and, in Section 5, we apply our lower bounds to prove strong polynomial lower bounds or nonadaptive dynamic data structure problems. We conclude in Section 6 with some open problems. 2 Preliminaries Communication complexity was irst studied or the two-party model in which the input is partitioned between two players, who compute a Boolean unction o their inputs [36]. At the end o the computation, both players now the value o the unction. For any unction : X Y Z, we use D() to denote the deterministic complexity o, which is the minimum over all deterministic protocols π that compute correctly on all inputs, o the maximum number o bits communicated during any execution o the protocol. I µ : X Y [0, 1] is a probability distribution and 0 < ɛ < 1, we use D ɛ µ() to denote the ɛ-error distributional complexity o or distribution µ, which is the minimum over all deterministic protocols that compute a unction g diering rom on a set o inputs with probability at most ɛ, o the maximum number o bits communicated during any execution o the protocol. Note that D ɛ µ() D() n+1 or any : X Y Z, µ : X Y [0, 1], and 0 < ɛ < 1, since one player can send its input to the other player, who responds with the answer. The computation o a randomized two-party protocol can be expressed as a unction o the players inputs, x and y, and a public (shared) sequence r o random bits that is provided to both players. A protocol π or has error probability ɛ i max{pr[π(x, y, r) does not compute (x, y)] x X, y Y } = ɛ, where the probability is taen over all choices o r. The ɛ-randomized complexity o, which we denote by R ɛ (), is the minimum over all randomized protocols or with error probability at most ɛ, o the maximum number o bits communicated during any execution o the protocol. Yao [35] gave the ollowing relationship between randomized and distributional communication complexities. It is oten called Yao s min-max principle. Theorem 4 For any unction and any 0 < ɛ < 1, R ɛ () = max µ { D ɛ µ () }. In act, the proo o Theorem 4 [20] applies to any reasonable non-uniorm model o computation with public coins. There are simple Boolean unctions that have very high deterministic complexity in the 2-party model, but have eicient 2-party randomized protocols. For example, consider the equality unction, EQ : {0, 1} n {0, 1} n {0, 1}, where { 1 i x = y EQ(x, y) = 0 i x y. 5

6 D(EQ) Θ(n), but R ɛ (EQ) O(1) or any constant 0 < ɛ < 1 2. Another example is the greater than unction, GT : {0, 1} n {0, 1} n {0, 1}, deined by { 1 i x > y GT(x, y) = 0 i x y, where x and y are interpreted as n-bit integers. D(GT) Θ(n), but R ɛ (GT) O(log n) or any constant 0 < ɛ < 1 2. [27]. Thus, in the context o two-player games, randomization can be substantially more powerul than determinism. The set-disjointness problem on n-bit strings, denoted by DISJ : {0, 1} n {0, 1} n {0, 1}, is deined by n DISJ(x, y) = (x[i] y[i]). i=1 In other words, i x and y are viewed as the characteristic vectors o two subsets o {1,..., n}, then DISJ(x, y) = 1 i and only i the two subsets are disjoint, i.e., or all i {1,..., n}, either x[i] = 0 or y[i] = 0. Babai, Franl and Simon [2] were the irst to prove a randomized lower bound o Ω( n) or setdisjointness. The lower bound was improved to Θ(n) by the celebrated result o Kalyanasundaram and Schnitger [15] and later simpliied by Razborov [31]. Subsequently, Bar-Yosse et.al. [3] gave an elegant inormation theoretic proo o the bound, which has been very inluential in the current development o inormation complexity. Newman [26] proved that any randomized two-party communication protocol (with public randomness) in which each player has an input in {0, 1} n can be simulated by a two-party protocol that uses O(log n) random bits. Implicit in Newman s proo is a more general result, which holds or any nonuniorm model o computation, such as communication protocols, boolean circuits, decision trees and non-uniorm Turing machines: Theorem 5 I there is a randomized computation or a unction with domain U and error probability at most ɛ < 1/2, then there is a randomized computation or that unction with the same cost and error probability O(ɛ) that uses only O(log log U ) random bits. Proo: Let F (u, r) denote the randomized computation o a unction with input u rom domain U and the (ininite) public sequence r o random bits. Suppose F has error probability at most ɛ, i.e. or all inputs u U, the probability that F (u, r) computes (u) is at least 1 ɛ, where the probability is taen over the choices, r, or the public binary sequence. We show that there exist δ O(ɛ) and t O(log U ) binary sequences r 1,..., r t such that, or each input u, i we choose a sequence r at random rom r 1,..., r t, then F (u, r) computes (u) with probability at least 1 δ. Suppose r 1,..., r t are chosen independently at random rom the space o binary sequences. For any input u U and any i {1,..., t}, Pr [F (u, r i ) computes (u)] 1 ɛ, so E [ # { i {1,..., t} F (u, r i ) computes (u) }] (1 ɛ)t. By the Cherno bound [18], or all 0 < δ < 1, Pr [ # { i {1,..., t} F (u, r i ) computes (u) } < (1 δ )(1 ɛ)t ] < e (1 ɛ)t(δ ) 2 /2. 6

7 Let δ = 1/2, let δ = (1 + ɛ)/2 O(ɛ), and let t = 8(1 ɛ) 1 ln U O(log U ) since ɛ < 1 2. Then (1 δ )(1 ɛ) = 1 δ and (1 ɛ)t(δ ) 2 /2 = ln U, so Pr [ # { i {1,..., t} F (u, r i ) computes (u) } < (1 δ)t ] < 1/ U. The union bound implies that Pr [ exists u U such that # { i {1,..., t} F (u, r i ) computes (u) } < (1 δ)t ] < 1. Hence, there exist choices o r 1,..., r t such that, or every input u U, # { i {1,..., t} F (u, r i ) computes (u) } (1 δ)t. On any input u U, the computation F chooses i {1,..., t} uniormly at random and perorms the computation F (u, r i ). Then Pr[F (u, i) computes (u)] 1 δ, so F (u, i) computes with error probability δ O(ɛ). 3 Upper Bounds in the A B (B C) model For any protocol π in the A B (B C) model, we deine CC A B (π) to be the worst case number o bits sent by Alice and CC B C (π) to be the worst case number o bits communicated between Bob and Charlie. 3.1 Alice can Derandomize We begin by showing that every randomized protocol or SEL 1 be eiciently derandomized. in the A B (B C) model can Theorem 6 Consider any Boolean unction : {0, 1} n {0, 1} n {0, 1}. Let π be a randomized protocol or SEL 1 in the A B ( ) ( ) (B C) model with CC A B π = m, CCB C π = l,. Then, there ex- ( such that CC A B π ) O((m+log +log n)(log )/ɛ 2 ), ( CC B C π ) O((l + log + log n)(log )/ɛ 2 ( ), and CC C B π ) O(q(log )/ɛ 2 ). ( ) CC C B π = q, and error probability at most 1 2 ɛ, or some constant 0 < ɛ < 1 2 ists a deterministic protocol π or SEL 1 Proo: By Theorem 5, we may assume that π uses only h O(log + log n) random bits. Let t (1 + 2ɛ)(ln )/ɛ 2 be an odd integer. Choose t strings r 1,..., r t independently at random rom {0, 1} h. Let x {0, 1} n, y {0, 1} n, and i {1,..., }. Then, or each j {1,..., t}, Pr[π(x, y, i, r j ) outputs (x i, y)] ɛ. Let δ = 1 1/(2ɛ + 1). Then (1 δ)( ɛ) = 1 2, so by the Cherno bound, Pr [ # { j {1,..., t} π(x, y, i, r j ) does not output (x i, y) } < t/2 ] < e ( 1 2 +ɛ)tδ2 /2 e ln = 1/. Hence, there is a nonzero probability that, or all i {1,..., }, π(x, y, i, r j ) outputs (x i, y) or the majority o j {1,..., t}. Thus, given x and y, Alice can ind a sequence o t Θ((log )/ɛ 2 ) strings r 1,..., r t {0, 1} h or which this is true. She sends these strings to Bob, together with the messages a 1,..., a t she sends in π(x, y, i, r j ) or all j {1,..., t}. Bob orwards the strings r 1,..., r t to Charlie. Then, or all j {1,..., t}, Bob and Charlie run π(x, y, i, r j ) with Alice s message a j and tae the output that is produced most oten. 7

8 Any two-party protocol or computing a Boolean unction : {0, 1} n {0, 1} n {0, 1} is a protocol or computing SEL 1 in the A B (B C) model in which Alice sends nothing. Thus, we immediately get the ollowing corollary: Corollary 7 Let : {0, 1} n {0, 1} n {0, 1} be any Boolean unction such that R ɛ () (log n) O(1), or some constant 0 < ɛ < 1/2. I n O(1), then there exists a deterministic protocol or SEL 1 in the A B (B C) model where Alice sends O ( log 2 n ) bits to Bob and Bob and Charlie communicate (log n) O(1) bits. It ollows rom Corollary 7 that both SEL 1 EQ and SEL 1 GT have eicient deterministic protocols in the A B (B C) model. These unctions reute Pǎtraşcu s conjecture or deterministic protocols. 3.2 Alice as a Teacher Next, we show that there exists a Boolean unction with very large randomized complexity in the two-party model, or which SEL 1 has eicient deterministic protocols in the A B (B C) model. We need some deinitions rom computational learning theory. For any set S o Boolean unctions over {0, 1} n, we associate a Boolean matrix M S, whose rows are indexed by {0, 1} n and whose columns are indexed by S, such that M S [x, ] = (x). A randomized algorithm L is said to learn S with conidence δ and accuracy ɛ rom m random examples drawn rom a distribution µ on {0, 1} n i, or each S and or x 1,..., x m {0, 1} n chosen independently rom the distribution µ, given (x 1, (x 1 )),..., (x m, (x m )), L outputs a Boolean hypothesis unction h : {0, 1} n {0, 1} that, with probability at least 1 δ, is ɛ-close to, i.e. i x is chosen rom µ, then Pr [ h(x) (x) ] ɛ. The Vapni-Chervonenis (VC) dimension, vc(m), o a matrix M is the largest number d such that M has a d 2 d sub-matrix all o whose columns are distinct, i.e., each vector in {0, 1} d appears exactly once as a column in the sub-matrix. The ollowing result, nown as the VC Theorem [16], shows the relevance o VC dimension to learning. Theorem 8 Let S be a set o Boolean unctions over {0, 1} n and let µ be an arbitrary distribution on {0, 1} n. Then there exists a randomized algorithm L that learns S with conidence δ and accuracy ɛ rom m random examples drawn rom µ, where ( 1 m O ɛ log 1 δ + vc ( ) MS ɛ For any Boolean unction : {0, 1} n {0, 1} n {0, 1}, let M denote the matrix, whose rows and columns are indexed by {0, 1} n, such that M [x, y] = (x, y). I, or each y {0, 1} n, we deine the Boolean unction y : {0, 1} n {0, 1} such that y (x) = (x, y) and we let S = { y y {0, 1} n }, then M S = M. Using an elegant argument, Kremer, Nisan and Ron [19] showed that, i M has small VC-dimension, then has small distributional communication complexity under product distributions (i.e. under distributions that can be expressed as the product o two distributions over {0, 1} n ). We exploit this connection to learning theory to prove the ollowing result. log 1 ɛ ). 8

9 Theorem 9 Let : {0, 1} n {0, 1} n {0, 1} be a Boolean unction and let 0 < ɛ < 1 2 be a constant. Then, there exists a randomized protocol π or SEL 1 in the A B (B C) model with error probability at most ɛ such that ( ) ( ) ( ) 1 CC A B π, CCB C π O ɛ vc( ) 1 M log ɛ log. Proo: Using Yao s min-max principle (Theorem 4) in the A B (B C) model, our tas reduces to showing that, or every distribution µ on {1,..., } ( {0, 1} n) {0, 1} n, there exists a deterministic protocol π µ or SEL 1 ( ) ( ) with error probability at most ɛ and CC A B πµ, CCB C πµ having the desired bound. We do this by irst constructing a randomized protocol that has the required error probability over its internal coin tosses and over µ. A standard averaging argument then yields the desired deterministic protocol. Let S denote the set o unctions { y y {0, 1} n }. For any inputs x = (x 1,..., x ) {0, 1} n and y {0, 1} n, Alice can determine the conditional distribution µ x,y induced on {1,..., }. By Theorem 8, there is a randomized algorithm L that learns the unction y S with conidence and accuracy ɛ/2 rom m random examples drawn rom µ x,y, where ( 2 m O ɛ log 2 ɛ + 2vc(M S) log 2 ). ɛ ɛ Alice draws m samples i 1,..., i m rom µ x,y and sends Bob a message containing ( i1, (x i1, y) ),..., ( i m, (x im, y) ). This requires communicating at most m(1 + log ) bits. Bob transmits this message to Charlie. In learning theoretic terms, Alice, the teacher, is trying to teach y to the learning algorithm Charlie. Charlie uses the randomized algorithm L to compute a hypothesis h consistent with Alice s m examples such that the probability h(x i ) (x i, y) is at most ɛ. Note that this probability is over the random coin tosses used by Alice to sample points and over the distribution µ x,y or i. Finally, Charlie completes the protocol by sending h(x i ) to Bob. By a standard averaging argument, Alice s coin tosses can be ixed such that the resulting deterministic protocol has error probability at most ɛ or distribution µ. The above theorem shows that i M has at most polylogarithmic VC-dimension, then SEL 1 has very eicient protocols in the A B (B C) model. Using earlier results o Ben-David et.al. [5] and Linial and Shraibman [24], Sherstov [33] showed (implicitly in the proo o Theorem 3.5) that there exists a unction with high randomized communication complexity in the two-party model such that M has low VC-dimension. Theorem 10 For any constant 0 < ɛ < 1, there are unctions such that M has VC-dimension O(1) and R ɛ () Ω(n). Now, we have everything in place to prove our irst main result. Theorem 2. There exists a Boolean unction with two-party randomized communication complexity Ω(n) such that, or n 2 (log n)o(1), SEL 1 has a deterministic protocol in the A B (B C) model, in which Alice sends Bob O ( log 2 ) bits and then Bob and Charlie communicate a total o O ( log 2 ) bits. 9

10 Proo: By Theorem 10, there is a unction such that R ɛ () Ω(n) and vc ( M ) = O(1). It ollows rom Theorem 9 that SEL 1 has a randomized protocol π in the A B (B C) model in which Alice sends O(log ) bits o advice and Bob and Charlie communicate O(log ) bits. Finally, applying Theorem 6, we derandomize π to obtain a deterministic protocol π such that CC A B ( π ) = O ( log 2 ) and CC B C ( π ) = O ( log 2 ). This disproves Conjecture 1, even or randomized protocols. 3.3 An Upper Bound or Set-Disjointness We construct a protocol or SEL 1 DISJ with o(n) communication complexity in the A B (B C) model. Throughout the construction, it is helpul to view the inputs x 1,..., x, and y as subsets o {1,..., n}. Theorem 3 There is a deterministic protocol or SEL 1 DISJ in the A B (B C) model, in which Alice sends at most n log bits, Bob sends at most 1 + n log bits, and Charlie sends at most n log n bits. Proo: Given x 1,..., x, and y, Alice repeatedly pics a set rom among x 1,..., x that is: disjoint rom y and contains a least n elements that are not in the union o the sets she has already piced. Then Alice send the indices o these sets to Bob. Note that each time Alice pics a set, the number o elements in the union o the sets she has piced increases by at least n. Since these sets are all subsets o {1,..., n}, Alice pics at most n sets. A set index can be represented using log bits, so Alice sends at most n log bits to Bob. Bob orwards the inormation he receives rom Alice to Charlie. Charlie computes the union o these sets and removes them rom x i, since none o them are in y. Let x denote the resulting set, so x y = x i y. I x contains at least n elements, then x is not disjoint rom y, since, otherwise, Alice would have piced more sets and sent more indices. In this case, Charlie sends 0 to Bob and the protocol terminates. I x contains ewer than n elements, then, Charlies send 1 to Bob, ollowed by each o the elements in x. Since each element can be represented using log n bits, Charlie sends at most n log n bits to Bob. In this case, Bob computes x y = x i y and sends the answer to Charlie. Thus Bob sends at most 1 + n log bits. 4 Lower Bounds in Restricted Models An interesting act is that our upper bounds do not use the ull power o the A B (B C) model. First, Alice sends ar ewer bits than she is allowed to. Second, Bob, the receiver o Alice s advice, is merely orwarding it to Charlie without processing it in any way. Third, the algorithms in Sections 3.2 and 3.3 have limited interaction between Bob and Charlie. We now discuss the limitations that these restrictions place on the power o the A B (B C) model. In Section 4.1, we prove our upper bound or set-disjointness cannot be substantially improved, unless we allow Alice to send more than n bits o advice, even i players interact arbitrarily. In Section 4.2, we complement this by showing the upper bound or set-disjointness cannot be improved i Bob and Charlie have limited interaction. 10

11 4.1 Lower Bounds via Strong Direct Product Theorems For any Boolean unction : {0, 1} n {0, 1} n {0, 1}, let () : {0, 1} n {0, 1} n {0, 1} denote the unction such that, or all x 1,..., x, y 1,..., y {0, 1} n, () (x 1,..., x, y 1,..., y ) = ((x 1, y 1 ),..., (x, y )). Suppose that every c-bit communication protocol or has probability o success σ < 1. Then a strong direct product theorem or states that any c-bit protocol or () has success probability that is exponentially small in. There is a rich history o both positive and negative results or strong direct product theorems in complexity theory, including Yao s amous XOR Lemma (see or example [11]). Shaltiel [32] initiated the study o strong direct product theorems in communication complexity and proved a strong direct product theorem or unctions where we have lower bounds via the discrepancy method over product distributions. This includes unctions such as the inner product unction. Lee, Shraibman, and Spale [23] strengthened Shaltiel s result by proving a strong direct product theorem or unctions that have lower bounds via the discrepancy method over any distribution. There is no nown lower bound or set-disjointness via the discrepancy method, although a weaer orm o a strong direct product theorem (with suboptimal parameters) was obtained by Beame, Pitassi, Segerlind and Wigderson [4]. Finally, Klauc [17] proved the ollowing optimal strong direct product theorem or set-disjointness. Theorem 11 There exist constants 0 < β < 1 and α > 0 such that, or all 1, every randomized protocol which computes DISJ () : {0, 1} n {0, 1} n {0, 1} using at most βn bits o communication has error probability greater than 1 2 α. Using this theorem, we obtain the ollowing lower bound or asymmetric set-disjointness in the A B (B C) model. A similar lower bound can also be obtained or any Boolean unction that has a strong direct product theorem. Theorem 12 There exist constants 0 < β < 1 and α > 0 such that, in any deterministic protocol n 1 or SELDISJ in the A B (B C) model, i Alice sends at most α n bits, then Bob and Charlie must communicate at least β n bits. Proo: Let α and β be constants that satisy Theorem 11 and let = n. To obtain a contradiction, suppose that there is a deterministic protocol or SEL 1 DISJ, where Alice sends α bits o advice to Bob, and then Bob and Charlie communicate c < β bits. Using this protocol, or every distribution µ on {0, 1} {0, 1}, we construct a deterministic c-bit protocol or DISJ () : {0, 1} {0, 1} {0, 1} with error probability at most ɛ = 1 2 α, i.e., Dµ ɛ (DISJ () ) c. Then Yao s min-max principle (Theorem 4) implies that R ɛ (DISJ () ) c. This contradicts Theorem 11, the direct product theorem or set-disjointness. Consider any distribution µ : {0, 1} {0, 1} [0, 1]. Given inputs x 1,..., x, y 1,..., y {0, 1}, we create inputs x 1,..., x, y {0, 1} n or SEL 1 DISJ as ollows: y = y 1 y and, or each i {1,..., }, x i = 0 (i 1) x i 0(n i). In particular, x i is all 0 s except or its i th bloc o bits, which is x i. Let µ be the resulting distribution on {0, 1}n {0, 1} n. Alice s α-bit message partitions the space {0, 1} n {0, 1} n into 2 α equivalence classes. Let C be an equivalence class with maximal weight under distribution µ. Then µ(c) 2 α. Since the A B (B C) protocol is deterministic, it answers correctly or every input ((x 1,..., x ), y) C and i {1,..., }. 11

12 Consider the ollowing two-party protocol or DISJ () : Given x 1,..., x, y 1,..., y {0, 1}, Bob and Charlie compute the answers to SEL 1 DISJ (i, (x 1,..., x ), y), or i = 1,...,, pretending, each time, that Alice sent the message or equivalence class C. They concatenate these one-bit answers to get a -bit answer to DISJ () (x 1,..., x, y 1,..., y ). Since Bob and Charlie communicate at most c bits to compute each one-bit answer, they communicate at most c < β 2 bits in total. Let C {0, 1} {0, 1} consist o all inputs (x 1,..., x, y 1,..., y ) rom which inputs ((x 1,..., x ), y) in C are produced. Then the two-party protocol correctly computes DISJ () or all inputs in C and µ (C ) 2 α. Thus Dµ ɛ (DISJ () ) c. This lower bound matches our upper bound to within actors o log n and log. 4.2 Lower Bounds via Compression In all our upper bounds, there is limited interaction between Bob and Charlie. We say that π is a 1.5 round (m, l, q)-protocol i it proceeds in the ollowing way: First, as usual, Alice sends m bits A = A(X, y) to Bob. Then there are two rounds o communication between Bob and Charlie. In the irst round, Bob communicates l bits B = B 1 (y, A) to Charlie that do not depend on i. (For example, Bob could orward l bits o Alice s message to Charlie.) In the second round, Charlie communicates q bits C(x 1,..., x, i, B) bac to Bob. Finally, using his nowledge o i, Bob computes the answer B 2 (y, i, A, C) = (X i, y). Note that there is no restriction on Alice s advice. The crucial restriction, beyond the act that there are only two rounds o communication ater the advice, is that Bob s communication to Charlie is independent o i. We say that this is only hal a round o communication. Interestingly, 1.5 round protocols have non-trivial power. The proo o Theorem 2, which reutes Pǎtraşcu s conjecture, employs a 1.5 round (O((log n) 2, O((log n) 2, 1)-protocol. For set-disjointness, Theorem 3 gives a 1.5 round ( n log, n log, n log n ) -protocol. It is un to veriy that unctions lie equality and greater-than can all be solved cheaply, without even using the 0.5 round communication rom Bob to Charlie, i.e. they both have deterministic (O(log n), 0, O(log n))- protocols. In this section, we show the ollowing limitations o 1.5 round protocols. Theorem 13 Let / log ω(n) and m n. For 1 q and 1 l n, every 1.5 round (m, l, q)- protocol or computing SEL 1 DISJ has l( 61nm + q ) 0.008n. I the protocol is randomized with error probability at most 1 2 ɛ, then it has ( l + log ) ( 61nm + q ) Ω ( ɛ 4 log 2 n). Note that l ( 61nm + q ) 0.008n implies that either m /61n or l (q+1) 0.008n. Since every 1.5 round (m, 0, q 0 )-protocol immediately gives a 1.5 round (m, 1, q 0 )-protocol, it ollows that every 1.5 round (m, 0, q 0 )-protocol or computing SEL 1 61nm DISJ has + q 0.008n, so either m Ω() or q Ω(n). I the protocol is randomized with error probability at most 1 2 ɛ, then either m Ω ( ɛ 4 log 2 ) or q Ω ( ɛ 4 log 2 n). The upper bound in Theorem 3 shows that Theorem 13 is tight to within logarithmic actors, when = n 2+δ or any constant δ > 0. The lower bound in Theorem 12 is incomparable, since it restricts the amount o advice Alice can send. Our next lower bound is or the well nown inner-product unction, IP(x, y) = n i=1 x iy i mod 2. The inner product unction is one o the hardest unctions in the standard two-party communication model. For example, Chor and Goldreich [8, 20] showed that, even or protocols with an inversesubexponential advantage over random guessing, inner product requires Ω(n) bits o communication 12

13 in the two party model. It is thus also a natural target or proving lower bounds on the cost o 1.5 round protocols in the A B (B C) model. Theorem 14 Let [ω(n), 2 o( n) ]. Consider any 1.5 round (m, l, q)-protocol or computing SEL 1 IP. I Alice communicates m (1 δ) bits or some δ > 0, then q + l (δ o(1))n. I the protocol is randomized with error probability at most 1 2 ɛ, then either m Ω( ɛ 2 log ) or q + l Ω ( ɛ 2 log n). Note that, i Alice communicates bits, she can send the answer or each i {1,..., }. Our lower bound or inner-product, l + q Ω(n), is stronger than our lower bound or setdisjointness, lq Ω(n). Recall that l and q are the number o bits communicated by Bob and Charlie, respectively. The naive algorithm in which Bob sends y to Charlie has l + q = n, which matches our lower bound or inner product. The algorithm or set-disjointness in Section 3.3 has lq O(n log log n), which is only logarithmic actors more than our lower bound. The main idea or proving both o these theorems is to ind an encoding o x 1,..., x using 1.5 round protocols. I the cost o the protocol is small, our encoding compresses n bits o inormation to ewer bits. However, this is impossible, since x 1,..., x has entropy n, We begin by proving Theorem 14, since compression can be done cleanly or the inner-product unction. Implementing compression or set-disjointness is more involved. Proo o Theorem 14: Assume that π is a deterministic 1.5 round (m, l, q)-protocol or computing SEL 1 IP, in which Alice communicates m (1 δ) bits. Our goal is to give a scheme or encoding x 1,..., x, where each x i is chosen uniormly rom {0, 1} n. Fix x 1,..., x. Because Alice does not now i and Bob s message B = B 1 (y, A(x 1,..., x, y)) to Charlie cannot depend on i, this message depends only on y. Hence, by averaging, there exists a message B ixed that Bob sends or at least 2 n l many y s. Thus, there exists a set, Y, o n l many linearly independent vectors such that Bob sends B ixed on each o them. Let Y be a set o l additional linearly independent vectors such that Y Y orms a basis o the vector space {0, 1} n. For each partial basis Y, we choose Y in some ixed way. Our encoding o x 1,..., x contains the ollowing: (a) the set Y, which can be represented using (n l) n bits; (b) Alice s message A(x 1,..., x, y) or each y Y, which can be represented using (n l) m bits; (c) or each index i {1,..., }, the q bit message C(x 1,..., x, B ixed, i) sent rom Charlie to Bob; and (d) extra inormation E consisting o the inner product o each x i with each y Y, which can be represented using l bits. A ey point is that Charlie s message does not depend on y Y. This is because Charlie does not see y and, or all y Y, Charlie receives the same message, B ixed, rom Bob. Given any such encoding o x 1,..., x, decoding can be done as ollows: First, simulate Bob in protocol π, or each i {1,..., } and each y Y. This is possible because y, i, Alice s message A(x 1,..., x, y), Bob s message B ixed, and Charlie s message C(x 1,..., x, B ixed, i) are all nown. Because π is a correct protocol or SEL 1 IP, the output o the protocol is IP(x i, y). From {IP(x i, y) i = 1,..., } and the inner products in E, x 1,..., x can be obtained by solving a system o linear equations o ull ran. Because no encoding o x 1,..., x can use less than n bits, it ollows that (n l) n + (n l) m + q + l n. Thus, q + l n (n l)(n+m). By assumption, 13

14 m (1 δ). Thereore q + l n (n l)( n + 1 δ) (δ n )n, which is in (δ o(1))n, since ω(n). We now extend the result to handle randomized protocols. Theorem 6 converts a randomized (m r, l r, q r )-protocol into a deterministic (m d, l d, q d )-protocol, with m d O((m r + log + log n)(log )/ɛ 2 ), l d O((l r + log + log n)(log )/ɛ 2 ), and q d O(q r (log )/ɛ 2 ). Setting δ = 0.5 gives that m d 0.5 or q d + l d (0.5 o(1))n. In the irst case, m r Ω ( ɛ 2 log m d log log n ), which is Ω ( ɛ 2 log ), since ω(n). In the second case, q r + l r Ω ( ɛ 2 log (q d + l d ) log log n ), which is Ω ( ɛ 2 log n), since ω(n) and (log ) 2 o(n). This proo can be easily adapted to prove a similar lower bound or the indexing unction, IN : {0, 1} n {1,..., n} {0, 1}, where IN ( x, y ) = x y. This unction outputs the y th bit o string x. Theorem 15 Consider any (m, 0, q) protocol or computing SEL 1 IN. I m (1 δ), or some δ > 0, then q δn. I the protocol is randomized with error probability at most 1 2 ɛ, then either m Ω ( ɛ 2 log ) or q Ω ( ɛ 2 log n). Proo: The indexing unction can be treated as a Boolean unction with domain {0, 1} n {0, 1} n, with the promise that y is an n-bit string with Hamming weight 1. In this case, IN ( x, y ) is the inner product o x and y. Then the proo is a simple specialization o the proo o Theorem 14. Since l = 0, Bob sends the same message (which consists o no bits) or every string y {0, 1} n with Hamming weight 1. These n strings are linearly independent. Let Y denote this set o strings. The encoding o x 1,..., x is the same, except or (a) and (e), which are not needed, since Y is ixed and Y is empty. It ollows that nm + q n, so m (1 δ) implies that q n nm/ n (1 δ)n = δn. The remainder o the proo, including the extension to randomized protocols, is the same as in the proo o Theorem 14. The inner-product o x i with any nown non-zero vector y provides 1 bit o inormation about x i. However, the act that DISJ(x i, y) = 0 does not provide much inormation about x i. This is the main source o complication when trying to apply the same approach to prove a lower bound or set-disjointness. On the other hand, the act that DISJ(x i, y) = 1 provides a lot o inormation about x i : all indices at which y is 1 are indices at which x i is 0. Thereore, to encode x 1,..., x eiciently, we would lie to choose a convenient set Y o y s such that DISJ(x i, y) = 1 or many i {1,..., }. Unortunately, i we choose vectors x and y uniormly at random rom {0, 1} n, then DISJ(x, y) = 0 with very high probability. Hence, we have to wor with a restricted set o vectors. This maes it delicate to ind the set Y. Let Γ x consist o the vectors in {0, 1} n with Hamming weight σ x and let Γ y consist o the vectors in {0, 1} n with Hamming weight σ y. We will consider only x 1,..., x Γ x and y Γ y. The ollowing act will be used in the proo o Claim 20 to show that, when σ x σ y is appropriately set, 85% o the vectors in Γ y do not intersect with any given vector in Γ x. Fact 16 Suppose σ x, σ y n/4. For each x Γ x, i y is chosen at random rom Γ y, then [ ] ( 4σ Pr y DISJ(x, y) = 1 exp xσ y ) n. 14

15 Proo: ( n σx ) ( σ ( y n ) = σ y Fix x Γ x. I y is chosen at random rom Γ y, the probability that x and y are disjoint is 1 σ x n ) ( 1 σ ) ( x 1 n 1 ) ( σ x 1 n σ y + 1 ) σ σy ( x > 1 4σ ) σy x, n σ y + 1 3n since σ y n/4. Now 1 t exp ( 3t ) or 0 t 1/3 and 4σ x /3n 1/3, since σ x n/4. Thereore ( ) 1 4σx σy 3n (exp( 4σx n )σy = exp( 4σxσy n ). Let Cover(x i, y) {1,..., n} denote the indices o x i that one learns are zero rom learning DISJ(x i, y). I DISJ(x i, y) = 1, then Cover(x i, y) = {j y j = 1} and, i DISJ(x i, y) = 0, then Cover(x i, y) =. For any Y Γ y, let Cover((x 1,..., x ), Y) = {(i, j) there exists y Y such that DISJ(x i, y) = 1 and y j = 1} = i {1,...,} y Y {i} Cover(x i, y). Next, we present the main lemma that enables x 1,..., x to be encoded eiciently. Lemma 17 Consider any deterministic 1.5 round (m, l, q)-protocol, where 1 l n/20. Let σ y = 5l, let σ x = 0.008n/l, and ix x 1,..., x Γ x. Then there exists a message B ixed and a set Y Γ y o size at most 30n such that, or each y Y, Bob sends B ixed to Charlie and Cover((x 1,..., x ), Y) 1 2 n. Proving this lemma needs technical wor. Beore we do that, let us see how the lower bound or disjointness ollows rom Lemma 17 via compression. We will need also the ollowing simple act: Fact 18 Let v 1,..., v, σ be positive integers such that v 1 + +v r and σ min {v i 1 i }. Then, ( ) ( ) vi r/. σ σ i=1 Proo: We will mae use o the inequality o arithmetic and geometric means: I a 1,..., a are non-negative real numbers, then ( ) a1 + + a a 1 a 2 a. Let j be any integer such that 0 j < σ and let a i = v i j or all i {1,..., }. Then, ( r ) (v i j) j ( r/ j), i=1 so, by the deinition o binomial coeicients, i=1 ( ) vi = 1 σ ( σ! ) σ 1 i=1 j=0 ( vi j ) 1 ( σ! ) σ 1 ( r/ j) j=0 ( ) r/ =. σ 15

16 Proo o Theorem 13: Let σ y = 5l and σ x = 0.008n/l. Fix any 1.5 round deterministic (m, l, q)- protocol π or SEL 1 DISJ and any x = x 1,..., x Γ x. I l > n/20, then l ( 61nm + q ) l > 0.05n 0.008n. Thus, we may assume that 1 l n/20. Then it is easy to veriy that σ x, σ y n/4. Let B ixed and Y be the message and subset o Γ y guaranteed by Lemma 17. Our encoding o x 1,..., x contains the ollowing: (a) the set Y, which can be represented using 30n n bits; (b) Bob s ixed message B ixed, which can be represented using l = 0.2σ y bits; (c) Alice s message A(x 1,..., x, y) or each y Y, which can be represented using 30n m bits; (d) or each i {1,..., }, the q bit message C(x 1,..., x, B ixed, i) sent rom Charlie to Bob; and (e) the remaining inormation E about x 1,..., x that is not learned rom Cover((x 1,..., x ), Y). For i {1,..., }, let S i = {j : (i, j) Cover((x 1,..., x ), Y). Observe that the indices or which x i has value one lie in S i. Let v i = S i. Thus, the number o possibilities or the locations o the ones o x 1,..., x is ( v 1 ) ( σ x v2 ) ( σ x v ) σ x. As Cover((x1,..., x ), Y) > 0.5n, we have v 1 + +v < 0.5n. Invoing Fact 18, inormation E can be transmitted in at most log ( ) 0.5n σ x bits. Given any such encoding o x 1,..., x, decoding can be done as ollows: First, simulate Bob in protocol π, or each i {1,..., } and each y Y. This is possible because y, i, Alice s message A(x 1,..., x, y), Bob s message B ixed, and Charlie s message C(x 1,..., x, B ixed, i) are all nown. Because π is a correct protocol or SEL 1 DISJ, the output o the protocol is DISJ(x i, y). From this, compute the indices in Cover((x 1,..., x ), Y) where x 1,..., x has zeroes. By deinition, E communicates the remaining inormation about x 1,..., x, so it is possible to decode correctly. Because no encoding o x 1,..., x can use less than its entropy H(x 1,..., x ), we have: Note that H(x 1,..., x ) = log ( n σ x ). Thus, 30n n + l + 30n m + q + H(E) H(x 1,..., x ). ( ( n ) ) σ H(x 1,..., x ) H(E) log x ( 0.5n ). σ x Using the observation that ( ) ( n σ x / 0.5n ) σ x 2 σ x, we obtain H(x 1,..., x ) H(E) σ x. Thus we have 30n n + l + 30n m + q σ x. Then, using the act that l n nm, we have 30n(n + m) + nm Since m n, l = 0.2σ y, and σ x σ y = 0.04n, it ollows that l ( 61nm + q ) l ( 30n(n + m) + nm + q σ x. + q ) lσ x = 0.2σ x σ y = 0.008n. As in the proo o Theorem 14, Theorem 6 can be used to extend the lower bound to randomized protocols, giving (l + log + log n) ( 61n (m + log + log n) + q) Ω( ɛ4 n). Since / log ω(n), log 2 it ollows that log + log n Θ(log ) and 61n log n) + q Θ(q). Hence (l + log ) ( 61nm 61n (log + log n) o(1). But q 1, so (log + + q ) Ω( ɛ4 log 2 n). 16

17 All that remains is to prove the existence o the set, Y(x 1,..., x ), promised by Lemma 17, or each x 1,..., x. We will do so in two stages, using the probabilistic method. First, we will construct an intermediate set, Y 0 (x 1,..., x ), with some nice properties. This will allow us to obtain our inal desired set by picing elements rom Y 0 at random. Lemma 19 Consider any deterministic 1.5 round (m, l, q)-protocol, where 1 l n/20. Let σ y = 5l and σ x = 0.008n/l. Then, or each x 1,..., x, there exists a set Y 0 (x 1,..., x ) such that: Bob sends the same message or each y Y 0 (x 1,..., x ), Y 0 (x 1,..., x ) 10 (0.8 ) σy Γ y, and {i {1,..., } Pr y Y0 [ DISJ(xi, y) = 1] > 0.2} > 0.7. Proo: Fix x 1,..., x. Choose B ixed at random, where the probability o choosing each message is proportional to the number o y s or which Bob sends the message. Let Y 0 denote the set o messages [ y Γ y or which Bob sends B ixed. Bob sends at most 2 l dierent messages. Hence, Pr Bixed Y0 < l Γ y ] < 1/4. It ollows that, or l suiciently large and, hence, or σ y = 5l 1 suiciently large, 4 2 l Γ y > 1 [ 4 (0.87)σy Γ y > 10 (0.8) σy Γ y. Thereore, Pr Bixed Y0 < 10 (0.8) σy Γ y ] < 1/4. The ollowing claim shows that, or a typical random message B ixed, or most messages y Y 0 and or most i {1,..., }, the sets represented by y and x i are disjoint. [ Claim 20 E Bixed {i {1,..., } Pr y Y0 [DISJ(x i, y) = 1] 0.2} ] 0.2. To prove the claim, consider any i {1,..., }. Let D i be the event that Pr y Y0 [ DISJ(xi, y) = 1 ] 0.2. Let a = Pr[D i ]. We will bound a rom above by computing Pr y Γy [ DISJ(xi, y) = 1 ] in two ways. First, note that choosing y at random rom Γ y is the same as irst choosing B ixed at random and then choosing y Y 0 at random. Hence, Pr y Γy [ DISJ(xi, y) = 1 ] = Pr[D i ] Pr y Y0 [DISJ(x i, y) = 1 D i ] + Pr[ D i ] Pr y Y0 [DISJ(x i, y) = 1 D i ] a (1 a) 1 = 1 0.8a. Since σ x, σ y n/4 and σ x σ y = 0.04n, Fact 16 implies that Pr y Γy [ DISJ(xi, y) = 1] e.16 > Hence 1 0.8a 0.85, which implies that a < < 0.2. This is true or all i {1,..., }. The claim now ollows rom the linearity o expectation. [ Applying Marov s inequality to this claim gives Pr Bixed {i {1,..., } Pr y Y0 [DISJ(x i, y) = 1] 0.2} ] /0.3 2/3. As 1/4 + 2/3 < 1, there is a non-zero probability that both Y 0 10 (0.8) σy Γ y and {i {1,..., } Pr y Y0 [ DISJ(xi, y) = 1] > 0.2} > 0.7. Let us now show why choosing Y 0 using Lemma 19 helps us construct Y. We will need one more act that ormalizes the ollowing natural intuition: i we tae a suiciently large subset o Γ y, then the distribution o the ones in the vectors o this subset is airly well spread out among {1,..., n}. For any such set S Γ y, let C(S) = { i {1,..., n} Pr y S [y i = 1] 1 2n}. 17