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Communication Complexity: An introduction, survey, and open problems

This talk provides an overview of communication complexity, its applications, and associated open problems. It covers communication protocols, function analysis, lower bounds, and fooling sets.

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Communication Complexity: An introduction, survey, and open problems

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  1. Communication Complexity:An introduction, survey, and open problems Paul Beame Barriers in Complexity II 8/29/10

  2. 2-party Communication Inputx  X Inputy  Y 010 11 0 Bob … … Alice f(x,y) Complexity = # of bits Alice and Bob need to exchange to compute f on XY

  3. Sample Applications

  4. Modeling data structure queries m memory cells,w bits each Query Alg Data Structure x U SU cell name  [m] cell contents [2w] . . . . . . Query time = # of rounds/2

  5. Other Applications • Area-time tradeoffs in VLSI • Few bits can flow across a small cut through the chip in unit time • Time-space tradeoffs. Space for data stream algorithms • Different portions of the computation see only part of the input and communicate via the memory • Boolean circuit depth. • Circuit for f viewed as a communication game between two players holding 0-inputs and 1-inputs for f • Constant-depth unbounded fan-in circuits • Circuits can be simulated by approximate computation over multivariate polynomials which can be evaluated efficiently by multiparty communication protocols • Proof rank • Costs for expressing preferences in combinatorial auctions • . . .

  6. This talk • Overview of basics of communication complexity • Most contained in excellent 1997 text “Communication Complexity” by E. Kushilevitz and N. Nisan [KN] • First formalized by Andrew Yao in 1979 • Many others involved in foundations • I will omit most citations that already appear in [KN] • Will gloss over some details on slides but slides will be posted on the web • Other talks today will discuss other aspects, new work, related areas, and applications

  7. Communication Protocols Defn: A protocol P with domain XY and range Z is a binary tree s.t. each internal node v is labeled by av:X[0,1] or bv:Y[0,1] and each leaf is labeled by some z∊Z. • Label of v is av Alice speaks • is bv Bob speaks • Protocol P computes a function f:XYZ

  8. 0 1 0 1 0 1 11 00 11 10 00 01 00 01 11 10 10 01 0 1 1 1 0 0 1 0 0 1 1 0 0 1 Bob 0 1 0 1 00 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 1 01 0 0 1 1 0 1 0 10 0 0 1 1 0 0 0 0 1 0 1 0 1 1 1 1 1 0 11 1 1 1 0 Alice 0 1 Communication Protocols Defn: A protocol P with domain XY and range Z is a binary tree s.t. each internal node v is labeled by av:X[0,1] or bv:Y[0,1] and each leaf is labeled by some z∊Z. Input 1111 Function matrix Mf for f

  9. D(f) = min C(P) • P computes f (Deterministic) Communication Complexity of f:XYZ Communication Complexity Defn: costC(P) = height of P = max # of bits sent on any path in P Defn: rounds(P) = max # of alternations between players on any path in P • Observation: Any function f:XY{0,1} has a trivial 1-round protocol of complexity log2X+1 • Alice sends input to Bob who computes answer

  10. 0 1 0 1 0 1 10 00 10 01 01 00 01 11 00 11 10 11 1 0 0 0 1 0 1 0 1 1 1 0 0 1 Bob 0 1 0 1 00 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 1 01 0 1 1 0 0 1 0 10 0 1 0 1 0 0 0 0 1 0 1 1 0 1 1 1 1 0 11 1 0 1 1 Alice 0 1 C(P)=4  D(f) ≤ 4 rounds(P)=3 But D(f) ≤ 3 by 1-round protocol Function matrix Mf for f

  11. Some functionsf:{0,1}n{0,1}n{0,1} • Equality and GreaterThanOrEqual • EQn(x,y)=1 iff x=y • GTn(x,y)=1 iff x≥yas binary integers • Inner Product modulo 2 • IPn(x,y)=<x,y>=(x1y1+x2y2+…+xnyn mod 2) • Set Disjointness (Intersection) • DISJn(x,y)=1 iff ixi=yi=1 (x,y [n]={1,…,n}) • Majorityn(x,y)=1 iff x1+y1+…+xn+yn≥n • Mediann(x,y)=1 iff the median of the set xy is odd D(EQn),D(GTn),D(IPn),D(DISJn)≥n D(Majorityn),D(Mediann) areO(log n)

  12. Analyzing Functions & Protocols

  13. 0 1 0 1 0 1 11 10 10 01 00 11 10 01 00 11 00 01 1 1 0 1 1 1 0 0 1 0 0 0 0 1 Bob 0 1 0 1 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 1 0 1 0 00 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 0 01 0 1 0 1 Alice 10 0 0 1 1 0 1 11 0 1 1 1 Defn: A (combinatorial) rectangle in XY is a set AB with AX and BY Lemma: The set of inputs Rv that reach any node v of a protocol is a combinatorial rectangle. Input 1111 Function matrix Mf for f

  14. 00 10 01 00 10 00 11 10 01 11 11 01 1 1 1 0 0 0 0 0 1 1 1 0 Bob 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 1 00 0 0 0 0 0 0 0 1 0 0 1 0 1 1 1 1 01 0 1 0 1 10 0 0 1 1 11 0 1 1 1 Defn: A rectangle is monochromatic for f iff all its elements have the same f value. It is a b-rectangle if this value is b. 0 1 Lemma: If protocol P computes f then its leaves partition Mf into monochromatic rectangles, one per leaf 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 Alice 0 1 Function matrix Mf for f

  15. Partition-based Lower Bounds Defn: Let Part(f) = min # of monoΧ rectangles in any partition of Mf Mono(f)= max fraction of Mf in any monoX rectangle Theorem:D(f)≥ log2Part(f) Note:Not every partition into monoX rectangles can be the result of a protocol Open: Is D(f) = O(log Part(f)) ? ≥ log2(1/Mono(f))

  16. yiyj xi b xjb Fooling Set Lower Bound Defn:(x1,y1),…,(xt,yt) is a b-fooling set for f iff (1) f(xi,yi)=b for all i (2) f(xi,yj)b or f(xj,yi)b for all ij Lemma:D(f) ≥ log2(b-fooling set size(f)) Proof: Every element of a b-fooling set must be in a different b-rectangle. b b

  17. Mf MR1 MR2 MR3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 =++ 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 0 1 0 1 0 0 0 0 0 0 1 0 Rank Lower Bound Lemma:D(f)≥log2 rank(Mf) rank(Mf) ≤ rank(MR1)+rank(MR2)+rank(MR3) = 1 + 1 + 1 ≤ Part(f) if partition is optimal

  18. 0 0 0 0 0 0 1 1 1 1 1 1 1 1 000 000 0 0 1 1 1 1 . . 0 1 . . 1 1 111 111 Consequences MEQn=I2n MGTn rank(MEQn)=rank(MGTn)=2n D(EQn),D(GTn) ≥ n (Diagonals are also 1-fooling sets)

  19. 1 1 0 0 0 1 1 1 1 0 0 0 0 000 0 1 001 0 1 010 . . 1 0 111 Consequences MDISJn 0-fooling set of size 2n: For every S[n] the pair (S, [n]-S). If ST then S([n]-T)  D(DISJn)≥n

  20. 1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 -1 -1 1 Consequences MIPnis the 0/1 version of H2n the full rank 2n2nHadamard matrix which has entries (-1)<x,y>=(-1)IPn(x,y) In fact MIPn= (J-H2n)/2 so rank(MIPn)=2n-1 D(IPn) ≥ n Lower bound n for f(x,y)=g(x1y1,…,xnyn) for every read-once g.

  21. How Good is the Log Rank Bound? Assume Boolean f: Lemma:rank(Mf)2≥ 1-fooling set size(f) Proof: Identity submatrix of MfMf Lemma:D(f) ≤ rank2(Mf)+1 Proof: Alice and Bob share row basis. Alice sends bits expressing her row of Mf in this basis to Bob log2rank(Mf) ≤ D(f) ≤ rank2(Mf)+1

  22. Log-rank conjecture Conjecture:D(f) is (log rank(Mf))O(1) Surprisingly this is equivalent to: Conjecture: For every undirected graphG, log (G) is (log rank(MG))O(1) where MG=adjacency matrix of G Theorem: [Nisan-Wigderson] log (1/Mono(f))is (log rank(Mf))O(1)

  23. Log-rank conjecture Conjecture:D(f) is (log rank(Mf))O(1) Known:[Nisan-Wigderson, Kushilevitz] Constant in exponent ≥log36 > 1.63 Proof uses properties of certain simplifications of DISJ function.

  24. Direct Sum Conjecture What if Alice and Bob want to compute f on k unrelated pairs of inputs ? • Call the new function fk • Clearly D(fk)≤k D(f) Conjecture:For all f, D(fk) isΩ(k D(f)) Lower bounds like log-rank satisfy this property A related direct sum conjecture for certain relations (based on composition) would yield strong circuit complexity lower bounds [Karchmer-Raz-Wigderson]

  25. Nondeterministic Communication Complexity

  26. Nondeterministic Communication Complexity Nondeterministic: • Target answer b • Players make guesses about what to send based on their input. • Who speaks is determined by conversation so far • One player verifies that the answer is b. (Don’t count sending b in the cost.) • Complexity Nb(f)

  27. Rectangle Covers Defn:Covb(f) = min # of b-rectangles needed to cover all b’s of Mf Lemma:Nb(f) =log2Covb(f) Proof: Each b-input is covered by a protocol rectangle, one per leaf Nb(f) ≥ log2Covb(f) Given b-rectangle cover: Alice guesses some b-rectangle R in cover consistent with x and sends its name. Bob checks that y is in columns of R.

  28. 0 0 0 0 Examples 0 1 0 1 0 0 1 1 0 1 1 1 N1(DISJn) = log2n • Any intersecting pair x and yhave some coordinate i such that xi=yi=1 Cov1(f) = n N0(DISJn) = n • Already seen 0-fooling set of size 2n • All pairs in a b-fooling set must be in different b-rectangles so Lemma:Nb(f) ≥ log2(b-fooling set size(f))

  29. 0 1 0 0 1 . 1 0 . 1 1 Examples N1(EQn) = n N0(EQn) = log2n+1 • 2n0-rectangles, one for each coordinate i where x and y could differ and for value of xi • “Diagonal” 1-fooling set of size 2n N0(GTn), N1(GTn) ≥ n • “Diagonal” 1-fooling set of size 2n • “Off-diagonal” 0-fooling set of size 2n-1

  30. Determinism vs Nondeterminism Lemma: For all Boolean f D(f) ≈ N0(f) N1(f) (Tight even for randomized protocols [Jayram-Kumar-Sivakumar]) Proof:Alice and Bob search for 0-rectangle containing (x,y). They maintain shared list of potential 0-rectangles. In each round Alice (Bob) names a 1-rectangle containing x(y) that intersects rows (columns) with ≤½ of remaining 0-rectangles, if one exists. Such a 1-rectangle rules out ≥½ of remaining 0-rectangles. Claim:f=1 Either Alice or Bob finds a 1-rectangle At most log2Cov0(f) =N0(f) rounds sending log2Cov1(f)+3=N1(f)+3 bits per round.

  31. Communication ComplexityClasses

  32. Communication Complexity Classes Analogy:polylog = efficient ≈ polynomial Consider families of functions f={fn} s.t. fn:{0,1}n {0,1}n{0,1} Defn: Pcc= { f : D(f) is logO(1)n} NPcc= { f : N1(f) is logO(1)n} coNPcc= { f : N0(f) is logO(1)n} Cor:Pcc = NPcc  coNPcc

  33. DISJ is NPcc-complete Defn:f is rectangular-reducible to g iff there is a pair of functions (a,b) s.t. fn(x,y)=1 iff g2polylog(n)(a(x),b(y))=1 Fact:NPcc contains DISJ and is closed under rectangular reductions Lemma: If f is in NPcc then f is rectangular reducible to DISJ Proof:|Cov1(fn)| is 2polylog(n) a(x) is 2polylog(n)-bit vector listing which 1-rectangles in the cover contain x b(y) is same bit vector for y DISJ(a(x),b(y))=1 iff x and y in some1-rectangle

  34. Randomized Communication Complexity

  35. ε-Error Randomized CC Alice and Bob use random bits during protocol. • Cost = # bits sent Correctness of computing f (2-sided error) • For each (x,y)XY,Pr[P(x,y)f(x,y)]≤ε< ½ Access to random bits • Private – each player flips their own coins • Complexity Rε(f) • Public– players access shared random string • Yields shared probability distribution over deterministic protocols • ComplexityRpubε(f) (≤Rε(f))

  36. Randomized Protocols for Equality R1/4 (EQn) is Θ(log n): Alice chooses random O(log n)-bit primep Sends (p, x mod p) to Bob who checks that x mod p = y mod p Correctness follows since Πp≤np> 2n Rpub1/4 (EQn) is 3: Shared random string: r1, r2 each n bits long Alice sends <x,r1>,<x,r2> to Bob who checks that these equal <y,r1>,<y,r2>

  37. Public vs Private Randomness Newman’s Lemma: Rε+δ(f) ≤Rpubε(f)+O(log n+log δ-1) Proof: By Chernoff bounds, an independent sampleS of size (n/δ)O(1) from the shared random strings approximates the protocol answers on every input of length n with error at most δ. Fix a shared sample S and use O(log n+log δ-1) random bits to choose an element of S. Alice flips these bits privately and sends them to Bob. Alice and Bob use the shared random string for the original protocol that these bits define.

  38. Randomized Protocols for GTnand Linear Inequalities • GTn(x,y)=1 iff x≥y • Protocols: • Find most significant bit i where xiyi • Use binary search on prefixes andrandomized protocol for EQm • Need to account for errors • Obvious cost O(log2n) • Using shared randomness and error accounting • R1/4(GTn)≤Rpub1/4(GTn)+ O(log n)=O(log n) • Similar ideas give efficient evaluation of linear inequalities

  39. Lower Bounds for Randomized Protocols

  40. Distributional CC Defn: For probability distnμ on XY define Dμε(f) = min {cost(P) : Pr(x,y)μ[P(x,y)f(x,y)]≤ε } Yao’s Lemma: Rpubε(f)=maxμDμε(f) Special case of von Neumann Minimax Theorem Useful direction: Rpubε(f)≥ Dμε(f) Observe:Rpubε(f) ≤ Cprobability distnπ over deterministic protocols Pof cost ≤ Cs.t. for all(x,y) in XY,PrPπ[P(x,y)f(x,y)]≤ε

  41. cost ≤ C deterministic protocols P1P2 .P4 … .. ….P80….. Pm (x,y) (x’,y’) … (x’’,y’’) inputs XY 0 0 . 0..0 0 10 … 0 0 0 0 0 Rpubε(f) ≤ C  Dμε(f) ≤ C Observe:Rpubε(f) ≤ Cprobability distnπ over deterministic protocols Pof cost ≤ Cs.t. for all(x,y) in XY,PrPπ[P(x,y)f(x,y)]≤ ε Table of Errors: Distribution π Averageπ ≤ε 0 0 .1. 0 0 0 10 … 0 . 1 0 ≤ε 0 1 . 0..0 0 10 … 0 0 0 0 1 Distribution μ ≤ε ≤ε Averageπ Averageμ Averageμ Averageπ so  Piof cost≤ C with Averageμerror≤ε

  42. Randomized Lower Bounds Strategy • Choose probability distribution μ on inputs that seems hard for f • Prove that under μany deterministic protocol of complexity C must make at least ε measure of errors in computing f • so Dμε(f) ≥ C • Apply Yao’s Lemma to conclude Rε(f) ≥ Rpubε(f) ≥ C

  43. Discrepancy Defn: Given f:XY{0,1}, distributionμ on XY, and a set R in XY • the discrepancy offon Runderμdiscμ(f,R)= |μ(Rf-1(1)) - μ(Rf-1(0))| • the discrepancy offunderμdiscμ(f)= maxR discμ(f,R) s.t. R is a rectangle on XY

  44. Mf for f with small discrepancy Discrepancy Defn: Given f:XY{0,1} and distributionμ on XY, the discrepancy offunderμ discμ(f)=maxR|μ(Rf-1(1)) - μ(Rf-1(0))| s.t. R is a rectangle on XY

  45. Discrepancy Defn: Given f:XY{0,1} and distributionμ on XY, the discrepancy offunderμ discμ(f)=maxR|μ(Rf-1(1)) - μ(Rf-1(0))| s.t. R is a rectangle on XY Small discrepancy  every rectangle either is small or is balanced w.r.t. f Small average error  most inputs are in protocol rectangles that have nearly constant f (not balanced) Both  most inputs are in small protocol rectangles  protocol has many rectangles  large Dμε(f)

  46. Discrepancy Defn: Given f:XY{0,1} and distributionμ on XY, the discrepancy offunderμ discμ(f)=maxR|μ(Rf-1(1)) - μ(Rf-1(0))| s.t. R is a rectangle on XY Theorem: Rε(f) ≥Dμε(f) ≥ log2((1-2ε)/discμ(f)) Lindsay’s Lemma: For uniform distribution μ, discμ(IPn)=2-n/2 Proof:Use fact that rows of MIPnare pairwise independent and apply Cauchy-Schwartz. Corollary:Rpubε(IPn) is Ω(n) for ε < ½

  47. Discrepancy Drawback: The lower bound is never larger than min(N0(f),N1(f))+1. Small discrepancy μ assigns roughly ½ its weight on f-1(0) and ½ on f-1(1) f-1(b) is covered by 2Nb(f)b-rectangles  There is a b-rectangle of weight ≥ 2-Nb(f)-1  Discrepancy doesn’t work well for DISJn since N1(DISJn) is O(log n)

  48. Theorem:Assume: μ(f-1(b))≥ 1/c and:every rectangle either is small (μ-measure ≤ β) or corrupted (has a ≥2cεμ-fraction of (1-b)-entries) Then: Rε(f) ≥Dμε(f) ≥ log2(β-1/c) -1 Proof: At most 1/2c of μ-measure of f-1(b) can be covered by “corrupted” b-rectangles since total error ≤ε The leaves at least 1/2c of μ-measure of f-1(b) to be covered by “small” rectangles of μ-measure ≤ β. At least β-1/2c rectangles needed. Corruption/Rectangle Bound

  49. Randomized CC of DISJn Theorem: There is an ε>0 and (non-product) distribution μ such that Dμε(DISJn) is Ω(n) Proof:μ only puts weight on inputs (x,y) such that |xy|=0 or 1 (not product). Show that any rectangle of density >2-δn must contain a constant fraction of 1-inputs. Note: Best bound possible for a product distribution is Ω(√n) Applications: Theorem[Raz-Wigderson] Bipartite Perfect Matching requires depth Ω(n) monotone circuits Theorem[Nisan],[Nisan-Segal] Combinatorial auctions require exponential communication even for simple preferences

  50. Generalized/Smooth Discrepancy Defn: A distribution μ on inputs defines a distance between functions: Δμ(f,g)=Prμ[fg] Theorem:[Klauck 01] If Δμ(f,g)≤δthen Rε(f) ≥Dμε(f)≥ log2((1-2ε+2δ)/discμ(g)) Applications:[Razborov 03],[Sherstov 08] Lower bounds for DISJn and other functions with large approximate degree

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