kb/data/en.wikipedia.org/wiki/Communication_complexity-4.md

title	chunk	source	category	tags	date_saved	instance
Communication complexity	5/8	https://en.wikipedia.org/wiki/Communication_complexity	reference	science, encyclopedia	2026-05-05T14:40:16.151030+00:00	kb-cron

=== Distributional complexity === One approach to studying randomized communication complexity is through distributional complexity. Given a joint distribution

    μ
  

{\displaystyle \mu }

on the inputs of both players, the corresponding distributional complexity of a function

    f
  

{\displaystyle f}

is the minimum cost of a deterministic protocol

    R
  

{\displaystyle R}

such that

    Pr
    [
    f
    (
    x
    ,
    y
    )
    =
    R
    (
    x
    ,
    y
    )
    ]
    ≥
    2
    
      /
    
    3
  

{\displaystyle \Pr[f(x,y)=R(x,y)]\geq 2/3}

, where the inputs are sampled according to

    μ
  

{\displaystyle \mu }

. Yao's minimax principle (a special case of von Neumann's minimax theorem) states that the randomized communication complexity of a function equals its maximum distributional complexity, where the maximum is taken over all joint distributions of the inputs (not necessarily product distributions!). Yao's principle can be used to prove lower bounds on the randomized communication complexity of a function: design the appropriate joint distribution, and prove a lower bound on the distributional complexity. Since distributional complexity concerns deterministic protocols, this could be easier than proving a lower bound on randomized protocols directly. As an example, let us consider the disjointness function DISJ: each of the inputs is interpreted as a subset of

    {
    1
    ,
    …
    ,
    n
    }
  

{\displaystyle \{1,\dots ,n\}}

, and DISJ(x,y)=1 if the two sets are disjoint. Razborov proved an

    Ω
    (
    n
    )
  

{\displaystyle \Omega (n)}

lower bound on the randomized communication complexity by considering the following distribution: with probability 3/4, sample two random disjoint sets of size

    n
    
      /
    
    4
  

{\displaystyle n/4}

, and with probability 1/4, sample two random sets of size

    n
    
      /
    
    4
  

{\displaystyle n/4}

with a unique intersection.

=== Information complexity === A powerful approach to the study of distributional complexity is information complexity. Initiated by Bar-Yossef, Jayram, Kumar and Sivakumar, the approach was codified in work of Barak, Braverman, Chen and Rao and by Braverman and Rao. The (internal) information complexity of a (possibly randomized) protocol R with respect to a distribution μ is defined as follows. Let

    (
    X
    ,
    Y
    )
    ∼
    μ
  

{\displaystyle (X,Y)\sim \mu }

be random inputs sampled according to μ, and let Π be the transcript of R when run on the inputs

    X
    ,
    Y
  

{\displaystyle X,Y}

. The information complexity of the protocol is

      IC
      
        μ
      
    
    ⁡
    (
    R
    )
    =
    I
    (
    Π
    ;
    Y
    
      |
    
    X
    )
    +
    I
    (
    Π
    ;
    X
    
      |
    
    Y
    )
    ,
  

{\displaystyle \operatorname {IC} _{\mu }(R)=I(\Pi ;Y|X)+I(\Pi ;X|Y),}

where I denotes conditional mutual information. The first summand measures the amount of information that Alice learns about Bob's input from the transcript, and the second measures the amount of information that Bob learns about Alice's input. The ε-error information complexity of a function f with respect to a distribution μ is the infimal information complexity of a protocol for f whose error (with respect to μ) is at most ε. Braverman and Rao proved that information equals amortized communication. This means that the cost for solving n independent copies of f is roughly n times the information complexity of f. This is analogous to the well-known interpretation of Shannon entropy as the amortized bit-length required to transmit data from a given information source. Braverman and Rao's proof uses a technique known as "protocol compression", in which an information-efficient protocol is "compressed" into a communication-efficient protocol. The techniques of information complexity enable the computation of the exact (up to first order) communication complexity of set disjointness to be

    1.4923
    …
    n
  

{\displaystyle 1.4923\ldots n}

. Information complexity techniques have also been used to analyze extended formulations, proving an essentially optimal lower bound on the complexity of algorithms based on linear programming that approximately solve the maximum clique problem. Omri Weinstein's 2015 survey surveys the subject.

== Quantum communication complexity == Quantum communication complexity tries to quantify the communication reduction possible by using quantum effects during a distributed computation. At least three quantum generalizations of communication complexity have been proposed; for a survey see the suggested text by G. Brassard. The first one is the qubit-communication model, where the parties can use quantum communication instead of classical communication, for example by exchanging photons through an optical fiber. In a second model the communication is still performed with classical bits, but the parties are allowed to manipulate an unlimited supply of quantum entangled states as part of their protocols. By doing measurements on their entangled states, the parties can save on classical communication during a distributed computation (see an application in Collapse of Randomized Communication Complexity). The third model involves access to previously shared entanglement in addition to qubit communication, and is the least explored of the three quantum models.

== Nondeterministic communication complexity == In nondeterministic communication complexity, Alice and Bob have access to an oracle. After receiving the oracle's word, the parties communicate to deduce

    f
    (
    x
    ,
    y
    )
  

{\displaystyle f(x,y)}

. The nondeterministic communication complexity is then the maximum over all pairs

    (
    x
    ,
    y
    )
  

{\displaystyle (x,y)}

over the sum of number of bits exchanged and the coding length of the oracle word. Viewed differently, this amounts to covering all 1-entries of the 0/1-matrix by combinatorial 1-rectangles (i.e., non-contiguous, non-convex submatrices, whose entries are all one (see Kushilevitz and Nisan or Dietzfelbinger et al.)). The nondeterministic communication complexity is the binary logarithm of the rectangle covering number of the matrix: the minimum number of combinatorial 1-rectangles required to cover all 1-entries of the matrix, without covering any 0-entries. Nondeterministic communication complexity occurs as a means to obtaining lower bounds for deterministic communication complexity (see Dietzfelbinger et al.), but also in the theory of nonnegative matrices, where it gives a lower bound on the nonnegative rank of a nonnegative matrix.