kb/data/en.wikipedia.org/wiki/Communication_complexity-3.md

---
title: "Communication complexity"
chunk: 4/8
source: "https://en.wikipedia.org/wiki/Communication_complexity"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T14:40:16.151030+00:00"
instance: "kb-cron"
---

=== Public coins versus private coins ===
Creating random protocols becomes easier when both parties have access to the same random string, known as a shared string protocol. However, even in cases where the two parties do not share a random string, it is still possible to use private string protocols with only a small communication cost.  Any shared string random protocol using any number of random string can be simulated by a private string protocol that uses an extra O(log n) bits.
Intuitively, we can find some set of strings that has enough randomness in it to run the random protocol with only a small increase in error.  This set can be shared beforehand, and instead of drawing a random string, Alice and Bob need only agree on which string to choose from the shared set. This set is small enough that the choice can be communicated efficiently. A formal proof follows.
Consider some random protocol P with a maximum error rate of 0.1. Let 
  
    
      
        R
      
    
    {\displaystyle R}
  
 be 
  
    
      
        100
        n
      
    
    {\displaystyle 100n}
  
 strings of length n, numbered 
  
    
      
        
          r
          
            1
          
        
        ,
        
          r
          
            2
          
        
        ,
        …
        ,
        
          r
          
            100
            n
          
        
      
    
    {\displaystyle r_{1},r_{2},\dots ,r_{100n}}
  
. Given such an 
  
    
      
        R
      
    
    {\displaystyle R}
  
, define a new protocol 
  
    
      
        
          P
          
            R
          
          ′
        
      
    
    {\displaystyle P'_{R}}
  
 that randomly picks some 
  
    
      
        
          r
          
            i
          
        
      
    
    {\displaystyle r_{i}}
  
 and then runs P using 
  
    
      
        
          r
          
            i
          
        
      
    
    {\displaystyle r_{i}}
  
 as the shared random string. It takes O(log 100n) = O(log n) bits to communicate the choice of 
  
    
      
        
          r
          
            i
          
        
      
    
    {\displaystyle r_{i}}
  
.
Let us define 
  
    
      
        p
        (
        x
        ,
        y
        )
      
    
    {\displaystyle p(x,y)}
  
 and 
  
    
      
        
          p
          
            R
          
          ′
        
        (
        x
        ,
        y
        )
      
    
    {\displaystyle p'_{R}(x,y)}
  
 to be the probabilities that 
  
    
      
        P
      
    
    {\displaystyle P}
  
 and 
  
    
      
        
          P
          
            R
          
          ′
        
      
    
    {\displaystyle P'_{R}}
  
  compute the correct value for the input 
  
    
      
        (
        x
        ,
        y
        )
      
    
    {\displaystyle (x,y)}
  
.
For a fixed 
  
    
      
        (
        x
        ,
        y
        )
      
    
    {\displaystyle (x,y)}
  
, we can use Hoeffding's inequality to get the following equation:

  
    
      
        
          Pr
          
            R
          
        
        [
        
          |
        
        
          p
          
            R
          
          ′
        
        (
        x
        ,
        y
        )
        −
        p
        (
        x
        ,
        y
        )
        
          |
        
        ≥
        0.1
        ]
        ≤
        2
        exp
        ⁡
        (
        −
        2
        (
        0.1
        
          )
          
            2
          
        
        ⋅
        100
        n
        )
        <
        
          2
          
            −
            2
            n
          
        
      
    
    {\displaystyle \Pr _{R}[|p'_{R}(x,y)-p(x,y)|\geq 0.1]\leq 2\exp(-2(0.1)^{2}\cdot 100n)<2^{-2n}}
  

Thus when we don't have 
  
    
      
        (
        x
        ,
        y
        )
      
    
    {\displaystyle (x,y)}
  
 fixed:

  
    
      
        
          Pr
          
            R
          
        
        [
        ∃
        (
        x
        ,
        y
        )
        :
         
        
          |
        
        
          p
          
            R
          
          ′
        
        (
        x
        ,
        y
        )
        −
        p
        (
        x
        ,
        y
        )
        
          |
        
        ≥
        0.1
        ]
        ≤
        
          ∑
          
            (
            x
            ,
            y
            )
          
        
        
          Pr
          
            R
          
        
        [
        
          |
        
        
          p
          
            R
          
          ′
        
        (
        x
        ,
        y
        )
        −
        p
        (
        x
        ,
        y
        )
        
          |
        
        ≥
        0.1
        ]
        <
        
          ∑
          
            (
            x
            ,
            y
            )
          
        
        
          2
          
            −
            2
            n
          
        
        =
        1
      
    
    {\displaystyle \Pr _{R}[\exists (x,y):\ |p'_{R}(x,y)-p(x,y)|\geq 0.1]\leq \sum _{(x,y)}\Pr _{R}[|p'_{R}(x,y)-p(x,y)|\geq 0.1]<\sum _{(x,y)}2^{-2n}=1}
  

The last equality above holds because there are 
  
    
      
        
          2
          
            2
            n
          
        
      
    
    {\displaystyle 2^{2n}}
  
 different pairs 
  
    
      
        (
        x
        ,
        y
        )
      
    
    {\displaystyle (x,y)}
  
. Since the probability does not equal 1, there is some 
  
    
      
        
          R
          
            0
          
        
      
    
    {\displaystyle R_{0}}
  
 so that for all 
  
    
      
        (
        x
        ,
        y
        )
      
    
    {\displaystyle (x,y)}
  
:

  
    
      
        
          |
        
        
          p
          
            
              R
              
                0
              
            
          
          ′
        
        (
        x
        ,
        y
        )
        −
        p
        (
        x
        ,
        y
        )
        
          |
        
        <
        0.1
      
    
    {\displaystyle |p'_{R_{0}}(x,y)-p(x,y)|<0.1}
  

Since 
  
    
      
        P
      
    
    {\displaystyle P}
  
 has at most 0.1 error probability, 
  
    
      
        
          P
          
            
              R
              
                0
              
            
          
          ′
        
      
    
    {\displaystyle P'_{R_{0}}}
  
 can have at most 0.2 error probability.

=== Collapse of randomized communication complexity ===
Let's say we additionally allow Alice and Bob to share some resource, for example a pair of entangled particles. Using that ressource, Alice and Bob can correlate their information and thus try to 'collapse' (or 'trivialize') communication complexity in the following sense.
Definition. A resource 
  
    
      
        R
      
    
    {\displaystyle R}
  
 is said to be "collapsing" if, using that resource 
  
    
      
        R
      
    
    {\displaystyle R}
  
, only one bit of classical communication is enough for Alice to know the evaluation 
  
    
      
        f
        (
        x
        ,
        y
        )
      
    
    {\displaystyle f(x,y)}
  
 in the worst case scenario for any Boolean function 
  
    
      
        f
      
    
    {\displaystyle f}
  
. 
The surprising fact of a collapse of communication complexity is that the function 
  
    
      
        f
      
    
    {\displaystyle f}
  
 can have arbitrarily large entry size, but still the number of communication bit is constant to a single one.
Some resources are shown to be non-collapsing, such as quantum correlations  or more generally almost-quantum correlations, whereas on the contrary some other resources are shown to collapse randomized communication complexity, such as the PR-box, or some noisy PR-boxes satisfying some conditions.