On the role of private coins in unbounded-round Information Complexity

Information Complexity Alexander Koza hinsky∗ ∗Mos ow State University, Fa ulty of Me hani s and Mathemati s kozla h mail.ru 2014 Abstra t We prove a version of "Reversed Newman Theorem" in ontext of information omplexity: every privateoin ommuni ation proto ol with information omplexity I and ommuni ation omplexity C an be repla ed by publi oin proto ol with the same behavior so that it's information omplexity does not ex eed O (√IC). This result holds for unboundedround ommuni ation whereas previous results in this area dealt with one-way proto ols. As an appli ation it gives an undire t way to prove a best-known ompression theorem in Information Complexity. 1 Introdu tion Information omplexity of ommuni ation proto ol π, denoted by ICμ(π), is the amount of information Ali e and Bob reveal about their inputs while omputing π in a assumption that input are distributed a ording μ. Information omplexity is useful foremost in ontext of a Dire t-Sum problem in Communi ation omplexity. Let us rstly des ribe the substan e of this problem. Fix a small onstant ǫ. Suppose that you are given an arbitrary fun tion f : X ×Y → {0, 1} and probability distribution μ on the set X × Y, (here X is orresponded to Ali e and Y is orresponded to Bob). De ne D ǫ (f) as follows: D ǫ (f) = inf π CC(π), where in mum ranges over all deterministi ommuni ation proto ols π whi h output 1 bit π(x, y), su h that μ {(x, y) |π(x, y) 6= f(x, y)} ≤ ǫ. Imagine then, that you task is to ompute n opies of f in parallel. Consider fun tion f : (X × Y) → {0, 1}n and probability distribution μ on the set (X × Y)n, whi h are de ned as follows: f ((x1, y1), . . . , (xn, yn)) = (f(x1, y1), . . . , f(xn, yn)) , 1