On the Communication Complexity of AND Functions

Abstract

Log-rank conjecture is one of challenging problems in communication complexity. It is known to hold for some special function classes such as XOR functions f(x ⊕ y) when the outer function f is monotone, symmetric, AC <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> , low F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -degree, linear threshold or read-k. However, for AND functions f(x ∧ y), Log-rank conjecture is only known to hold when the outer function is a monotone function or a function with small alternating numbers. In this paper, we show that Log-rank conjecture holds for AND functions whose outer functions are read-once F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -degree polynomials, constant-F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -degree polynomials, and symmetric functions. For read-once and symmetric AND functions, their proof are based on the elementary analysis of Möbius sparsity of the outer functions and designing efficient protocols for computing the corresponding AND functions in terms of the Möbius sparsity of their outer functions. For a constant- F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -degree AND function f(x ∧ y), we use the concept of the variable rank to design a decision tree for computing f where the tree depth is bounded above by a polynomial in the logarithm of the Möbius sparsity of f. The communication protocol can be obtained easily from the constructed decision tree. Our results deepen the understanding of the log-rank conjecture for AND functions.