Communication Complexity of Sum-Type Functions Invariant under Translation

Abstract

The communication complexity of a function f denotes the number of bits that two processors have to exchange in order to compute f ( x , y ), when each processor knows one of the variables x and y , respectively. In this paper the deterministic communication complexity of sum-type functions, such as the Hamming distance and the Lee distance, is examined. Here f : X × X → G , where X is a finite set and G is an Abelian group, and the sum-type function f n : X n × X n → G is defined by f n (( x 1 , ..., x n ), ( y 1 , ..., y n )) = Σ n i =1 f ( x i , y i ) Since the functions examined are also translation-invariant, their function matrices are simultaneously diagonalizable and the corresponding eigenvalues can be calculated. This allows to apply a rank lower bound for the communication complexity. The best results are obtained for G = Z /2 Z . For prime numbers | X | in this case the communication complexity of all non-trivial sum-type functions is determined exactly. Exact results are also obtained for the parity of the Hamming distance and the parity of the Lee distance. For the Hamming distance and the Lee distance exact results are only obtained for special parameters n and | X |.