Trade-offs between communication and space

Abstract

This paper initiates the study of communication complexity when the processors have limited work space. The following trade-offs between the number C of communications steps and space S are proved: 1. 1. For multiplying two n × n matrices in the arithmetic model with two-way communication, CS = Θ(n 3). 2. 2. For convolution of two degree n polynomials in the arithmetic model with two-way communication, CS = Θ(n 2). 3. 3. For multiplying an n × n matrix by an n-vector in the Boolean model with one-way communication, CS = Θ(n 2). In contrast, the discrete Fourier transform and sorting can be accomplished in O(n) communication steps and O(log n) space simultaneously, and the search problems of Karchmer and Wigderson associated with any language in NC k can be solved in O(log k n) communication steps and O(log k n) space simultaneously.