Communication complexity concerns the following scenario. There are two players with unlimited computational power, each of whom holds an n bit input, say x and y. Neither knows the other's input, and they wish to collaboratively compute f(x, y) where the function f:{0, 1}n × {0, 1}n → {0, 1} is known to both. Furthermore, they had foreseen this situation (e.g., one of the parties could be a spacecraft and the other could be the base station on earth), so they had already–before they knew their inputs x, y–agreed upon a protocol for communication. The cost of this protocol is the number of bits communicated by the players for the worst-case choice of inputs x, y.

Researchers have studied many modifications of this basic scenario, including randomized protocols, nondeterministic protocols, and average-case protocols. Furthermore, lower bounds on communication complexity have uses in a variety of areas, including lower bounds for parallel and VLSI computation, circuit lower bounds, polyhedral theory, data structure lower bounds, and more. Communication complexity has been one of the most successful models studied in complexity, as it strikes the elusive balance of being simple enough so that we can actually prove strong lower bounds, but general enough so we can obtain important applications of these lower bounds.