Lower bounds from complex function theory

We employ elementary results from the theory of several complex variables to obtain a quadratic lower bound on the complexity of computing the mean distance between points in the plane. This problem has 2N inputs and a single output and we show that exactly N(N-1)/2 square roots must be computed by any program over +, -, ×, ÷,) √, log and comparisons, even allowing an arbitrary field of constants. The argument is based on counting the total number of sheets of the Riemann surface of the analytic continuation to the complex domain of the (real) function computed by any algorithm which solves the problem. While finding an exact answer requires O(N2) operations, we show that an ε-approximate solution can be obtained in O(N) time for any ε ≫ 0, even if no square roots are permitted.