Greengard's N-Body Algorithm is not Order N

Greengard's N -body algorithm claims to compute the pairwise interactions in a system ofN particles in O(N) time for a xed precision. In this paper, we show that the choice of precision is not independent of N and has a lower bound of logN . We use this result to show that Greengard's algorithm is not O(N). The N -body problem constitutes simulating the motion of N particles under the in uence of mutual gravitational interactions. Since the problem cannot be solved in closed form, a discrete approximation has to be used. The force on every particle due to the rest of the system is computed and this information is used to update the system over a small interval of time t. This constitutes one iteration in the solution of an N -body problem. A straightforward computation of the interactions takes O(N) time per iteration, which is prohibitive since physicists want to simulate the motion of large collections of particles over long time scales. A number of algorithms have been designed to reduce the time complexity per iteration. The main principle behind these algorithms is approximating the force between two collections of particles that are far apart, without having to compute every pairwise interaction. With the notable exception of Greengard [1], most researchers paid little attention to a rigorous worst-case analysis of the complexity of their algorithms. Greengard's algorithm remains the only algorithm so far with a proven worst-case complexity of O(N). We limit the discussion to the two-dimensional version of Greengard's algorithm for convenience and simplicity. The rst step in Greengard's algorithm is creating a hierarchical subdivision of the space containing the This work is supported by the Applied Mathematical Sciences Program of the Ames Laboratory-USDOE under contract No. W-7405-ENG-82.