Abstract
The N-body problem is to simulate the motion of N particles under the influence of mutual force fields based on an inverse square law. Greengards algorithm claims to compute the cumulative force on each particle in O(N) time for a fixed precision irrespective of the distribution of the particles. In this paper, we show that Greengards algorithm is distribution dependent and has a lower bound of (N log 2 N) in two dimensions and (N log 4 N) in three dimensions. We analyze the Greengard and Barnes-Hut algorithms and show that they are unbounded for arbitrary distributions. We also present a truly distribution independent algorithm for the N-body problem that runs in O(N log N) time for any fixed dimension.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Aluru, Distribution-independent hierarchical N-body methods, Ph.D. thesis, Iowa State University, 1994.
S. Aluru, G.M. Prabhu and J. Gustafson, Truly distribution-independent hierarchical algorithms for the N-body problem, Proc. Supercomputing '94 (1994) 420–428.
R.J. Anderson, Tree Data Structures for N-Body Simulations, Proc. Annual Symposium on Foundations of Computer Science, (1996) 224–233.
A. W. Appel, An efficient program for many-body simulation, SIAM J. Sci. Stat. Comput., 6 (1985) 85–103.
J. Barnes, A modified tree code: Don't laugh; It runs, J. Comput. Phys., 87 (1990) 161–170.
J. Barnes and P. Hut, A hierarchical O(N log N) force-calculation algorithm, Nature, 324 (1986) 446–449.
S. Bhatt, M. Chen, C.Y. Len and P. Liu, Abstractions for parallel N-body simulations, Tech. Rep. DCS/TR-895, Yale University, 1992.
P.B. Callahan and S.R. Kosaraju, A decomposition of multidimensional point sets with applications to κ-nearest neighbors and n-body potential fields, J. of the ACM, 42(1) (1995) 67–90.
K. Esselink, The order of Appel's algorithm, Info. Proc. Letters, 41 (1992) 141–147.
L. Greengard, The rapid evaluation of potential fields in particle systems, MIT Press, Cambridge, MA, 1988.
L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comp. Phys., 73 (1987) 325–348.
L. Hernquist, Vectorization of tree traversals, J. Comp. Phys., 87 (1990) 137–147.
R.W. Hockney and J.W. Eastwood, Computer simulation using particles, McGraw-Hill, New York, 1981.
J. Katzenelson, Computational structure of the N-body problem, SIAM. J. Sci. Stat. Comput., 10 (1989) 787–915.
J. Makino, Vectorization of a treecode, J. Comp. Phys., 87 (1990) 148–160.
E.M. Mc Creight, Priority Search Trees, SIAM J. Comput (1985) 257–268.
R.H. Miller and K.H. Prendergast, Stellar dynamics in a discrete phase space, Astrophys. J., 151 (1968) 699–709.
R.H. Miller, K.H. Prendergast and W.J. Quirk, Numerical experiments on spiral structure, Astrophys. J., 161 (1970) 903–916.
J.K. Salmon, Parallel hierarchical N-body methods, Ph.D. thesis, California Institute of Technology, 1990.
J.P. Singh, Parallel hierarchical N-body methods and their implications for multiprocessors, Ph.D. thesis, Stanford University, 1993.
J.P. Singh, C. Holt, T. Totsuka, A. Gupta and J.L. Hennesy, Load balancing and data locality in hierarchical N-body methods, Journal of Parallel and Distributed Computing, to appear.
M.S. Warren and J.K. Salmon, Astrophysical N-body simulations using hierarchical tree data structures, Proc. Supercomputing '92 (1992) 570–576.
M.S. Warren and J.K. Salmon, A parallel hashed oct-tree N-body algorithm, Proc. Supercomputing '93 (1993) 1–12.
F. Zhao and L. Johnsson, The parallel multipole method on the connection machine, SIAM. J. Sci. Stat. Comput., 12 (1991) 1420–1437.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Aluru, S., Gustafson, J., Prabhu, G. et al. Distribution-Independent Hierarchical Algorithms for the N-body Problem. The Journal of Supercomputing 12, 303–323 (1998). https://doi.org/10.1023/A:1008047806690
Issue Date:
DOI: https://doi.org/10.1023/A:1008047806690