This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Constructing Euclidean Minimum Spanning Trees and All Nearest Neighbors on Reconfigurable Meshes
August 1996 (vol. 7 no. 8)
pp. 806-817

Abstract—A reconfigurable mesh, R-mesh for short, is a two-dimensional array of processors connected by a grid-shaped reconfigurable bus system. Each processor has four I/O ports that can be locally connected during execution of algorithms. This paper considers the d-dimensional Euclidean Minimum Spanning Tree (EMST) and the All Nearest Neighbors (ANN) problem. Two results are reported. First, we show that a minimum spanning tree of n points in a fixed d-dimensional space can be constructed in O(1) time on a $\sqrt {n^3}\times \sqrt {n^3}$ R-mesh. Second, all nearest neighbors of n points in a fixed d-dimensional space can be constructed in O(1) time on an n×n R-mesh. There is no previous O(1) time algorithm for the EMST problem; ours is the first such algorithm. A previous R-mesh algorithm exists for the two-dimensional ANN problem; we extend it to any d-dimensional space. Both of the proposed algorithms have a time complexity independent of n but growing with d. The time complexity is O(1) if d is a constant.

[1] Y. Ben-Asher,D. Peleg,R. Ramaswami,, and A. Schuster,“The power of reconfiguration,” J. of Parallel and Distributed Computing, vol. 13, no. 2, pp. 139-153, Oct. 1991.
[2] V. Bokka, H. Gurla, S. Olariu, and J.L. Schwing, "Constant Time Convexity Problems on Dense Reconfigurable Meshes," J. Parallel and Distributed Computing, vol. 27, pp. 86-99, 1995.
[3] F.Y. Chin, J. Lam, and I. Chen, "Efficient Parallel Algorithms for Some Graph Problems," Comm. ACM, vol. 25, no. 9, pp. 659-665, 1982.
[4] R. Courant and F. John, Introduction to Calculus and Analysis, vol. 2. New York: John Wiley&Sons, 1974.
[5] J. Jang,H. Park,, and V.K. Prasanna,“A fast algorithm for computing histogram on reconfigurable mesh,” Proc. Frontiers of Massively Parallel Computation, pp. 244-251, 1992.
[6] J. Jang,H. Park,, and V. K. Prasanna,“An optimal multiplication algorithm on reconfigurable mesh,” Proc. Symp. Parallel and Distributed Processing, pp. 384-391, 1992.
[7] J. Jang and V.K. Prasanna,“An optimal sorting algorithm on reconfigurable mesh,” Proc. Int’l Parallel Processing Symp., pp. 130-137, Mar. 1992.
[8] J. Jang and V. Prasanna, "A Fast Sorting Algorithm on Higher Dimension Reconfigurable Meshes," Proc. 26th Conf. Information Sciences and Systems, 1992.
[9] J. Jang and V.K. Prasanna,“An optimal sorting algorithm on reconfigurable mesh,” Proc. Int’l Parallel Processing Symp., pp. 130-137, Mar. 1992.
[10] J.W. Jaromczyk and M. Kowaluk, "Constructing the Relative Neighborhood Graph in 3-Dimensional Euclidean Space," Discrete Applied Mathematics, vol. 31, pp. 181-191, 1991.
[11] J. Jenq and S. Sahni,“Reconfigurable mesh algorithms for image shrinking, expanding, clustering, and template matching,” Proc. Int’l Parallel Processing Symp., pp. 208-215, 1991.
[12] J. Jenq and S. Sahni,“Histogramming on a reconfigurable mesh computer,” Proc. Int’l Parallel Processing Symp., pp. 425-432, 1992.
[13] R. Lin, S. Olariu, J. Schwing, and J. Zhang, "A VLSI-Optimal Constant Time Sorting on a Reconfigurable Mesh," Proc. Ninth European Workshop on Parallel Computing, pp. 1-16,Madrid, Spain, 1992.
[14] R. Miller,V.K. Prasanna Kumar,D.I. Reisis,, and Q.F. Stout,“Meshes with reconfigurable buses,” MIT Conf. on Advanced Research in VLSI, pp. 163-178, 1988.
[15] R. Miller,V.K. Prasanna Kumar,D.I. Reisis, and Q.F. Stout,“Parallel computations on reconfigurable meshes,” IEEE Trans. on Computers, pp. 678-692, June 1993.
[16] R. Miller and Q.F. Stout, "Mesh Computer Algorithms for Computational Geometry," IEEE Transactions on Computers, vol. 38, no. 3, pp. 321-340, Mar. 1989.
[17] M. Nigam and S. Sahni, "Sorting n Numbers on n x n Reconfigurable Meshes With Buses," Proc. Int'l Parallel Processing Symp., pp. 174-181, 1993.
[18] M. Nigam and S. Sahni, "Computational Geometry on a Reconfigurable Mesh," Proc. Int'l Parallel Processing Symp., pp. 258-261, 1994.
[19] M. Nigam and S. Sahni, "Triangulation on a Reconfigurable Mesh With Buses," Proc. Int'l Conf. Parallel Processing, pp. III-251-257, 1994.
[20] S. Olariu, J.L. Schwing, and J. Zhang, "Fast Computer Vision Algorithms for Reconfigurable Meshes," Proc. Int'l Parallel Processing Symp., pp. 258-261, 1992.
[21] F.P. Preparata and M.I. Shamos, Computational Geometry. Springer-Verlag, 1985.
[22] C. Savage and J. JaJa, "Fast, Efficient Parallel Algorithms for Some Graph Problems," SIAM J. Computing, vol. 10, no. 4, pp. 682-691, 1981.
[23] G.T. Toussaint, "The Relative Neighbourhood Graph of a Finite Planar Set," Pattern Recognition, vol. 12, pp. 261-268, 1980.
[24] J.L. Trahan, R. Vaidyanathan, and C. Subbaraman, "Constant Time Graph and Poset Algorithms on the Reconfigurable Multiple Bus Machine," Proc. Int'l Conf. Parallel Processing, pp. III-210-213, 1994.
[25] B. F. Wang and G. H. Chen,“Constant time algorithms for the transitive closure problem and some related graph problems on processor arrays with reconfigurable bus systems,” IEEE Trans. on Parallel and Distributed Systems, vol. 1, no. 4, pp. 500-507, 1991.
[26] B.F. Wang,G.H. Chen,, and F.C. Lin,“Constant time sorting on a processor array with a reconfigurable bus systems,” Information Processing Letters, vol. 34, pp. 187-192, 1990.
[27] C.C. Weems, S.P. Levitan, A.R. Hanson, E.M. Riseman, J.G. Nash, and D.B. Sheu, "The Image Understanding Architecture," Int'l J. Computer Vision, vol. 2, pp. 251-282, 1989.

Index Terms:
Parallel algorithms, reconfigurable meshes, computational geometry.
Citation:
Ten H. Lai, Ming-Jye Sheng, "Constructing Euclidean Minimum Spanning Trees and All Nearest Neighbors on Reconfigurable Meshes," IEEE Transactions on Parallel and Distributed Systems, vol. 7, no. 8, pp. 806-817, Aug. 1996, doi:10.1109/71.532112
Usage of this product signifies your acceptance of the Terms of Use.