Time-optimal nearest-neighbor computations on enhanced meshes
A simple polygon P is said to be unimodal if for every vertex of P, the Euclidian distance function to the other vertices of P is unimodal. The study of unimodal polygons has emerged as a fruitful area of computational and discrete geometry. Is the well-known that nearest and furthest neighbor computations are a recurring theme in pattern recognition, VLSI design, computer graphics, and image processing, among others. Our contribution is to propose time-optimal algorithms for constructing the Euclidian Minimum Spanning Tree, the Relative Neighborhood Graph, as well as the Symmetric Further Neighbor Graph of an n-vertex unimodal polygon on meshes with multiple broadcasting. We begin by establishing a Ω(log n) time lower bound for solving arbitrary instances of size n of these problems. This lower bound holds for both the CREW-PRAM and for the mesh with multiple broadcasting. We obtain our time lower bound results for the CREW-PRAM by using a novel technique involving geometric constructions. These constructions allow us to reduce the well-known OR problem to each of the geometric problems of interest. We then port these time lower bounds to the mesh with multiple broadcasting using simulation results.
Next, we show that the time lower bound is tight by exhibiting algorithms for these tasks running in O(log n) time on a mesh with multiple broadcasting of size n×n.
Keywordsunimodal polygons mesh with multiple broadcasting pattern recognition nearest-neighbor norms computational geometry computational morphology parallel algorithms EMST RNG SFNG
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- 1.A. Aggarwal, Optimal bounds for finding maximum on array of processots with k global buses, IEEE Transactions on Computers, C-35 (1986) 62–64.Google Scholar
- 2.A. Aggarwal and R. C. Melville, Fast computation of the modality of polygons, Journal of Algorithms, 7 (1986) 369–381.Google Scholar
- 3.G. S. Almasi and A. Gottlieb, Highly parallel computing, Second Edition, Benjamin/Cummings, Redwood City, California, 1994.Google Scholar
- 4.D. H. Ballard and C. M. Brown, Computer Vision, Prentice-Hall, Englewood Cliffs, New Jersey, 1982.Google Scholar
- 5.D. Bhagavathi, P. J. Looges, S. Olariu, J. L. Schwing, and J. Zhang, A fast selection algorithm on meshes with multiple broadcasting, Proc. International Conference on Parallel Processing, 1992, St-Charles, Illinois, III-10–17.Google Scholar
- 6.D. Bhagavathi, S. Olariu, W. Shen, and L. Wilson, A Time-optimal multiple search algorithm on enhanced meshes, with applications, Proc. Fourth Canadian Computational Geometry Conference, St-Johns, August 1992, 359–364.Google Scholar
- 8.D. Bhagavathi, S. Olariu, W. Shen, and L. Wilson, A unifying look at semigroup computations on meshes with multiple broadcasting, Proc. PARLE'93, Munich, Germany, June 1993, LNCS 694, 561–570.Google Scholar
- 9.D. Bhagavathi, V. Bokka, H. Gurla, S. Olariu, I. Stojmenović, J. Schwing, and J. Zhang, Time-optimal solution to visibility-related problems on meshes with multiple broadcasting, Proc. of ASAP'93, Venice, Italy, October 1993, 226–237.Google Scholar
- 10.S. H. Bokhari, Finding maximum on an array processor with a global bus, IEEE Transactions on Computers C-33 (1984) 133–139.Google Scholar
- 13.R. O. Duda and P. E. Hart, Pattern Classification and Scene Analysis, Wiley and Sons, New York, 1973.Google Scholar
- 14.J. JáJá, An introduction to parallel algorithms, Addison-Wesley, Reading, MA, 1992.Google Scholar
- 15.V. P. Kumar and C. S. Raghavendra, Array processor with multiple broadcasting, Journal of Parallel and Distributed Computing, 2, (1987) 173–190.Google Scholar
- 22.S. Olariu, The morphology of convex polygons, Computers and Mathematics, with Applications, 24, (1992), 59–68.Google Scholar
- 23.S. Olariu, J. L. Schwing, and J. Zhang, Optimal convex hull algorithms on enhanced meshes, BIT 33 (1993), 396–410.Google Scholar
- 24.S. Olariu and I. Stojmenović, Time-optimal proximity problems on meshes with multiple broadcasting, Proc. International Parallel Processing Symposium, Cancun, Mexico, 1994, to appear.Google Scholar
- 25.D. Parkinson, D. J. Hunt, and K. S. MacQueen, The AMT DAP 500, 33rd IEEE Comp. Soc. International Conf., 1988, 196–199.Google Scholar
- 26.F. P. Preparata and M. I. Shamos, Computational Geometry — An Introduction, Springer-Verlag, Berlin, 1988.Google Scholar