This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Optimal Algorithms for the Multiple Query Problem on Reconfigurable Meshes, with Applications
September 2001 (vol. 12 no. 9)
pp. 875-887

Abstract—The main contribution of this work is to show that a number of fundamental and seemingly unrelated problems in database design, pattern recognition, robotics, computational geometry, and image processing can be solved simply and elegantly by stating them as instances of a unifying algorithmic framework that we call the Multiple Query problem. The Multiple Query problem (MQ, for short) is a 5-tuple ({\cal{Q}}, {\cal{A}}, {\cal{D}}, \phi, \oplus), where {\cal{Q}} is a set of queries, {\cal{A}} is a set of items, \cal D is a set of solutions, \phi:{\cal{Q}}\times {\cal{A}}\rightarrow {\cal{D}} is a function, and \oplus is a commutative and associative binary operator over \cal D. The input to the MQ problem consists of a sequence Q=\langle q_1, q_2, \ldots, q_m\rangle of m queries from {\cal{Q}} and of a sequence A=\langle a_1, a_2, \ldots a_n\rangle of n items from {\cal{A}}. The goal is to compute, for every query q_i (1\leq i\leq m) its solution defined as \phi(q_i,A) = \phi(q_i,a_1)\oplus \phi(q_i,a_2) \oplus \cdots \oplus \phi(q_i,a_n). We begin by discussing a generic algorithm that solves a large class of MQ problems in O(\sqrt{m}+f(n)) time on a reconfigurable mesh of size \sqrt{n}\times \sqrt{n}, where f(n) is the time necessary to compute the expression d_1 \oplus d_2\oplus \cdots \oplus d_n with d_i\in{\cal{D}} on such a platform. We then go on to show that the MQ framework affords us an optimal algorithm for the multiple point location problem on a reconfigurable mesh of size \sqrt{n}\times\sqrt{n}. Given a set A of n points and a set Q of m(m\leq n) points in the plane, our algorithm reports, in O(\sqrt{m}+\log\log n) time, all points of Q that lie inside the convex hull of A. Quite surprisingly, our algorithm solves the multiple point location problem without computing the convex hull of A which, in itself, takes \Omega(\sqrt{n}) time on a reconfigurable mesh of size \sqrt{n}\times\sqrt{n}. Finally, we prove an \Omega(\sqrt{m}+g(n)) time lower bound for nontrivial MQ problems, where g(n) is the lower bound for evaluating the expression d_1 \oplus d_2\oplus \cdots \oplus d_n with d_i\in{\cal{D}}, on a reconfigurable mesh of size \sqrt{n}\times\sqrt{n}.

[1] S.G. Akl and K.A. Lyons, Parallel Computational Geometry, Prentice Hall, Englewood Cliffs, N.J., 1993.
[2] D.H. Ballard and C.M. Brown, Computer Vision, Prentice Hall, Upper Saddle River, N.J., 1982.
[3] Y.-C. Chen and W.-T. Cheng, “Reconfigurable Mesh Algorithms for Summing up Binary Values and Its Applications,” Proc. Fourth Symp. Frontiers of Massively Parallel Computation, pp. 427-433, 1992.
[4] R.O. Duda and P.E. Hart, Pattern Classification and Scene Analysis. New York: Wiley and Sons, 1973.
[5] J.D. Foley,A. van Dam,S.K. Feiner,, and J.F. Hughes,Computer Graphics: Principles and Practice,Menlo Park, Calif.: Addison-Wesley, 1990.
[6] J. Jang, H. Park, and V.K. Prasanna, "A Fast Algorithm for Computing a Histogram on Reconfigurable Mesh," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, no. 2, pp. 97-106, Feb. 1995.
[7] J. Jenq and S. Sahni,“Histogramming on a reconfigurable mesh computer,” Proc. Int’l Parallel Processing Symp., pp. 425-432, 1992.
[8] C.S. Jeong and D.T. Lee, “Parallel Geometric Algorithms on a Mesh-Connected Computer,” Algorithmica, vol. 5, pp. 155-177, 1990.
[9] J.-P. Laumond, "Obstacle Growing in a Non-Polygonal World," Information Processing Letters, vol. 25, pp. 41-50, 1987.
[10] H. Li and M. Maresca,“Polymorphic-torus network,” IEEE Trans. on Computers, vol. 38, no. 9, pp. 1345-1351, Sept. 1989.
[11] T. Lozano-Perez, “Spatial Planning: A Configurational Space Approach,” IEEE Trans. Computers, vol. 32, pp. 108-119, 1983.
[12] M. Maresca, "Polymorphic Processor Arrays," IEEE Trans. Parallel and Distributed Systems, vol. 4, pp. 490-506, 1993.
[13] M. Maresca, H. Li, and P. Baglietto, “Hardware Support for Fast Reconfigurability in Processor Arrays,” Proc. Int'l Conf. Parallel Processing, vol. I, pp. 282-289, 1993.
[14] 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.
[15] K. Nakano, “A Bibliography of Published Papers on Dynamically Reconfigurable Architectures,” Parallel Processing Letters, vol. 5, pp. 111-124, 1995.
[16] K. Nakano and S. Olariu, “An Efficient Algorithm for Row Minima Computations on Basic Recofigurable Meshes,” IEEE Trans. Parallel and Distributed Systems, vol. 9, pp. 561-569, 1998.
[17] K. Nakano and K. Wada, ”Integer Summing Algorithms on Reconfigurable Meshes,“ Theoretical Computer Science, vol. 197, pp. 57-77, 1998.
[18] S. Olariu, J.L. Schwing, and J. Zhang, “Optimal Convex Hull Algorithms on Enhanced Meshes,” BIT vol. 33, pp. 396-410, 1993.
[19] 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.
[20] B.T. Preas and M.J.Lorenzetti eds. Physical Design Automation of VLSI Systems, Menlo Park: Benjamin/Cummings, 1988.
[21] F.P. Preparata and M.I. Shamos, Computational Geometry. Springer-Verlag, 1985.
[22] J. Serra, Image Analysis and Mathematical Morphology. New York: Academic Press, 1982.
[23] C. Thompson and H. Kung,“Sorting on a mesh connected parallel computer,”Commun. ACM, vol. 20, pp. 263–271, 1977.

Index Terms:
Query processing, reconfigurable mesh, database design, pattern recognition, image processing, computational geometry, robotics, morphology, parallel algorithms.
Citation:
Venkatavasu Bokka, Koji Nakano, Stephen Olariu, James L. Schwing, Larry Wilson, "Optimal Algorithms for the Multiple Query Problem on Reconfigurable Meshes, with Applications," IEEE Transactions on Parallel and Distributed Systems, vol. 12, no. 9, pp. 875-887, Sept. 2001, doi:10.1109/71.954618
Usage of this product signifies your acceptance of the Terms of Use.