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Time-Optimal Visibility-Related Algorithms on Meshes with Multiple Broadcasting
July 1995 (vol. 6 no. 7)
pp. 687-703

Abstract—Given a collection of objects in the plane along with a viewpoint ω, the visibility problem involves determining the portion of each object that is visible to an observer positioned at ω. The visibility problem is central to various application areas including computer graphics, image processing, VLSI design, and robot navigation, among many others. The main contribution of this work is to provide time-optimal solutions to this problem for several classes of objects, namely ordered line segments, disks, and iso-oriented rectangles in the plane. In addition, our visibility algorithm for line segments is at the heart of time-optimal solutions for determining, for each element in a given sequence of real numbers, the position of the nearest larger element within that sequence, triangulating a set of points in the plane, determining the visibility pairs among a set of vertical line segments, and constructing the dominance and visibility graphs of a set of iso-oriented rectangles in the plane. All the algorithms in this paper involve an input of size n and run in O(log n) time on a mesh with multiple broadcasting of size n×n. This is the first instance of time-optimal solutions for these problems on this architecture.

[1] M.J. Atallah,R. Cole,, and M.T. Goodrich,“Cascading divide-and-conquer: A technique for designing parallelalgorithms,” SIAM J. Computing, vol. 18, no. 3, pp. 499-532, 1989.
[2] A. Bar-Noy and D. Peleg, "Square Meshes Are Not Always Optimal," IEEE Trans. Computers, vol. 40, pp. 196-204, 1991
[3] K.E. Batcher,“Design of massively parallel processor,” IEEE Trans. Computers, vol. 29, pp. 836-840, 1980.
[4] O. Berkman,D. Breslauer,Z. Galil,B. Schieber,, and U. Vishkin,“Highly parallelizable problems,” Proc. Ann. Symp. Theory of Computing, pp. 770-780, 1989.
[5] D. Bhagavathi,H. Gurla,R. Lin,S. Olariu,J.L. Schwing,, and J. Zhang,“Time- and VLSI-optimal sorting on meshes with multiple broadcasting,” Proc. Int’l Conf. Parallel Processing,St-Charles, Ill., vol. III, pp. 196-201, Aug. 1993.
[6] D. Bhagavathi, P.J. Looges, S. Olariu, J.L. Schwing, and J. Zhang, "A Fast Selection Algorithm on Meshes with Multiple Broadcasting," IEEE Trans. Parallel and Distributed Systems, vol. 5, pp. 772-778, 1994.
[7] D. Bhagavathi, S. Olariu, W. Shen, and L. Wilson, "A Time-Optimal Multiple Search Algorithm on Enhanced Meshes, with Applications," J. Parallel and Distributed Computing, vol. 22, pp. 113-120, 1994.
[8] D. Bhagavathi,S. Olariu,J.L. Schwing,, and J. Zhang,“Convex polygon problems on meshes with multiple broadcasting,” Parallel Processing Letters, vol. 2, pp. 249-256, 1992.
[9] D. Bhagavathi, S. Olariu, W. Shen, and L. Wilson, "A Unifying Look at Semigroup Computations on Meshes with Multiple Broadcasting," Proc. Parallel Architectures and Languages Europe, LNCS 694, pp. 561-569,München, Germany, June 1993.
[10] D. Bhagavathi,V. Bokka,H. Gurla,S. Olariu,J.L. Schwing,, and Zhang,“Square meshes are not optimal for convex hull computation,” Proc. Int’l Conf. Parallel Processing,St. Charles, Ill., vol. III, pp. 307-311, Aug. 1993.
[11] S.H. Bokhari,“Finding maximum on an array processor with a global bus,” IEEE Trans. Computers, vol. 33, pp. 133-139, 1984.
[12] I.-W. Chan, and D.K. Friesen,“An optimal parallel algorithm for the vertical segment visibility reporting problem,” Proc. ICCI’91, Lecture Notes in Computer Science, vol. 497, pp. 323-334, Springer-Verlag, 1991.
[13] Y.C. Chen,W.T. Chen,G.-H. Chen,, and J.P. Sheu,“Designing efficient parallel algorithms on mesh connected computers with multiple broadcasting,” IEEE Trans. Parallel and Distributed Systems, vol. 1, 1990.
[14] S.A. Cook, C. Dwork, and R. Reischuk, "Upper and Lower Time Bounds for Parallel Random Access Machines without Simultaneous Writes," SIAM J. Computing, vol. 15, pp. 87-97, 1986.
[15] R.O. Duda and P.E. Hart,Pattern Classification and Scene Analysis, John Wiley and Sons, New York 1973.
[16] J.D. Foley,A. van Dam,S.K. Feiner,, and J.F. Hughes,Computer Graphics: Principles and Practice,Menlo Park, Calif.: Addison-Wesley, 1990.
[17] Computer Architecture for Spatially Distributed Data, H. Freeman and G. Pieroni, eds., Springer-Verlag, Heidelberg, Berlin, 1985.
[18] J. J'aJ'a, An Introduction to Parallel Algorithms.New York: Addison-Wesley, 1992.
[19] V.K. Prasanna Kumar, and C.S. Raghavendra,“Array processor with multiple broadcasting,” J. of Parallel and Distributed Computing, vol. 4, pp. 173-190, 1987.
[20] V. Prasanna-Kumar and D.I. Reisis, "Image Computations on Meshes with Multiple Broadcast," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, pp. 1,194-1,201, 1989.
[21] J.-P. Laumond, "Obstacle Growing in a Non-Polygonal World," Information Processing Letters, vol. 25, pp. 41-50, 1987.
[22] H. Li and M. Maresca,“Polymorphic-torus network,” IEEE Trans. on Computers, vol. 38, no. 9, pp. 1345-1351, Sept. 1989.
[23] R. Lin,S. Olariu,J.L. Schwing,, and J. Zhang,“Simulating enhanced meshes, with applications,” Parallel Processing Letters, vol. 3, pp. 59-70, 1993.
[24] E. Lodi and L. Pagli, "A VLSI Solution to the Vertical Segment Visibility Problem," IEEE Trans. Computers, vol. 35, pp. 923-928, 1986.
[25] T. Lozano-Perez,“Spatial planning: a configurational space approach,” IEEE Trans. Computers, vol. 32, pp. 108-119, 1983.
[26] F. Luccio,S. Mazzone,, and C.K. Wong,“A note on visibility graphs,” Discrete Math., vol. 64, pp. 209-219, 1987.
[27] A.A. Malik,“An efficient algorithm for generation of constraint graph for compaction,” Proc. Int’l Conf. CAD, pp. 130-133, 1987.
[28] M. Maresca and H. Li,“Connection autonomy in SIMD computers: a VLSI implementation,”J. Parallel Distribut. Comput., vol. 7, pp. 302–320, 1989.
[29] C. Mead and L. Conway, Introduction to VLSI Systems, Addison-Wesley, Reading, Mass., 1980.
[30] S. Olariu,J.L. Schwing,, and J. Zhang,“Optimal convex hull algorithms on enhanced meshes,” BIT, vol. 33, pp. 396-410, 1993.
[31] S. Olariu and I. Stojmenovic, "Time-Optimal Proximity Algorithms on Meshes with Multiple Broadcasting," Proc. Eighth Int'l Parallel Processing Symp., pp. 94-101,Cancun, Mexico, Apr. 1994.
[32] S. Olariu and I. Stojmenovi$c \acute$,“Time-optimal nearest-neighbor computations on enhanced meshes,” Proc. PARLE, Patras, Greece, July 1994 (to appear).
[33] D. Parkinson,D.J. Hunt,, and K.S. MacQueen,“The AMT DAP 500,” 33rd IEEE Computer Soc. Int’l Conf., pp. 196-199, 1988.
[34] T. Pavlidis,Computer Graphics, Computer Science Press, Potomac, Md., 1978.
[35] Physical Design Automation of VLSI Systems, B.T. Preas and M.J. Lorenzetti, eds., Benjamin/Cummings, Menlo Park, Calif., 1988.
[36] F.P. Preparata and M.I. Shamos, Computational Geometry. Springer-Verlag, 1985.
[37] J. Rothstein,“Bus Automata, Brains, and Mental Models,” IEEE Trans. on Systems, Man, and Cybernetics, vol. 18, no. 4, pp. 522-531, Apr. 1988.
[38] M. Schlag,F. Luccio,P. Maestrini,D.T. Lee,, and C.K. Wong,“A visibility problem in VLSI layout compaction,” Advances in Computing Research, F.P. Preparata, ed., vol. 2, pp. 259-282, 1985.
[39] H.S. Stone, High-Performance Computer Architecture.Reading, Mass.: Addison-Wesley, 1990.
[40] G.T. Toussaint,Computational Geometry, Elsevier Science Publishers, North-Holland, Amsterdam, 1985.
[41] G.T. Toussaint,“Movable separability of sets,” Computational Geometry, Elsevier Science Publishers, North-Holland, Amsterdam, 1985.
[42] D. Vernon,Machine Vision, Automated Visual Inspection, and Robot Vision, Prentice Hall, Englewood Cliffs, N.J., 1991.
[43] C.A. Wang and Y.H. Tsin, "An O(log n) Time Parallel Algorithm for Triangulating a Set of Points in the Plane," Information Processing Letters, vol. 25, pp. 55-60, Apr. 1987.
[44] W.H. Wolf and A.E. Dunlop,“Symbolic layout and compaction,” Physical Design Automation of VLSI Systems, B.T. Preas and M.J. Lorenzetti, eds., Benjamin/Cummings, Menlo Park, Calif., pp. 211-281, 1988.

Index Terms:
Visibility, triangulation, compaction, dominance graph, visibility graph, robotics, image processing, computer graphics, VLSI design, computational geometry, meshes with multiple broadcasting, parallel algorithms, time-optimal algorithms.
Citation:
Dharmavani Bhagavathi, Venkata V. Bokka, Himabindu Gurla, Stephan Olariu, James L. Schwing, Ivan Stojmenovic, Jingyuan Zhang, "Time-Optimal Visibility-Related Algorithms on Meshes with Multiple Broadcasting," IEEE Transactions on Parallel and Distributed Systems, vol. 6, no. 7, pp. 687-703, July 1995, doi:10.1109/71.395398
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