Abstract
In this paper we give a parallel algorithm for constructing the Voronoi diagram of a polygonal scene, i.e., a set of line segments in the plane such that no two segments intersect except possibly at their endpoints. Our algorithm runs in O(log2 n) time using O(n) processors in the CREW PRAM model.
Research supported by NSF Grant CCR-8810568.
Research supported in part by NSF grants DCR-84-01898, DCR-84-01633, and CCR-8703458.
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References
A. Aggarwal, B. Chazelle, L. Guibas, C. Ó'Dúnaling, and C. Yap, “Parallel Computational Geometry,” Algorithmica, Vol. 3, No. 3, 1988, 293–328.
M.J. Atallah, R. Cole, and M.T. Goodrich, “Cascading Divide-and-Conquer: A Technique for Designing Parallel Algorithms,” SIAM J. Comput., in press.
M.J. Atallah and M.T. Goodrich, “Efficient Parallel Solutions to Some Geometric Problems,” J. of Par. and Dist. Comp., Vol. 3, 1986, 492–507.
M.J. Atallah and M.T. Goodrich, “Parallel Algorithms for Some Functions of Two Convex Polygons,” Algorithmica, Vol. 3, 1988, 535–548.
M.J. Atallah and U. Vishkin, “Finding Euler Tours in Parallel,” J. Comp. and System Sci., Vol. 29, 1985, 330–337.
G. Bilardi and A. Nicolau, “Adaptive Bitonic Sorting: An Optimal Parallel Algorithm for Shared Memory Machines,” TR 86-769, Dept. of Comp. Sci., Cornell Univ., August 1986.
H. Blum, “A Transformation for Extracting New Descriptors of Shape,” Proc. Symp. Models for Perception of Speech and Visual Form, W. Whaten-Dunn, ed., Cambridge, MA: M.I.T. Press, 1967, 362–380.
A. Chow, “Parallel Algorithms for Geometric Problems,” Ph.D. thesis, Comp. Sci. Dept., Univ. of Illinois at Urbana-Champaign, 1980.
R. Cole, “Parallel Merge Sort,” SIAM J. Computing, Vol. 17, No. 4, August 1988, 770–785.
R. Cole and M.T. Goodrich, “Optimal Parallel Algorithms for Polygon and Point-Set Problems,” Algorithmica, in press.
R.L. Drysdale, III, “Generlaized Voronoi Diagrams and Geometric Searching,” Computer Science Report STAN-CS-79-705, Stanford Univ., Ph.D. thesis, 1979.
S. Fortune, “A Sweepline Algorithm for Voronoi Diagrams,” Proc. 2nd ACM Symp. on Computational Geometry, 1986, 313–322.
M.T. Goodrich, “Efficient Parallel Techniques for Computational Geometry,” Ph.D. thesis, Dept. of Computer Science, Purdue Univ., August 1987.
M.T. Goodrich, “Finding the Convex Hull of a Sorted Point Set in Parallel,” Info. Proc. Letters, Vol. 26, December 1987, 173–179.
L. Guibas, L. Ramshaw, and J. Stolfi, “A Kinetic Framework for Computational Geometry,” Proc. 24th IEEE Symp. on Found. of Comp. Sci., 1983, 100–111.
R.M. Karp and V. Ramachandran, “A Survey of Parallel Algorithms for Shared-Memory Machines,” to appear in Handbook of Theoretical Computer Science, North-Holland.
D.G. Kirkpatrick, “Efficient Computation of Continuous Skeletons,” Proc. 20th IEEE Symp. on Foundations of Computer Science, 1979, 18–27.
C.P. Kruskal, L. Rudolph, and M. Snir, “The Power of Parallel Prefix,” Proc. 1985 IEEE Int. Conf. on Parallel Processing, 180–185.
R.E. Ladner and M.J. Fischer, “Parallel Prefix Computation,” J. ACM, October 1980, 831–838.
D.T. Lee, “Medial Axis Transformation of a Planar Shape,” IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol. PAMI-4, No. 4, July 1982, 363–369.
D.T. Lee and R.L. Drysdale, III,“Generalization of Voronoi Diagrams in the Plane,” SIAM Journal on Computing, Vol. 10, No. 1, Feb. 1981, 73–87.
J.S.B. Mitchell, “On Maximum Flows in Polyhedral Domains,” Proc. 4th ACM Symp. on Computational Geometry, 1988, 341–351.
C. Ó'Dúnlaing and C. Yap, “A ‘Retraction’ Method for Planning the Motion of a Disc, J. Algorithms, Vol. 6, 1985, 104–111.
F.P. Preparata, “The Medial Axis of a Simple Polygon,” Proc. 6th Symp. on Mathematical Foundations of Computer Science, 1977, 443–450.
M.I. Shamos, “Geometric Complexity,” Proc. 7th ACM Symp. on Theory of Computing, 1975, 224–233.
G.T. Toussaint, “Solving Geometric Problems with Rotating Calipers,” Proc. IEEE MELECON '83, Athens, Greece, May 1983.
H. Wagener, “Optimally Parallel Algorithms for Convex Hull Determination,” unpublished manuscript, September 1985.
J.C. Wyllie, “The Complexity of Parallel Computation,” Ph.D. thesis, Technical Report TR 79-387, Department of Computer Science, Cornell University, 1979.
C.K. Yap, “An O(n log n) Algorithm for the Voronoi Diagram of a Set of Simple Curve Segments,” Discrete and Computational Geometry, Vol. 2, 1987, 365–393.
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Goodrich, M.T., Ó'Dúnlaing, C., Yap, C.K. (1989). Constructing the Voronoi diagram of a set of line segments in parallel. In: Dehne, F., Sack, J.R., Santoro, N. (eds) Algorithms and Data Structures. WADS 1989. Lecture Notes in Computer Science, vol 382. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51542-9_3
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DOI: https://doi.org/10.1007/3-540-51542-9_3
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