Abstract
Given ann-vertex simple polygon we address the following problems: (i) find the shortest path between two pointss andd insideP, and (ii) compute the shortestpath tree between a single points and each vertex ofP (which implicitly represents all the shortest paths). We show how to solve the first problem inO(logn) time usingO(n) processors, and the more general second problem inO(log2 n) time usingO(n) processors, and the more general second problem inO(log2 n) time usingO(n) processors for any simple polygonP. We assume the CREW RAM shared memory model of computation in which concurrent reads are allowed, but no two processors should attempt to simultaneously write in the same memory location. The algorithms are based on the divide-and-conquer paradigm and are quite different from the known sequential algorithms
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Aggarwal A, Chazelle B, Guibas L, O'Dunlaing C, Yap C (1985) Parallel computational geometry. Proc 25th Ann Symp Foundat Comput Sci, pp 468–477
Atallah MJ, Cole R, Goodrich MT (1987) Cascading divideand-conquer: a technique for designing parallel algorithms. Proc. 28 th Ann Symp Foundat Comput Sci, pp 151–160
Atallah MJ, Goodrich MT (1988) Efficient parallel solutions to some geometric problems. J Parallel Distrib Comput 3:492–507
Atallah MJ, Goodrich MT (1986) Parallel algorithms for some functions of two convex polygons. 24th Ann Allerton Conf Commun Control Comput, Urbana-Champaign (October 1986), pp 758–767
Chazelle B (1982) A theorem on polygon cutting with applications. Proc 23rd Ann Symp Foundat Comput Sci, pp 339–349
Chow A (1980) Parallel algorithms for geometric problems. PhD Diss, Comput Sci Dept, Univ Illinois at Urbana-Champaign
ElGindy H (1985) Hierarchical decomposition of polygons with applications. PhD Diss, School Comput Sci, McGill Univ (June 1985)
ElGindy H (1986) A parallel algorithm for the shortest path problem in monotone polygons. Tech Rep MS-CIS-86-49, Dept Comput Inf Sci, School Eng Appl Sci, Univ Pennsylvania (June 1986)
Goodrich MT (1988) Triangulation a simple polygon in parallel. J Algorithms (in press)
Guibas L, Hershberger J, Leven D, Sharir M, Tarjan R (1986) Linear time algorithms for visibility and shortest path problems inside simple polygons. Proc 2nd Symp Comput Geom, York Heights (June 1986), pp 1–13
Lee DT, Preparata FP (1984) Euclidean shortest paths in the presence of rectilinear barriers. Networks 14:393–410
Lee DT, Preparata FP (1984) Computational geometry- a survey. IEEE Trans Comput C 33:872–1101
Lozano-Perez T, Wesley MA (1979) An algorithm for planning collision-free paths among polyhedral obstacles. Commun ACM 22:560–570
Pan V, Reif J (1985) Efficient parallel solution of linear systems. Proc 17th ACM Symp Theor Comput, pp 143–152
Preparata FP, Shamos MI (1985) Computational geometry: an introduction. Springer, Berlin Heidelberg New York
Quinn MJ, Deo N (1984) Parallel graph algorithms. Comput Surv 16:319–346
Savage C (1984) Parallel algorithms for graph theoretic problems. PhD Diss, Math Dept, Univ. Illinois, Urbana, III
Tarjan RE, Van Wyk CJ (1986) AnO(n loglogn)-time algorithm for triangulating simple polygons. Tech Rep CS-TR-052-86, Dept Comput Sci, Princeton Univ
Tarjan RE, Vishkin U (1985) An efficient parallel biconnectivity algorithm. SIAM J Comput 14:862–874
Welzl E (1985) Constructing the visibility graph forn line segments inO(n 2) time. Inf Proc Lett 20:167–171
Author information
Authors and Affiliations
Additional information
Research supported by the Faculty of Graduate Studies and Research (McGill University) grant 276-07
Rights and permissions
About this article
Cite this article
ElGindy, H., Goodrich, M. Parallel algorithms for shortest path problems in polygons. The Visual Computer 3, 371–378 (1988). https://doi.org/10.1007/BF01901194
Issue Date:
DOI: https://doi.org/10.1007/BF01901194