Computing homotopic shortest paths in the plane

We address the problem of computing homotopic shortest paths in the presence of obstacles in the plane. Problems on homotopy of paths received attention very recently [Cabello et al., in: Proc. 18th Annu. ACM Sympos. Comput. Geom., 2002, pp. 160–169; Efrat et al., in: Proc. 10th Annu. European Sympo...

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Published in:	Journal of algorithms Vol. 49; no. 2; pp. 284 - 303
Main Author:	Bespamyatnikh, Sergei
Format:	Journal Article
Language:	English
Published:	San Diego, CA Elsevier Inc 01.11.2003 Elsevier
Subjects:	Algorithmics. Computability. Computer arithmetics Applied sciences Classical combinatorial problems Combinatorics Combinatorics. Ordered structures Computer science; control theory; systems Convex and discrete geometry Exact sciences and technology Geometry Graph theory Homotopy Mathematics Output-sensitive algorithm Sciences and techniques of general use Shortest path Theoretical computing Shortest path Homotopy Output-sensitive algorithm Output sensitive algorithm Graph theory Computer theory Computing
ISSN:	0196-6774, 1090-2678
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Abstract	We address the problem of computing homotopic shortest paths in the presence of obstacles in the plane. Problems on homotopy of paths received attention very recently [Cabello et al., in: Proc. 18th Annu. ACM Sympos. Comput. Geom., 2002, pp. 160–169; Efrat et al., in: Proc. 10th Annu. European Sympos. Algorithms, 2002, pp. 411–423]. We present two output-sensitive algorithms, for simple paths and non-simple paths. The algorithm for simple paths improves the previous algorithm [Efrat et al., in: Proc. 10th Annu. European Sympos. Algorithms, 2002, pp. 411–423]. The algorithm for non-simple paths achieves O(log 2 n) time per output vertex which is an improvement by a factor of O( n/log 2 n) of the previous algorithm [Hershberger, Snoeyink, Comput. Geom. Theory Appl. 4 (1994) 63–98], where n is the number of obstacles. The running time has an overhead O( n 2+ ε ) for any positive constant ε. In the case k< n 2+ ε , where k is the total size of the input and output, we improve the running to O(( n+ k+( nk) 2/3)log O(1) n).
AbstractList	We address the problem of computing homotopic shortest paths in the presence of obstacles in the plane. Problems on homotopy of paths received attention very recently [Cabello et al., in: Proc. 18th Annu. ACM Sympos. Comput. Geom., 2002, pp. 160–169; Efrat et al., in: Proc. 10th Annu. European Sympos. Algorithms, 2002, pp. 411–423]. We present two output-sensitive algorithms, for simple paths and non-simple paths. The algorithm for simple paths improves the previous algorithm [Efrat et al., in: Proc. 10th Annu. European Sympos. Algorithms, 2002, pp. 411–423]. The algorithm for non-simple paths achieves O(log 2 n) time per output vertex which is an improvement by a factor of O( n/log 2 n) of the previous algorithm [Hershberger, Snoeyink, Comput. Geom. Theory Appl. 4 (1994) 63–98], where n is the number of obstacles. The running time has an overhead O( n 2+ ε ) for any positive constant ε. In the case k< n 2+ ε , where k is the total size of the input and output, we improve the running to O(( n+ k+( nk) 2/3)log O(1) n).
Author	Bespamyatnikh, Sergei
Author_xml	– sequence: 1 givenname: Sergei surname: Bespamyatnikh fullname: Bespamyatnikh, Sergei email: besp@utd.edu organization: Department of Computer Science, University of Texas at Dallas, Box 830688, Richardson, TX 75083, USA
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SubjectTerms	Algorithmics. Computability. Computer arithmetics Applied sciences Classical combinatorial problems Combinatorics Combinatorics. Ordered structures Computer science; control theory; systems Convex and discrete geometry Exact sciences and technology Geometry Graph theory Homotopy Mathematics Output-sensitive algorithm Sciences and techniques of general use Shortest path Theoretical computing
Title	Computing homotopic shortest paths in the plane
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