Computing homotopic shortest paths in the plane

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).