Computing homotopic shortest paths in the plane

We address the problem of computing homotopic shortest paths in presence of obstacles in the plane. The problems on homotopy of the paths received attention very recently [3, 8]. We present two output-sensitive algorithms, for simple paths and non-simple paths. The algorithm for simple paths improves the previous algorithm [8]. The algorithm for non-simple paths achieves O(log n) time per output vertex which is an improvement by a factor of O(n/ log n) of the previous algorithm [13] where n is the number of obstacles.