Shortest paths in line arrangements

Anton Likhtarov
The problem of finding a shortest Euclidean path in an arrangement of lines between two points in the arrangement has been extensively studied; however, the best known exact solution takes quadratic time, and it's not known if a subquadratic time algorithm exists. While I did not succeed in improving these bounds, I examined instead the problem of efficiently finding the approximate shortest path where the runtime depends on the bound of the relative error in the path length. I present an
more » ... I present an algorithm for computing this approximate shortest path. The algorithm uses the geometric structure of the arrangement; I show that certain lines are never used by the shortest path, while other lines could be ignored without making the path much longer. My work includes a number of lemmas that provide simple proofs for related problems (such as shortest path in two intersecting pencils of lines), and could have applications in future work on this problem.
doi:10.14288/1.0389809 fatcat:fdi7tlpay5fmri2fniv4jkgkki