UBC Theses and Dissertations
Shortest paths in line arrangements Likhtarov, Anton
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 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.
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