Approximating the shortest path in line arrangements

@inproceedings{Hart2001ApproximatingTS,
  title={Approximating the shortest path in line arrangements},
  author={David Hart},
  booktitle={CCCG},
  year={2001}
}
Suppose one has a line arrangement and one wants to find a shortest path from one point lying on a line of the arrangement to another such point. The best known time bound for computing this isO(n2). We develop an algorithm that finds a1 + approximation of the shortest path in time O(n logn+ (minfn; 1 2 g)1 log 1 ).