Taking a walk in a planar arrangement

We present a randomized algorithm for computing portions of an arrangement of n arcs in the plane, each pair of which intersect in at most t points. We use this algorithm to perform online walks inside such an arrangement (i.e., compute all the faces that a curve, given in an online manner, crosses), and to compute a level in an arrangement, both in an output-sensitive manner. The expected running time of the algorithm is O(λ_t+2(m + n) log n), where m is the number of intersections between the walk and the given arcs. No similarly efficient algorithm is known for the general case of arcs. For the case of lines and for certain restricted cases involving line segments, our algorithm improves the best known algorithm of [15] by almost a logarithmic factor.