Online point location in planar arrangements and its applications

Recently, Har-Peled [HP2] presented a new randomized technique for online c onstruction of the zone of a curve in a planar arrangement of arcs. In this paper we present several applications of this technique, which yield impro ved solutions to a variety of problems. These applications include: (i) an efficient mechanism for performing online point-location queries in an arra ngement of arcs; (ii) an efficient algorithm for computing an approximation to the minimum weight Steiner tree of a set of points, where the weight is the number of intersections between the tree edges and a given collection of area; (iii) a subquadratic algorithm for cutting a set of pseudo-parabol as into pseudo-segments; (iv) an algorithm for cutting a set of line segmen ts ("rods") in 3-space to eliminate all cycles in the vertical depth order; and (v) a near-optimal algorithm for reporting all bichromatic intersectio ns between a set R of red arcs and a set B of blue area, where the unions o f the arcs in each set are both connected.