Online point location in planar arrangements and its applications

Recently, Har-Peled [HP99b] presented a new randomized technique for online construction of the zone of a curve in a planar arrangement of arcs. In this paper, we present several applications of this technique, which yield improved solutions to a variety of problems. These applications include: (i) an efficient mechanism for performing online point location queries in an arrangement 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 arcs; (iii) a subquadratic algorithm for cutting a set of pseudo-parabolas into pseudo-segments; (iv) an algorithm for cutting a set of line segments (`rods') in 3-space so as to eliminate all cycles in the vertical depth order; and (v) a near-optimal algorithm for reporting all bichromatic intersections between a set R of red arcs and a set B of blue arcs, where the unions of the arcs...

• Type: Conference
• Year: 2001
• Where: SODA
• Authors: Sariel Har-Peled, Micha Sharir
• Keywords: Algorithm, Algorithms, Point Location Queries, Red Arcs, SODA 2001