Recently, Har-Peled [HP2] 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 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 in each set are both connected.
Received December 22, 1999, and in revised form August 25, 2000. Online publication May 11, 2001.
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Har-Peled, S., Sharir, M. Online Point Location in Planar Arrangements and Its Applications. Discrete Comput Geom 26, 19–40 (2001). https://doi.org/10.1007/s00454-001-0026-y