Pseudo-Line Arrangements: Duality, Algorithms, and Applications

Abstract

A finite collection of x-monotone unbounded Jordan curves in the plane is called a family of pseudo-lines if every pair of curves intersect in at most one point, and the two curves cross each other there. Let L be such a collection of n pseudo-lines, and let P be a set of m points in $\reals^2$. Extending a result of Goodman [Discrete Math., 32 (1980), pp. 27--35], we define a duality transform that maps L to a set L* of points in $\reals^2$ and P to a set P* of (x-monotone) pseudo-lines in $\reals^2$, so that the incidence and the above-below relations between the points and the pseudo-lines are preserved. We present an efficient algorithm for computing the dual arrangement {\eus A}$(P^*)$ under an appropriate model of computation. We also present a dynamic data structure for reporting, in $O(m^\eps + k)$ time, all k points of P that lie below a query arc, which is either a circular arc or a portion of the graph of a polynomial of fixed degree. This result is needed for computing the dual arrangement for certain classes of pseudo-lines arising in several applications, but is also interesting in its own right. We present a few applications of our dual arrangement algorithm, such as computing incidences between points and pseudo-lines and computing a subset of faces in a pseudo-line arrangement. Next, we present an efficient algorithm for cutting a set of circles into arcs so that every pair of arcs intersect in at most one point, i.e., the resulting arcs constitute a collection of pseudo-segments. By combining this algorithm with our algorithm for computing the dual arrangement of pseudo-lines, we obtain efficient algorithms for several problems involving arrangements of circles or circular arcs, such as reporting or counting incidences between points and circles and computing a set of marked faces in arrangements of circles.