Pseudo-line arrangements: Duality, algorithms, and applications
Published
Conference Paper
A collection L of n 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 P be a set of m points in R2. We define a duality transform that maps L to a set L-of points in R2 and P to a set P∗of pseudo-lines in E2, so that the incidence and the "above-below" relationships between the points and pseudo-lines are preserved. We present an efficient algorithm for computing the dual arrangement A(P∗) under an appropriate model of computation. We also propose a dynamic data structure for reporting, in 0(me + fc) 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 our 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 a number of problems involving arrangements of circles or circular arcs, such as detecting, counting, or reporting incidences between points and circles.
Duke Authors
Cited Authors
- Agarwal, PK; Sharir, M
Published Date
- January 1, 2002
Published In
- Proceedings of the Annual Acm Siam Symposium on Discrete Algorithms
Volume / Issue
- 06-08-January-2002 /
Start / End Page
- 800 - 809
International Standard Book Number 10 (ISBN-10)
- 089871513X
Citation Source
- Scopus