TY - GEN

T1 - Pseudo-line arrangements

T2 - 13th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2002

AU - Agarwal, Pankaj K.

AU - Sharir, Micha

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84968877244&partnerID=8YFLogxK

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AN - SCOPUS:84968877244

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 800

EP - 809

BT - Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2002

PB - Association for Computing Machinery

Y2 - 6 January 2002 through 8 January 2002

ER -