Computing shortest transversals of sets (extended abstract)

Given a family of objects in the plane, the line transversal problem is to compute a line that intersects every member of the family. In this paper we examine a variation of the line transversal problem that involves computing a shortest line segment that intersects every member of the family. In particular, we give O(n log n) time algorithms for computing a shortest transversal of a family of n lines and of a family of n line segments. We also present an O(n log2 n) time algorithm for computing a shortest transversal of a family of polygons with a total of n vertices. In general, finding a line transversal for a family of n objects takes fl(n log n) time. This time bound holds for a family of n line segments thus our shortest transversal algorithm for this family is optimal.

[1]  David Avis,et al.  Algorithms for high dimensional stabbing problems , 1990, Discret. Appl. Math..

[2]  Arnold L. Rosenberg,et al.  Stabbing line segments , 1982, BIT.

[3]  Chanderjit Bajaj,et al.  On the duality of intersection and closest points , 1983 .

[4]  M. Keil,et al.  A Simple Algorithm for Determining the Envelope of a Set of Lines , 1991, Inf. Process. Lett..

[5]  Leonidas J. Guibas,et al.  The upper envelope of piecewise linear functions: Algorithms and applications , 2015, Discret. Comput. Geom..

[6]  Leonidas J. Guibas,et al.  Topologically sweeping an arrangement , 1986, STOC '86.

[7]  Rephael Wenger,et al.  Ordered stabbing of pairwise disjoint convex sets in linear time , 1991, Discret. Appl. Math..

[8]  Herbert Edelsbrunner,et al.  Computing the Extreme Distances Between Two Convex Polygons , 1985, J. Algorithms.

[9]  Hiroshi Imai,et al.  Weighted Orthogonal Linear L∞-Approximation and Applications , 1989, WADS.

[10]  Mikhail J. Atallah,et al.  Efficient Algorithms for Common Transversals , 1987, Inf. Process. Lett..

[11]  Joseph O'Rourke,et al.  An on-line algorithm for fitting straight lines between data ranges , 1981, CACM.

[12]  David Avis,et al.  Lower Bounds for Line Stabbing , 1989, Inf. Process. Lett..

[13]  Martin E. Dyer,et al.  Linear Time Algorithms for Two- and Three-Variable Linear Programs , 1984, SIAM J. Comput..

[14]  Herbert Edelsbrunner.  Finding Transversals for Sets of Simple Geometric Figures , 1985, Theor. Comput. Sci..

[15]  Nimrod Megiddo,et al.  Linear-time algorithms for linear programming in R3 and related problems , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[16]  G. Toussaint,et al.  Finding the minimum vertex distance between two disjoint convex polygons in linear time , 1985 .

[17]  Rephael Wenger,et al.  Stabbing pairwise disjoint translates in linear time , 1989, SCG '89.

[18]  Jan van Leeuwen,et al.  Maintenance of Configurations in the Plane , 1981, J. Comput. Syst. Sci..

[19]  John Hershberger,et al.  Finding the Upper Envelope of n Line Segments in O(n log n) Time , 1989, Inf. Process. Lett..

[20]  V. Klee,et al.  Helly's theorem and its relatives , 1963 .

[21]  Kevin Q. Brown.  Geometric transforms for fast geometric algorithms , 1979 .