Linear Approximation of Simple Objects

Let S be a set of m convex polygons in the plane with a total number of n vertices. Each polygon has a positive weight. This paper presents algorithms to solve the weighted minmax approximation and the weighted minsum approximation problems. For the first problem, a line minimizing the maximum weighted distance to the polygons can be found in O(n2 log n) time and O(n2) space. The time and space complexities can be reduced to O(n log n) and O(n), respectively, when the weights are equal. For the second problem, a line minimizing the sum of the weighted distances to the polygons can be found in O(n2 log n) time and O(n) space. For both problems, we also obtain similar results for sets of n circles or line segments.

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

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

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

[4]  James G. Morris,et al.  Linear facility location -- Solving extensions of the basic problem , 1983 .

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

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

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

[8]  Michael Ian Shamos,et al.  Geometric intersection problems , 1976, 17th Annual Symposium on Foundations of Computer Science (sfcs 1976).

[9]  John Hershberger,et al.  Sweeping arrangements of curves , 1989, SCG '89.

[10]  M. Iri,et al.  Polygonal Approximations of a Curve — Formulations and Algorithms , 1988 .

[11]  Godfried T. Toussaint,et al.  Computing the Width of a Set , 1988, IEEE Trans. Pattern Anal. Mach. Intell..

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

[13]  Leonidas J. Guibas,et al.  The exact fitting problem for points , 1991 .

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

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