Weighted Rectilinear Approximation of Points in the Plane (original) (raw)

Fitting rectilinear polygonal curves to a set of points in the plane

European Journal of Operational Research, 2001

In this paper two problems of ®tting rectilinear polygonal curves to a set of points in the plane according to the minimax approximation are considered. The constraints are, respectively, on the number of vertices and length of the polygonal curve. In both cases ecient algorithms are developed.

Approximating a set of points by a step function

2006

Let S be a set of n points in the plane. We derive algorithms for approximating S by a step function of size k < n, i.e., by an x-monotone rectilinear polyline R with k < n horizontal segments. We use the vertical distance to measure the quality of the approximation, i.e., the maximum distance from a point in S to the horizontal segment directly above or below it. We consider two types of problems: min-e, where the goal is to minimize the error for a given number of horizontal segments k and min-#, where the goal is to minimize the number of segments for a given allowed error e. After O (n) preprocessing time, we solve instances of the latter in O (min{k log n, n}) time per instance. We can then solve the former problem in O (min{n 2 , nk log n}) time. Both algorithms require O (n) space. Our second contribution is an approximation algorithm for the min-e problem that computes a solution within a factor of 3 of the optimal error for k segments, or with at most the same error as the k-optimal but using 2k À 1 segments. Furthermore, experiments on real data show even better results than what is guaranteed by the theoretical bounds. Both approximations run in O (n log n) time and O (n) space.

Computational-geometric methods for polygonal approximations of a curve

1986

In cartography, computer graphics, pattern recognition, etc., we often encounter the problem of approximating a given finer piecewise linear curve by another coarser piecewise linear curve consisting of fewer line segments. In connection with this problem, a number of papers have been published, but it seems that the problem itself has not been well modelled from the standpoint of specific applications, nor has a nice algorithm, nice from the computational-geometric viewpoint, been proposed. In the present paper, we first consider (i) the problem of approximating a piecewise linear curve by another whose vertices are a subset of the vertices of the former, and show that an optimum solution of this problem can be found in a polynomial time. We also mention recent results on related problems by several researchers including the authors themselves. We then pose (ii) a problem of covering a sequence of n points by a minimum number of rectangles with a given width, and present an O(n log n)-time algorithm by making use of some fundamental established techniques in computational geometry. Furthermore, an O(mn(log n)2)-time algorithm is presented for finding the minimum width w such that a sequence of n points can be covered by at most m rectangles with width w. Finally, (iii) several related problems are discussed. 9

Minimax and Maximin Fitting of Geometric Objects to Sets of Points

2011

This thesis addresses several problems in the facility location sub-area of computational geometry. Let S be a set of n points in the plane. We derive algorithms for approximating S by a step function curve of size k < n, i.e., by an x-monotone orthogonal polyline R with k < n horizontal segments. We use the vertical distance to measure the quality of the approximation, i.e., the maximum distance from a point in S to the horizontal segment directly above or below it. We consider two types of problems: min-ε, where the goal is to minimize the error for a given number of horizontal segments k and min-#, where the goal is to minimize the number of segments for a given allowed error ε. After O(n) preprocessing time, we solve instances of the latter in O(min{k log n, n}) time per instance. We can then solve the former problem in O(min{n 2 , nk log n}) time. Both algorithms require O(n) space. The second contribution is a heuristic for the min-ε problem that computes a solution within a factor of 3 of the optimal error for k segments, or with at most the same error as the k-optimal but using 2k − 1 segments. Furthermore, experiments on real data show even better results than what is guaranteed by the theoretical bounds. Both approximations run in O(n log n) time and O(n) space. Then, we present an exact algorithm for the weighted version of this problem that runs in O(n 2) time and generalize the heuristic to handle weights at the expense of an additional log n factor. At this point, a randomized I thank my advisor Mario Lopez for his many years of close friendship and mentoring. His patience, advice, and consistent encouragement were absolutely invaluable to me in completing this dissertation and degree. I also thank my committee members, Alvaro Arias, Chris GauthierDickey, Scott Leutenegger, and Ronald DeLyser, for their time, advice, and comments. In addition, I would like to thank Drs. GauthierDickey and DeLyser for the many grammatical and presentation suggestions to my thesis. I thank again my advisor Mario Lopez for inviting me to continue my graduate school experience with doctoral work. I also want to thank the faculty at the departments of Computer Science and Mathematics whose courses I have taken at the University of Denver throughout all my degrees. I am grateful to all of you. Last, but not least, I wholeheartedly thank my graduate student colleagues and coauthors, Mohammed Al-Bow and Riquelmi Cardona, for their friendship, collaboration, conversations, and all the time we have spent together. I also thank Jeff Edgington for leading the way and showing me the path to graduation and freedom. All of you friends and colleagues, whose names are too many to mention here, have imparted your knowledge and wisdom to me, and I thank you all. Most importantly, I thank my parents, Boris and Rita Mayster, and my brother, Dmitriy Mayster, for their love and attention.

Algorithms for computing shape preserving spline approximations to data

Numerische Mathematik, 1985

We treat the problem of approximating data that are sampled with error from a function known to be convex and increasing. The approximating function is a polynomial spline with knots at the data points. This paper presents results (analogous to those in [7] and [9]) that describe some approximation properties of polynomial splines and algorithms for determining the existence of a "shape-preserving" approximant for given data.

An algorithm for best approximation of a line by lattice points in three dimensions

1995

In this paper we present an algorithm for finding successive best approximations of a line by lattice points in three dimensions. Our algorithm is primarily an extension of the work of Furtwängler [13], which has been generalised for arbitrary radius functions, lattices and initial bases. We show that, after a finite number of initialisation iterations, the algorithm will produce all best approximations to the line above a certain height. Conversely, we show that, after initialisation, all convergents of the algorithm are best approximations (with one possible exception). We also provide a numerical example to illustrate the algorithm.

Approximate Line Nearest Neighbor in High Dimensions

Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, 2009

We consider the problem of approximate nearest neighbors in high dimensions, when the queries are lines. In this problem, given n points in R d , we want to construct a data structure to support efficiently the following queries: given a line L, report the point p closest to L. This problem generalizes the more familiar nearest neighbor problem. From a practical perspective, lines, and low-dimensional flats in general, may model data under linear variation, such as physical objects under different lighting. For approximation 1 + , we achieve a query time of d 3 n 0.5+t , for arbitrary small t > 0, with a space of d 2 n O(1/ 2 +1/t 2). To the best of our knowledge, this is the first algorithm for this problem with polynomial space and sub-linear query time.

Maintaining the Approximate Width of a Set of Points in the Plane

The width of a set of n points in the plane is the smallest distance between two parallel lines that enclose the set. We maintain the set of points under insertions and deletions of points and we are able to report an approximation of the width of this dynamic point set. Our data structure takes linear space and allows for reporting the approximation with relative accuracy in O(p 1= log n) time; and the update time is O(log 2 n). The method uses the tentative prune-and-search strategy of Kirkpatrick and Snoeyink.

All Approximating Segments for a Sequence of Points

2014

In this paper, we consider the problem of approximating a sequence of n points by a line segment in such a way that the distance of each point from this segment is not greater than a given constant. Furthermore, the distance between the rst(last) input point and the start(end)-point of the approximating segment must not be greater than the given constant. This is a sub-problem in solving unrestricted line simplication and minimum-link path problems. We propose an O(n logn) algorithm for computing a representation of these segments and we prove that the lower time complexity of nding all such segments (in a specic representation) is (n logn) on the algebraic computation tree model which means that our algorithm is optimal.