Computational-geometric methods for polygonal approximations of a curve
Computer Vision, Graphics, and Image Processing
Algorithms for vertical and orthogonal L1 linear approximation of points
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
Linear programming and convex hulls made easy
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Fitting polygonal functions to a set of points in the plane
CVGIP: Graphical Models and Image Processing
Orthogonal weighted linear L1 and L∞ approximation and applications
Discrete Applied Mathematics
On approximating polygonal curves in two and three dimensions
CVGIP: Graphical Models and Image Processing
Off-line maintenance of planar configurations
Journal of Algorithms
Linear Programming in Linear Time When the Dimension Is Fixed
Journal of the ACM (JACM)
A new algorithm for fitting a rectilinear x-monotone curve to a set of points in the plane
Pattern Recognition Letters
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Rectilinear approximation of a set of points in the plane
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
Fitting a Step Function to a Point Set
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
A randomized algorithm for weighted approximation of points by a step function
COCOA'10 Proceedings of the 4th international conference on Combinatorial optimization and applications - Volume Part I
A deterministic algorithm for fitting a step function to a weighted point-set
Information Processing Letters
A note on searching line arrangements and applications
Information Processing Letters
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We consider the problem of weighted rectilinear approximation on the plane and offer both exact algorithms and heuristics with provable performance bounds. Let S = {(pi, wi)} be a set of n points pi in the plane, with associated distance-modifying weights wi 0. We present algorithms for finding the best fit to S among x-monotone rectilinear polylines R with a given number k n of horizontal segments. We measure the quality of the fit by the greatest weighted vertical distance, i.e., the approximation error is max1≤i≤n widv(pi, R), where dv(pi, R) is the vertical distance from pi to R. We can solve for arbitrary k optimally in O(n2) or approximately in O(n log2 n) time. We also describe a randomized algorithm with an O(n log2 n) expected running time for the unweighted case and describe how to modify it to handle the weighted case in O(n log3 n) expected time. All algorithms require O(n) space.