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Approximating Points by a Piecewise Linear Function

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Abstract

Approximating points by piecewise linear functions is an intensively researched topic in computational geometry. In this paper, we study, based on the uniform error metric, an array of variations of this problem in 2-D and 3-D, including points with weights, approximation with violations, using step functions or more generally piecewise linear functions. We consider both the min-# (i.e., given an error tolerance ϵ, minimizing the size k of the approximating function) and min-ϵ (i.e., given a size k of the approximating function, minimizing the error tolerance ϵ) versions of the problems. Our algorithms either improve on the previously best-known solutions or are the first known results for the respective problems. Our approaches are based on interesting geometric observations and algorithmic techniques. Some data structures we develop are of independent interest and may find other applications.

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Notes

  1. A preliminary version of this paper [19] gave an O(nlogn,log3 n) time result, which has been found incorrect.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN, 46556, USA

    Danny Z. Chen & Haitao Wang

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  1. Danny Z. Chen
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  2. Haitao Wang
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Correspondence to Haitao Wang.

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This research was supported in part by NSF under Grants CCF-0515203 and CCF-0916606.

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Chen, D.Z., Wang, H. Approximating Points by a Piecewise Linear Function. Algorithmica 66, 682–713 (2013). https://doi.org/10.1007/s00453-012-9658-y

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