How to Get Close to the Median Shape
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909994309386240 |
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| author | Har-Peled, Sariel |
| author_facet | Har-Peled, Sariel |
| contents | $\renewcommand{\Re}{\mathbb{R}}\newcommand{\eps}{\varepsilon}\newcommand{\poly}{\mathrm{poly}} $In this paper, we study the problem of $L_1$-fitting a shape to a set of $n$ points in $\Re^d$ (where $d$ is a fixed constant), where the target is to minimize the sum of distances of the points to the shape, or the sum of squared distances. We present a general technique for computing a $(1 + \eps ) $-approximation for such a problem, with running time $O(n + \poly( \log n, 1/\eps))$, where $\poly(\log n, 1/\eps)$ is a polynomial of constant degree of $\log n$ and $1/\eps$ (the power of the polynomial is a function of $d$). The new algorithm runs in linear time for a fixed $\eps>0$, and is the first subquadratic algorithm for this problem.
Applications of the algorithm include best fitting either a circle, a sphere, or a cylinder to a set of points when minimizing the sum of distances (or squared distances) to the respective shape. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12529 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | How to Get Close to the Median Shape Har-Peled, Sariel Computational Geometry $\renewcommand{\Re}{\mathbb{R}}\newcommand{\eps}{\varepsilon}\newcommand{\poly}{\mathrm{poly}} $In this paper, we study the problem of $L_1$-fitting a shape to a set of $n$ points in $\Re^d$ (where $d$ is a fixed constant), where the target is to minimize the sum of distances of the points to the shape, or the sum of squared distances. We present a general technique for computing a $(1 + \eps ) $-approximation for such a problem, with running time $O(n + \poly( \log n, 1/\eps))$, where $\poly(\log n, 1/\eps)$ is a polynomial of constant degree of $\log n$ and $1/\eps$ (the power of the polynomial is a function of $d$). The new algorithm runs in linear time for a fixed $\eps>0$, and is the first subquadratic algorithm for this problem. Applications of the algorithm include best fitting either a circle, a sphere, or a cylinder to a set of points when minimizing the sum of distances (or squared distances) to the respective shape. |
| title | How to Get Close to the Median Shape |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2601.12529 |