How to Get Close to the Median Shape
Sariel Har-Peled
TL;DR
The paper introduces a general technique for $(1+\varepsilon)$-approximate $L_1$ and $L_2$ fitting of parameterized shapes to point sets in fixed dimension, achieving $O(n + \mathrm{poly}(\log n, 1/\varepsilon))$ time and, for fixed $\varepsilon$, linear time. It achieves this via a two-pronged approach: a 1D coreset construction to compress 1D fitting problems and a level-based reduction that replaces a large family of surfaces with a small, manageable set of patches while preserving the objective within $(1+\varepsilon)$. The main result follows by combining these reductions with a slower but robust approximation algorithm, yielding a scalable, randomized algorithm for circle/sphere/cylinder fitting and for partial-data 1-median clustering. This work demonstrates subquadratic performance for a broad class of nonlinear shape-fitting problems and provides a foundation for practical, fast geometric fitting in low dimensions, albeit with large hidden constants. It also opens avenues for extensions to other geometric objects and to core-set-based pipelines.
Abstract
$\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.