How to Get Close to the Median Shape

TL;DR

This paper introduces a fast, near-linear time algorithm for approximating the best fit shape (circle, sphere, or cylinder) to a set of points in fixed dimensions, minimizing the sum of distances or squared distances.

Contribution

It presents a novel general technique for $(1 + ext{eps})$-approximate $L_1$-shape fitting with linear time complexity in the number of points, improving over previous methods.

Findings

First subquadratic algorithm for $L_1$ shape fitting.

Achieves $(1 + ext{eps})$-approximation in $O(n + ext{poly}( ext{log} n, 1/ ext{eps}))$ time.

Applicable to fitting circles, spheres, and cylinders with minimized distances.

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.