Efficiently approximating color-spanning balls

Abstract

Suppose n colored points with k colors in ${\mathbb{R}}^d$ are given. The Smallest Color-Spanning Ball (SCSB) is the smallest ball containing at least one point of each color. As the computation of the SCSB in $L_p$ metric ($p\ge 1$) is time-consuming, we focus on approximately computing the SCSB in near-linear time. Initially, we propose a 3-approximation algorithm running in $O(n\log n)$ time. This algorithm is then utilized to present a $(1+\epsilon )$-approximation algorithm with the running time of $O({(\frac{1}{\epsilon })}^{d}n\log n)$. We improve the running time to $O({(\frac{1}{\epsilon })}^{d}n)$ using randomized techniques. Afterward, spanning colors with two balls is studied. For a special case where $d=1$, there is an algorithm with $O(n^2)$ time. We demonstrate that for any $\epsilon >0$ under the assumption that SETH is true, no approximation algorithm running in $O(n^{2-\epsilon })$ time exists for the problem even in one-dimensional space. Nevertheless, we consider the $L_{\infty }$ metric where a ball is an axis-parallel hypercube and present a $(1+\epsilon )$-approximation algorithm running in $O({(\frac{1}{\epsilon })}^{2d}(\frac{n^2}{k}){\log }^{2}n)$ time which is remarkable when k is large. This time can be reduced to $O((\frac{1}{\epsilon })\frac{n^2}{k}\log n)$ when $d=1$.