Improved Search of Relevant Points for Nearest-Neighbor Classification

Given a training set $P \subset \mathbb{R}^d$, the nearest-neighbor classifier assigns any query point $q \in \mathbb{R}^d$ to the class of its closest point in $P$. To answer these classification queries, some training points are more relevant than others. We say a training point is relevant if its omission from the training set could induce the misclassification of some query point in $\mathbb{R}^d$. These relevant points are commonly known as border points, as they define the boundaries of the Voronoi diagram of $P$ that separate points of different classes. Being able to compute this set of points efficiently is crucial to reduce the size of the training set without affecting the accuracy of the nearest-neighbor classifier. Improving over a decades-long result by Clarkson, in a paper by Eppstein an output-sensitive algorithm was proposed to find the set of border points of $P$ in $O( n² + nk² )$ time, where $k$ is the size of such set. In this paper, we improve this algorithm to have time complexity equal to $O( nk² )$ by proving that the first steps of their algorithm, which require $O( n² )$ time, are unnecessary.