Proximity Algorithms for Nearly-Doubling Spaces
نویسندگان
چکیده
We introduce a new problem in the study of doubling spaces: Given a point set S and a target dimension d∗, remove from S the fewest number of points so that the remaining set has doubling dimension at most d∗. We present a bicriteria approximation for this problem, and extend this algorithm to solve a group of proximity problems.
منابع مشابه
Space-Time Tradeoffs for Proximity Searching in Doubling Spaces
We consider approximate nearest neighbor searching in metric spaces of constant doubling dimension. More formally, we are given a set S of n points and an error bound ε > 0. The objective is to build a data structure so that given any query point q in the space, it is possible to efficiently determine a point of S whose distance from q is within a factor of (1 + ε) of the distance between q and...
متن کاملAdaptive Metric Dimensionality Reduction
We study data-adaptive dimensionality reduction in the context of supervised learning in general metric spaces. Our main statistical contribution is a generalization bound for Lipschitz functions in metric spaces that are doubling, or nearly doubling, which yields a new theoretical explanation for empirically reported improvements gained by preprocessing Euclidean data by PCA (Principal Compone...
متن کاملAlgorithmic Interpretations of Fractal Dimension
We study algorithmic problems on subsets of Euclidean space of low fractal dimension. These spaces are the subject of intensive study in various branches of mathematics, including geometry, topology, and measure theory. There are several well-studied notions of fractal dimension for sets and measures in Euclidean space. We consider a definition of fractal dimension for finite metric spaces whic...
متن کاملNon-Archimedean fuzzy metric spaces and Best proximity point theorems
In this paper, we introduce some new classes of proximal contraction mappings and establish best proximity point theorems for such kinds of mappings in a non-Archimedean fuzzy metric space. As consequences of these results, we deduce certain new best proximity and fixed point theorems in partially ordered non-Archimedean fuzzy metric spaces. Moreover, we present an example to illustrate the us...
متن کاملDiameter Approximate Best Proximity Pair in Fuzzy Normed Spaces
The main purpose of this paper is to study the approximate best proximity pair of cyclic maps and their diameter in fuzzy normed spaces defined by Bag and Samanta. First, approximate best point proximity points on fuzzy normed linear spaces are defined and four general lemmas are given regarding approximate fixed point and approximate best proximity pair of cyclic maps on fuzzy normed spaces. U...
متن کامل