An Output-Sensitive Approach for the L 1/L ∞ k-Nearest-Neighbor Voronoi Diagram
نویسندگان
چکیده
This paper revisits the k-nearest-neighbor (k-NN) Voronoi diagram and presents the first output-sensitive paradigm for its construction. It introduces the k-NN Delaunay graph, which corresponds to the graph theoretic dual of the k-NN Voronoi diagram, and uses it as a base to directly compute the k-NN Voronoi diagram in R. In the L1, L∞ metrics this results in O((n + m) log n) time algorithm, using segment-dragging queries, where m is the structural complexity (size) of the k-NN Voronoi diagram of n point sites in the plane. The paper also gives a tighter bound on the structural complexity of the k-NN Voronoi diagram in the L∞ (equiv. L1) metric, which is shown to be O(min{k(n− k), (n− k)}).
منابع مشابه
On the Most Likely Voronoi Diagramand Nearest Neighbor Searching
We consider the problem of nearest-neighbor searching among a set of stochastic sites, where a stochastic site is a tuple (si,⇡i) consisting of a point si in a d-dimensional space and a probability ⇡i determining its existence. The problem is interesting and non-trivial even in 1-dimension, where the Most Likely Voronoi Diagram (LVD) is shown to have worst-case complexity ⌦(n). We then show tha...
متن کاملThe Complexity of Finding Minimal Voronoi Covers with Applications to Machine Learning
Our goal in this paper is to examine the application of Voronoi diagrams, a fundamental concept of computational geometry, to the nearest neighbor algorithm used in machine learning. We consider the question “Given a planar polygonal tessellation T and an integer k, is there a set of k points whose Voronoi diagram contains every edge in T?” We show that this question is NP-hard. We encountered ...
متن کاملVoronoi Projection-Based Fast Nearest-Neighbor Search Algorithms: Box-Search and Mapping Table-Based Search Techniques
In this paper we consider fast nearest-neighbor search techniques based on the projections of Voronoi regions. The Voronoi diagram of a given set of points provides an implicit geometric interpretation of nearest-neighbor search and serves as an important basis for several proximity search algorithms in computational geometry and in developing structure-based fast vector quantization techniques...
متن کامل(Approximate) Conic Nearest Neighbors and the induced Voronoi Diagram
For a given point set in Euclidean space we consider the problem of finding (approximate) nearest neighbors of a query point but restricting only to points that lie within a fixed cone with apex at the query point. Apart from being a rather natural question to ask, solutions to this problem have applications in surface reconstruction and dimension detection. We investigate the structure of the ...
متن کاملNon-zero probability of nearest neighbor searching
Nearest Neighbor (NN) searching is a challenging problem in data management and has been widely studied in data mining, pattern recognition and computational geometry. The goal of NN searching is efficiently reporting the nearest data to a given object as a query. In most of the studies both the data and query are assumed to be precise, however, due to the real applications of NN searching, suc...
متن کامل