On the Most Likely Voronoi Diagram and Nearest Neighbor Searching

Let [Formula: see text] be a set of stochastic sites, where each site is a tuple [Formula: see text] consisting of a point [Formula: see text] in [Formula: see text]-dimensional space and a probability [Formula: see text] of existence. Given a query point [Formula: see text], we define its most likely nearest neighbor (LNN) as the site with the largest probability of being [Formula: see text]’s nearest neighbor. The Most Likely Voronoi Diagram (LVD) of [Formula: see text] is a partition of the space into regions with the same LNN. We investigate the complexity of LVD in one dimension and show that it can have size [Formula: see text] in the worst-case. We then show that under non-adversarial conditions, the size of the [Formula: see text]-dimensional LVD is significantly smaller: (1) [Formula: see text] if the input has only [Formula: see text] distinct probability values, (2) [Formula: see text] on average, and (3) [Formula: see text] under smoothed analysis. We also describe a framework for LNN search using Pareto sets, which gives a linear-space data structure and sub-linear query time in 1D for average and smoothed analysis models as well as the worst-case with a bounded number of distinct probabilities. The Pareto-set framework is also applicable to multi-dimensional LNN search via reduction to a sequence of nearest neighbor and spherical range queries.

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Parallel kNN Queries for Big Data Based on Voronoi Diagram Using MapReduce

Advances in Data Mining and Database Management - Handbook of Research on Innovative Database Query Processing Techniques ◽

10.4018/978-1-4666-8767-7.ch014 ◽

2015 ◽

pp. 392-414

Author(s):

Wei Yan

Keyword(s):

Big Data ◽

Voronoi Diagram ◽

Spatial Databases ◽

Nearest Neighbor ◽

Programming Model ◽

Dimensional Space ◽

Data Sets ◽

Two Dimensional ◽

K Nearest Neighbor ◽

K Nearest Neighbors

In cloud computing environments parallel kNN queries for big data is an important issue. The k nearest neighbor queries (kNN queries), designed to find k nearest neighbors from a dataset S for every object in another dataset R, is a primitive operator widely adopted by many applications including knowledge discovery, data mining, and spatial databases. This chapter proposes a parallel method of kNN queries for big data using MapReduce programming model. Firstly, this chapter proposes an approximate algorithm that is based on mapping multi-dimensional data sets into two-dimensional data sets, and transforming kNN queries into a sequence of two-dimensional point searches. Then, in two-dimensional space this chapter proposes a partitioning method using Voronoi diagram, which incorporates the Voronoi diagram into R-tree. Furthermore, this chapter proposes an efficient algorithm for processing kNN queries based on R-tree using MapReduce programming model. Finally, this chapter presents the results of extensive experimental evaluations which indicate efficiency of the proposed approach.

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TWO-DIMENSIONAL RANGE SEARCH BASED ON THE VORONOI DIAGRAM

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195905001634 ◽

2005 ◽

Vol 15 (02) ◽

pp. 151-166

Author(s):

TAKESHI KANDA ◽

KOKICHI SUGIHARA

Keyword(s):

Voronoi Diagram ◽

Dimensional Space ◽

Search Problem ◽

Two Dimensional ◽

Worst Case ◽

Average Case ◽

Range Search ◽

Almost All ◽

Set Of Points ◽

Dimensional Range

This paper studies the two-dimensional range search problem, and constructs a simple and efficient algorithm based on the Voronoi diagram. In this problem, a set of points and a query range are given, and we want to enumerate all the points which are inside the query range as quickly as possible. In most of the previous researches on this problem, the shape of the query range is restricted to particular ones such as circles, rectangles and triangles, and the improvement on the worst-case performance has been pursued. On the other hand, the algorithm proposed in this paper is designed for a general shape of the query range in the two-dimensional space, and is intended to accomplish a good average-case performance. This performance is actually observed by numerical experiments. In these experiments, we compare the execution time of the proposed algorithm with those of other representative algorithms such as those based on the bucketing technique and the k-d tree. We can observe that our algorithm shows the better performance in almost all the cases.

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Parallel kNN Queries for Big Data Based on Voronoi Diagram Using MapReduce

Geospatial Research ◽

10.4018/978-1-4666-9845-1.ch029 ◽

2016 ◽

pp. 644-665

Author(s):

Wei Yan

Keyword(s):

Big Data ◽

Voronoi Diagram ◽

Spatial Databases ◽

Nearest Neighbor ◽

Programming Model ◽

Dimensional Space ◽

Data Sets ◽

Two Dimensional ◽

K Nearest Neighbor ◽

K Nearest Neighbors

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A worst-case analysis of nearest neighbor searching by projection

Automata, Languages and Programming - Lecture Notes in Computer Science ◽

10.1007/3-540-10003-2_92 ◽

1980 ◽

pp. 470-482 ◽

Cited By ~ 11

Author(s):

Christos H. Papadimitriou ◽

Jon Louis Bentley

Keyword(s):

Nearest Neighbor ◽

Case Analysis ◽

Worst Case Analysis ◽

Worst Case ◽

Nearest Neighbor Searching

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Dynamic Distribution-Sensitive Point Location

ACM Transactions on Algorithms ◽

10.1145/3487403 ◽

2022 ◽

Vol 18 (1) ◽

pp. 1-63

Author(s):

Siu-Wing Cheng ◽

Man-Kit Lau

Keyword(s):

Voronoi Diagram ◽

Nearest Neighbor ◽

Query Point ◽

Point Location ◽

Construction Time ◽

Expected Time ◽

Polygonal Region ◽

Convex Subdivision ◽

Incremental Construction ◽

Dynamic Data Structure

We propose a dynamic data structure for the distribution-sensitive point location problem in the plane. Suppose that there is a fixed query distribution within a convex subdivision S , and we are given an oracle that can return in O (1) time the probability of a query point falling into a polygonal region of constant complexity. We can maintain S such that each query is answered in O opt (S) ) expected time, where opt ( S ) is the expected time of the best linear decision tree for answering point location queries in S . The space and construction time are O(n log 2 n ), where n is the number of vertices of S . An update of S as a mixed sequence of k edge insertions and deletions takes O(k log 4 n) amortized time. As a corollary, the randomized incremental construction of the Voronoi diagram of n sites can be performed in O(n log 4 n ) expected time so that, during the incremental construction, a nearest neighbor query at any time can be answered optimally with respect to the intermediate Voronoi diagram at that time.

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A γ dose distribution evaluation technique using the k-d tree for nearest neighbor searching

Medical Physics ◽

10.1118/1.3480964 ◽

2010 ◽

Vol 37 (9) ◽

pp. 4868-4873 ◽

Cited By ~ 14

Author(s):

Jiankui Yuan ◽

Weimin Chen

Keyword(s):

Dose Distribution ◽

Nearest Neighbor ◽

Evaluation Technique ◽

Nearest Neighbor Searching

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Integrated 3D Spatial Data Model of Object-Oriented Digital Landslide

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.204-208.4872 ◽

2012 ◽

Vol 204-208 ◽

pp. 4872-4877

Author(s):

Da Xi Ma ◽

Xiao Hong Liu ◽

Li Wei Ma

Keyword(s):

Spatial Data ◽

Data Model ◽

Dimensional Space ◽

Three Dimensional ◽

Data Modeling ◽

Object Oriented ◽

Landslide Modeling ◽

Space Data ◽

Spatial Data Model ◽

Three Dimensional Space

By analyzing the attributes of three-dimensional space data model, the integrated 3D spatial data adopts object-oriented method for digital landslide modeling. It achieves spatial data modeling for landslide geological entity. An experimental case is given to indicate the feasibility of this approach for spatial data modeling.

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KVGCN: A KNN Searching and VLAD Combined Graph Convolutional Network for Point Cloud Segmentation

Remote Sensing ◽

10.3390/rs13051003 ◽

2021 ◽

Vol 13 (5) ◽

pp. 1003

Author(s):

Nan Luo ◽

Hongquan Yu ◽

Zhenfeng Huo ◽

Jinhui Liu ◽

Quan Wang ◽

...

Keyword(s):

Point Cloud ◽

Nearest Neighbor ◽

Semantic Segmentation ◽

K Nearest Neighbor ◽

Topological Graph ◽

Convolutional Network ◽

Cloud Data ◽

Nearest Neighbor Searching ◽

Point Cloud Segmentation ◽

Local Feature Extraction

Semantic segmentation of the sensed point cloud data plays a significant role in scene understanding and reconstruction, robot navigation, etc. This work presents a Graph Convolutional Network integrating K-Nearest Neighbor searching (KNN) and Vector of Locally Aggregated Descriptors (VLAD). KNN searching is utilized to construct the topological graph of each point and its neighbors. Then, we perform convolution on the edges of constructed graph to extract representative local features by multiple Multilayer Perceptions (MLPs). Afterwards, a trainable VLAD layer, NetVLAD, is embedded in the feature encoder to aggregate the local and global contextual features. The designed feature encoder is repeated for multiple times, and the extracted features are concatenated in a jump-connection style to strengthen the distinctiveness of features and thereby improve the segmentation. Experimental results on two datasets show that the proposed work settles the shortcoming of insufficient local feature extraction and promotes the accuracy (mIoU 60.9% and oAcc 87.4% for S3DIS) of semantic segmentation comparing to existing models.

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