scholarly journals Dynamic Distribution-Sensitive Point Location

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.

scholarly journals On the Most Likely Voronoi Diagram and Nearest Neighbor Searching

2016 ◽  
Vol 26 (03n04) ◽  
pp. 151-166 ◽  
Author(s):  
Subhash Suri ◽  
Kevin Verbeek
Keyword(s):  
Voronoi Diagram ◽  
Nearest Neighbor ◽  
Dimensional Space ◽  
Query Point ◽  
Worst Case ◽  
Smoothed Analysis ◽  
Bounded Number ◽  
Nearest Neighbor Searching ◽  
Space Data ◽  
Analysis Models

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.

A SIMPLE ON-LINE RANDOMIZED INCREMENTAL ALGORITHM FOR COMPUTING HIGHER ORDER VORONOI DIAGRAMS

1992 ◽  
Vol 02 (04) ◽  
pp. 363-381 ◽  
Author(s):  
FRANZ AURENHAMMER ◽  
OTFRIED SCHWARZKOPF
Keyword(s):  
Voronoi Diagram ◽  
Structural Changes ◽  
Nearest Neighbor ◽  
Voronoi Diagrams ◽  
Simple Algorithm ◽  
Incremental Algorithm ◽  
K Nearest Neighbor ◽  
On Line ◽  
Incremental Construction ◽  
Primal Space

We present a simple algorithm for maintaining order-k Voronoi diagrams in the plane. By using a duality transform that is of interest in its own right, we show that the insertion or deletion of a site involves little more than the construction of a single convex hull in three-space. In particular, the order-k Voronoi diagram for n sites can be computed in time [Formula: see text] and optimal space [Formula: see text] by an on-line randomized incremental algorithm. The time bound can be improved by a logarithmic factor without losing much simplicity. For k≥ log 2 n, this is optimal for a randomized incremental construction; we show that the expected number of structural changes during the construction is ⊝(nk2). Finally, by going back to primal space, we obtain a dynamic data structure that supports k-nearest neighbor queries, insertions, and deletions in a planar set of sites. The structure promises easy implementation, exhibits a satisfactory expected performance, and occupies no more storage than the current order-k Voronoi diagram.

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

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.

eDAR Algorithm for Continuous KNN Queries Based on Pine

2011 ◽  
pp. 154-174
Author(s):  
Maytham Safar ◽  
Dariush Ebrahimi
Keyword(s):  
Shortest Path ◽  
Nearest Neighbor ◽  
Query Point ◽  
Spatial Network ◽  
K Nearest Neighbor ◽  
Network Expansion ◽  
Shortest Distance ◽  
Stationary Objects ◽  
Split Point ◽  
Euclidean Distances

The continuous K nearest neighbor (CKNN) query is an important type of query that finds continuously the KNN to a query point on a given path. We focus on moving queries issued on stationary objects in Spatial Network Database (SNDB) The result of this type of query is a set of intervals (defined by split points) and their corresponding KNNs. This means that the KNN of an object traveling on one interval of the path remains the same all through that interval, until it reaches a split point where its KNNs change. Existing methods for CKNN are based on Euclidean distances. In this paper we propose a new algorithm for answering CKNN in SNDB where the important measure for the shortest path is network distances rather than Euclidean distances. We propose DAR and eDAR algorithms to address CKNN queries based on the progressive incremental network expansion (PINE) technique. Our experiments show that the eDAR approach has better response time, and requires fewer shortest distance computations and KNN queries than approaches that are based on VN3 using IE.

A proposition of querying scheme with network Voronoi diagram in bichromatic reverse k-nearest neighbor

2017 ◽  
Vol 13 (1) ◽  
pp. 62-75 ◽  
Author(s):  
Yusuke Gotoh ◽  
Chiori Okubo
Keyword(s):  
Voronoi Diagram ◽  
Design Methodology ◽  
Processing Time ◽  
Nearest Neighbor ◽  
Spatial Networks ◽  
K Nearest Neighbor ◽  
Influence Zone ◽  
Content Type ◽  
Future Direction ◽  
Practical Implications

Purpose This study aims to propose and evaluate a searching scheme for a bichromatic reverse k-nearest neighbor (BRkNN) that has objects and queries in spatial networks. In this proposed scheme, the author’s search for the BRkNN of the query using an influence zone for each object with a network Voronoi diagram (NVD). Design/methodology/approach The author’s analyze and evaluate the performance of the proposed searching scheme. Findings The contribution of this paper is that it confirmed that the proposed searching scheme gives shorter processing time than the conventional linear search. Research limitations/implications A future direction of this study will involve making a searching scheme that reduces the processing time when objects move automatically on spatial networks. Practical implications In BRkNN, consider two groups in a convenience store, where several convenience stores, which are constructed in Groups A and B, operate in a given region. The author’s can use RNN is RkNN when k = 1 (RNN) effectively to set a new store considering the Euclidean and road distances among stores and the location relationship between Groups A and B. Originality/value In the proposed searching scheme, the author’s search for the BRkNN of the query for each object with an NVD using the influence zone, which is the region where an object in the spatial network recognizes the nearest neighbor for the query.

scholarly journals REVERSE NEAREST NEIGHBOR QUERIES IN FIXED DIMENSION

2011 ◽  
Vol 21 (02) ◽  
pp. 179-188 ◽  
Author(s):  
OTFRIED CHEONG ◽  
ANTOINE VIGNERON ◽  
JUYOUNG YON
Keyword(s):  
Data Structure ◽  
Nearest Neighbor ◽  
Dimensional Euclidean Space ◽  
Query Point ◽  
Reverse Nearest Neighbor ◽  
Input Point ◽  
Fixed Dimension ◽  
Nearest Point ◽  
Point Set ◽  
Nearest Neighbor Queries

Reverse nearest neighbor queries are defined as follows: Given an input point set P, and a query point q, find all the points p in P whose nearest point in P ∪ {q} \ {p} is q. We give a data structure to answer reverse nearest neighbor queries in fixed-dimensional Euclidean space. Our data structure uses O(n) space, its preprocessing time is O(n log n), and its query time is O( log n).

A LINEAR-TIME RANDOMIZED ALGORITHM FOR THE BOUNDED VORONOI DIAGRAM OF A SIMPLE POLYGON

1996 ◽  
Vol 06 (03) ◽  
pp. 263-278 ◽  
Author(s):  
ROLF KLEIN ◽  
ANDRZEJ LINGAS
Keyword(s):  
Voronoi Diagram ◽  
Delaunay Triangulation ◽  
Spanning Tree ◽  
Linear Time ◽  
Randomized Algorithm ◽  
Simple Polygon ◽  
Minimal Spanning Tree ◽  
Expected Time

For a polygon P, the bounded Voronoi diagram of P is a partition of P into regions assigned to the vertices of P. A point p inside P belongs to the region of a vertex v if and only if v is the closest vertex of P visible from p. We present a randomized algorithm that builds the bounded Voronoi diagram of a simple polygon in linear expected time. Among other applications, we can construct within the same time bound the generalized Delaunay triangulation of P and the minimal spanning tree on P’s vertices that is contained in P.

ABSTRACT VORONOI DIAGRAMS WITH DISCONNECTED REGIONS

2014 ◽  
Vol 24 (04) ◽  
pp. 347-372 ◽  
Author(s):  
CECILIA BOHLER ◽  
ROLF KLEIN
Keyword(s):  
Hausdorff Metric ◽  
Voronoi Diagrams ◽  
Construction Technique ◽  
Voronoi Region ◽  
Expected Time ◽  
Randomized Incremental Construction ◽  
Concrete Type ◽  
Trapezoidal Decomposition ◽  
Number Of Faces ◽  
Incremental Construction

Abstract Voronoi diagrams, AVDs for short, are based on bisecting curves enjoying simple combinatorial properties, rather than on the geometric notions of sites and distance. They serve as a unifying concept. Once the bisector system of any concrete type of Voronoi diagram is shown to fulfill the AVD axioms, structural results and efficient algorithms become available without further effort; for example, the first optimal algorithms for constructing nearest Voronoi diagrams of disjoint convex objects, or of line segments under the Hausdorff metric, have been obtained this way. One of these axioms stated that all Voronoi regions must be pathwise connected, a property quite useful in divide&conquer and randomized incremental construction algorithms. Yet, there are concrete Voronoi diagrams where this axiom fails to hold. In this paper we consider, for the first time, abstract Voronoi diagrams with disconnected regions. By combining a randomized incremental construction technique with trapezoidal decomposition we obtain an algorithm that runs in expected time [Formula: see text], where s is the maximum number of faces a Voronoi region in a subdiagram of three sites can have, and where mj denotes the average number of faces per region in any subdiagram of j sites. In the connected case, where s = 1 = mj , this results in the known optimal bound [Formula: see text].

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