Higher-order Voronoi Diagrams Research Articles

Unbounded Regions of High-Order Voronoi Diagrams of Lines and Line Segments in Higher Dimensions

AbstractWe study the behavior at infinity of the farthest and the higher-order Voronoi diagram of n line segments or lines in a d-dimensional Euclidean space. The unbounded parts of these diagrams can be encoded by a Gaussian map on the sphere of directions $$\mathbb {S}^{d-1}$$ S d - 1 . We show that the combinatorial complexity of the Gaussian map for the order-k Voronoi diagram of n line segments and lines is $$O(\min \{k,n-k\}n^{d-1})$$ O ( min { k , n - k } n d - 1 ) , which is tight for $$n-k=O(1)$$ n - k = O ( 1 ) . This exactly reflects the combinatorial complexity of the unbounded features of these diagrams. All the d-dimensional cells of the farthest Voronoi diagram are unbounded, its $$(d-1)$$ ( d - 1 ) -skeleton is connected, and it does not have tunnels. A d-cell of the Voronoi diagram is called a tunnel if the set of its unbounded directions, represented as points on its Gaussian map, is not connected. In a three-dimensional space, the farthest Voronoi diagram of $$n \ge 2$$ n ≥ 2 lines in general position has exactly $$n(n-1)$$ n ( n - 1 ) three-dimensional cells. The Gaussian map of the farthest Voronoi diagram of line segments and lines can be constructed in $$O(n^{d-1} \alpha (n))$$ O ( n d - 1 α ( n ) ) time, for $$d\ge 4$$ d ≥ 4 , while if $$d=3$$ d = 3 , the time drops to worst-case optimal $$\Theta (n^2)$$ Θ ( n 2 ) . We extend the obtained results to bounded polyhedra and clusters of points as sites.

Analyzing the Squared Distance-to-Measure Gradient Flow System with k-Order Voronoi Diagrams

Point cloud data arises naturally from 3-dimensional scanners, LiDAR sensors, and industrial computed tomography (i.e. CT scans) among other sources. Most point clouds obtained through experimental means exhibit some level of noise, inhibiting mesh reconstruction algorithms and topological data analysis techniques. To alleviate the problems caused by noise, smoothing algorithms are often employed as a preprocessing step. Moving least squares is one such technique, however, many of these techniques are designed to work on surfaces in $$\mathbb {R}^3$$ . As interesting point clouds can naturally live in higher dimensions, we seek a method for smoothing higher dimensional point clouds. To this end, we turn to the distance to measure function. In this paper, we provide a theoretical foundation for studying the gradient flow induced by the squared distance to measure function, as introduced by Chazal, Cohen-Steiner, and Merigot. In particular, we frame the gradient flow as a Filippov system and find a method for solving the squared distance to measure gradient flow, induced by the uniform empirical measure, using higher order Voronoi diagrams. In contrast to some existing techniques, this gradient flow provides a smoothing algorithm which computationally scales with dimensionality.

Solving some vector subset problems by Voronoi diagrams

We propose a general approach to solving some vector subset problems in a Euclidean space that is based on higher-order Voronoi diagrams. In the case of a fixed space dimension, this approach allows us to find optimal solutions to these problems in polynomial time which is better than the runtime of available algorithms.

Tessellations in GIS: Part II–making changes

We attempt to describe the role of tessellated models of space within the discipline of Geographic Information Systems (GIS) – a speciality coming largely out of Geography and Land Surveying, where there was a strong need to represent information about the land’s surface within a computer system rather than on the original paper maps. We look at some of the basic operations in GIS, including dynamic and kinetic applications. We examine issues of topology and data structures, and produced a tessellation model that may be widely applied both to traditional “object” and “field” data types. The Part I of this study examined object and field spatial models, the Voronoi extension of objects, and the graphs that express the resulting adjacencies. The required data structures were also briefly described, along with 2D and 3D structures and hierarchical indexing. The importance of graph duality was emphasized. Here, this second paper builds on the structures described in the first, and examines how these may be modified: change may often be associated with either viewpoint or time. Incremental algorithms permit additional point insertion, and applications involving the addition of skeleton points, for map scanning, contour enrichment or watershed delineation and simulation. Dynamic algorithms permit skeleton smoothing, and higher order Voronoi diagram applications, including Sibson interpolation. Kinetic algorithms allow collision detection applications, free-Lagrange flow modeling, and pen movement simulation for map drawing. If desired these methods may be extended to 3D. Based on this framework, it can be argued that tessellation models are fundamental to our understanding and processing of geographical space, and provide a coherent framework for understanding the “space” in which we exist.

Map segmentation for geospatial data mining through generalized higher-order Voronoi diagrams with sequential scan algorithms

Segmentation is one popular method for geospatial data mining. We propose efficient and effective sequential-scan algorithms for higher-order Voronoi diagram districting. We extend the distance transform algorithm to include complex primitives (point, line, and area), Minkowski metrics, different weights and obstacles for higher-order Voronoi diagrams. The algorithm implementation is explained along with efficiencies and error. Finally, a case study based on trade area modeling is described to demonstrate the advantages of our proposed algorithms.

Voronoi Tessellation Based Interpolation Method for Wi-Fi Radio Map Construction

The fingerprint-based approach for positioning in WLAN has been drawing great attention these days. However, the approach usually requires tremendous time and efforts to collect location fingerprints for the target area. In this paper, we propose an interpolation method based on Voronoi tessellation to significantly reduce such calibration efforts and to improve accuracy. Unlike other interpolation methods, our method refines the propagation model for each cell of the target area tessellated by a higher-order Voronoi diagram. Consequently, our method can take into account the signal fading caused by walls and obstacles more accurately. The proposed method significantly outperformed other interpolation methods in accuracy.

AN ORDER-k VORONOI APPROACH TO GEOSPATIAL CONCEPT MANAGEMENT WITHIN CONCEPTUAL SPACES

Representing concepts and their boundaries is a core task in concept management. We introduce a flexible framework for representing concept boundaries and tessellations through the complete higher order Voronoi diagrams within conceptual spaces. The framework provides different levels of concept granularity modelling vagueness and indeterminacy of concept boundaries, effectively supports complex concepts of multi-pieced regions with holes, and robustly supports concept learning and formation. We also outline a generic data structure that systematically supports our concept management framework. Experimental results demonstrate the robustness and flexibility of the proposed framework.

Arrival-Time Order Location Revisited

Abstract A fast method to estimate an earthquake epicentral region from the very first P -wave arrivals is relevant in the context of early warning and rapid response systems. Arrival-time order location (AOL) uses the sequence in which an array of sensors detects an event to constrain the epicentral region based on simple geometrical relations. No velocity model is needed. This article recasts AOL using the theory of spatial tessellations and describes a solution based on higher order Voronoi diagrams. The implementation can take advantage of modern computational geometry algorithms. The required calculations are fast and are suitable for a real-time system.

Computing generalized higher-order Voronoi diagrams on triangulated surfaces

We present an algorithm for computing exact shortest paths, and consequently distance functions, from a generalized source (point, segment, polygonal chain or polygonal region) on a possibly non-convex triangulated polyhedral surface. The algorithm is generalized to the case when a set of generalized sites is considered, providing their distance field that implicitly represents the Voronoi diagram of the sites. Next, we present an algorithm to compute a discrete representation of the distance function and the distance field. Then, by using the discrete distance field, we obtain the Voronoi diagram of a set of generalized sites (points, segments, polygonal chains or polygons) and visualize it on the triangulated surface. We also provide algorithms that, by using the discrete distance functions, provide the closest, furthest and k-order Voronoi diagrams and an approximate 1-Center and 1-Median.

Higher-order Voronoi diagrams on triangulated surfaces

We study the complexity of higher-order Voronoi diagrams on triangulated surfaces under the geodesic distance, when the sites may be polygonal domains of constant complexity. More precisely, we show that on a surface defined by n triangles the sum of the combinatorial complexities of the order- j Voronoi diagrams of m sites, for j = 1 , … , k , is O ( k 2 n 2 + k 2 m + k n m ) , which is asymptotically tight in the worst case.

Euclidean push–pull partial covering problems

This paper considers a bicriteria model to locate a semi-obnoxious facility within a convex polygon. Assuming that a given number of closest points and farthest points may be neglected in the analysis, it considers simultaneously the resulting push and pull partial covering criteria with Euclidean distances. Although both objectives are neither concave or convex, low complexity polynomial algorithms to find all the efficient solutions and the tradeoffs involved are developed by way of higher-order Voronoi diagrams. Comparison of the tradeoff for full covering and partial covering enables decision makers to understand to what extent the maximin and minimax criteria are improved at the expense of neglecting some points. The extensions to different sets of points pulling and pushing the facility and to weighted points are discussed. All methods are illustrated via small scale examples.

Concatenation-Based Greedy Heuristics for the Euclidean Steiner Tree Problem

We present a class of O(n log n) heuristics for the Steiner tree problem in the Euclidean plane. These heuristics identify a small number of subsets with few, geometrically close, terminals using minimum spanning trees and other well-known structures from computational geometry: Delaunay triangulations, Gabriel graphs, relative neighborhood graphs, and higher-order Voronoi diagrams. Full Steiner trees of all these subsets are sorted according to some appropriately chosen measure of quality. A tree spanning all terminals is constructed using greedy concatenation. New heuristics are compared with each other and with heuristics from the literature by performing extensive computational experiments on both randomly generated and library problem instances.

Constructing Levels in Arrangements and Higher Order Voronoi Diagrams

We give simple randomized incremental algorithms for computing the Amk-level in an arrangement of n lines in the plane or in an arrangement of n planes in $\Reals^3$. The expected running time of our algorithms is $O(nk+n\alpha(n)\log n)$ for the planarcase and O(nk2 + n log3n) for the three-dimensional case. Both bounds are optimal unless k is very small. The algorithm generalizes to computing the Amk-level in an arrangement of discs or x-monotone Jordan curves in the plane. Our approach can also compute the k-level; this yields a randomized algorithm for computing the order-k Voronoi diagram of n points in the plane in expected time O(k(n-k)log n + n log3n).

FINDING k FARTHEST PAIRS AND k CLOSEST/FARTHEST BICHROMATIC PAIRS FOR POINTS IN THE PLANE

We study the problem of enumerating k farthest pairs for n points in the plane and the problem of enumerating k closest/farthest bichromatic pairs of n red and n blue points in the plane. We propose a new technique for geometric enumeration problems which iteratively reduces the search space by a half and provides efficient algorithms. As applications of this technique, we develop algorithms, using higher order Voronoi diagrams, for the above problems, which run in O(min{n2, n log n+k4/3log n/log1/3 k}) time and O(n+k4/3/log1/3 k+k log n) space for general Lp metric with p≠2, and O(min{n2, n log n+k4/3}) time and O(n+k4/3+k log n) space for L2 metric. For the problem of enumerating k closest/farthest bichromatic pairs, we shall also discuss the case where we have different numbers of red and blue points. To the authors’ knowledge, no nontrivial algorithms have been known for these problems and our algorithms are faster than trivial ones when k=o(n3/2).

A semidynamic construction of higher-order voronoi diagrams and its randomized analysis

Thek-Delaunay tree extends the Delaunay tree introduced in [1], and [2]. It is a hierarchical data structure that allows the semidynamic construction of the higher-order Voronoi diagrams of a finite set ofn points in any dimension. In this paper we prove that a randomized construction of thek-Delaunay tree, and thus of all the order≤k Voronoi diagrams, can be done inO(n logn+k 3n) expected time and O(k2n) expected storage in the plane, which is asymptotically optimal for fixedk. Our algorithm extends tod-dimensional space with expected time complexityO(k ⌈(d+1)/2⌉+1 n ⌊(d+1)/2⌋) and space complexityO(k ⌈(d+1)/2⌉ n ⌊(d+1)/2⌋). The algorithm is simple and experimental results are given.

On levels in arrangements and voronoi diagrams

This paper gives efficient, randomized algorithms for the following problems: (1) construction of levels of order 1 tok in an arrangement of hyperplanes in any dimension and (2) construction of higher-order Voronoi diagrams of order 1 tok in any dimension. A new combinatorial tool in the form of a mathematical series, called a ? series, is associated with an arrangement of hyperplanes inRd. It is used to study the combinatorial as well as algorithmic complexity of the geometric problems under consideration.

Constructing Arrangements of Lines and Hyperplanes with Applications

A finite set of lines partitions the Euclidean plane into a cell complex. Similarly, a finite set of $(d - 1)$-dimensional hyperplanes partitions d-dimensional Euclidean space. An algorithm is presented that constructs a representation for the cell complex defined by n hyperplanes in optimal $O(n^d )$ time in d dimensions. It relies on a combinatorial result that is of interest in its own right. The algorithm is shown to lead to new methods for computing $\lambda $-matrices, constructing all higher-order Voronoi diagrams, halfspatial range estimation, degeneracy testing, and finding minimum measure simplices. In all five applications, the new algorithms are asymptotically faster than previous results, and in several cases are the only known methods that generalize to arbitrary dimensions. The algorithm also implies an upper bound of $2^{cn^d } $, c a positive constant, for the number of combinatorially distinct arrangements of n hyperplanes in $E^d $.