Farthest-point Voronoi Diagram Research Articles

Farthest-Point Voronoi Diagrams in the Presence of Rectangular Obstacles

We present an algorithm to compute the geodesic $$L_1$$ farthest-point Voronoi diagram of m point sites in the presence of n rectangular obstacles in the plane. It takes $$O(nm+n \log n + m\log m)$$ construction time using O(nm) space. This is the first optimal algorithm for constructing the farthest-point Voronoi diagram in the presence of obstacles. We can construct a data structure in the same construction time and space that answers a farthest-neighbor query in $$O(\log (n+m))$$ time.

An Optimal Deterministic Algorithm for Geodesic Farthest-Point Voronoi Diagrams in Simple Polygons

Given in the plane a set S of m point sites in a simple polygon P of n vertices, we consider the problem of computing the geodesic farthest-point Voronoi diagram for S in P. It is known that the problem has an $$\Omega (n+m\log m)$$ time lower bound. Previously, a randomized algorithm was proposed [Barba, SoCG 2019] that solves the problem in $$O(n+m\log m)$$ expected time. The previous best deterministic algorithms solve the problem in $$O(n\log \log n+ m\log m)$$ time [Oh, Barba, and Ahn, SoCG 2016] or in $$O(n+m\log m+m\log ^2\!n)$$ time [Oh and Ahn, SoCG 2017]. In this paper, we present a deterministic algorithm that takes $$O(n+m\log m)$$ time, which is optimal. This answers affirmatively an open question posed by Mitchell in the Handbook of Computational Geometry two decades ago.

On the Minimum Consistent Subset Problem

Let $P$ be a set of $n$ colored points in the plane. Introduced by Hart (1968), a consistent subset of $P$, is a set $S\subseteq P$ such that for every point $p$ in $P\setminus S$, the closest point of $p$ in $S$ has the same color as $p$. The consistent subset problem is to find a consistent subset of $P$ with minimum cardinality. This problem is known to be NP-complete even for two-colored point sets. Since the initial presentation of this problem, aside from the hardness results, there has not been a significant progress from the algorithmic point of view. In this paper we present the following algorithmic results: 1. The first subexponential-time algorithm for the consistent subset problem. 2. An $O(n\log n)$-time algorithm that finds a consistent subset of size two in two-colored point sets (if such a subset exists). Towards our proof of this running time we present a deterministic $O(n \log n)$-time algorithm for computing a variant of the compact Voronoi diagram; this improves the previously claimed expected running time. 3. An $O(n\log^2 n)$-time algorithm that finds a minimum consistent subset in two-colored point sets where one color class contains exactly one point; this improves the previous best known $O(n^2)$ running time which is due to Wilfong (SoCG 1991). 4. An $O(n)$-time algorithm for the consistent subset problem on collinear points; this improves the previous best known $O(n^2)$ running time. 5. A non-trivial $O(n^6)$-time dynamic programming algorithm for the consistent subset problem on points arranged on two parallel lines. To obtain these results, we combine tools from planar separators, additively-weighted Voronoi diagrams with respect to convex distance functions, point location in farthest-point Voronoi diagrams, range trees, paraboloid lifting, minimum covering of a circle with arcs, and several geometric transformations.

The Geodesic Farthest-Point Voronoi Diagram in a Simple Polygon

Given a set of point sites in a simple polygon, the geodesic farthest-point Voronoi diagram partitions the polygon into cells, at most one cell per site, such that every point in a cell has the same farthest site with respect to the geodesic metric. We present an $$O(n\log \log n+ m\log m)$$-time algorithm to compute the geodesic farthest-point Voronoi diagram of m point sites in a simple n-gon. This improves the previously best known algorithm by Aronov et al. (Discrete Comput Geom 9(3):217–255, 1993). In the case that all point sites are on the boundary of the simple polygon, we can compute the geodesic farthest-point Voronoi diagram in $$O((n+m) \log \log n)$$ time.

Voronoi Diagrams for a Moderate-Sized Point-Set in a Simple Polygon

Given a set of sites in a simple polygon, a geodesic Voronoi diagram of the sites partitions the polygon into regions based on distances to sites under the geodesic metric. We present algorithms for computing the geodesic nearest-point, higher-order and farthest-point Voronoi diagrams of m point sites in a simple n-gon, which improve the best known ones for \(m \le n/{\text {polylog}}n\). Moreover, the algorithms for the geodesic nearest-point and farthest-point Voronoi diagrams are optimal for \(m \le n/{\text {polylog}}n\). This partially answers a question posed by Mitchell in the Handbook of Computational Geometry.

Two Efficient Hashing Schemes for High-Dimensional Furthest Neighbor Search

The $c$ -Approximate Furthest Neighbor ( $c$ -AFN) search is a fundamental problem in many applications. However, existing hashing schemes for $c$ -AFN search are designed for internal memory. The old techniques for external memory, such as furthest point Voronoi diagram and the tree-based methods, are only suitable for the low-dimensional case. In this paper, we introduce a novel concept of the Reverse Locality-Sensitive Hashing (RLSH) family which is directly designed for $c$ -AFN search. Accordingly, we propose a new reverse query-aware LSH function, which is a random projection coupled with query-aware interval identification. Based on the reverse query-aware LSH functions, we introduce a novel Reverse Query-Aware LSH scheme named RQALSH for high-dimensional $c$ -AFN search over external memory. Our theoretical studies show that RQALSH enjoys a guarantee on query quality. In addition, in order to further speed up RQALSH, we propose a heuristic variant named RQALSH $^*$ which applies a data-dependent objects selection to largely reduce the number of data objects. In the experiment, we compare with two state-of-the-art hashing schemes QDAFN and DrusillaSelect which have been adapted for external memory. Extensive experiments on four real datasets show that our proposed RQALSH and RQALSH $^*$ schemes significantly outperform these two methods.

Data Structures for Halfplane Proximity Queries and Incremental Voronoi Diagrams

We consider preprocessing a set S of n points in convex position in the plane into a data structure supporting queries of the following form: given a point q and a directed line $$\ell $$ in the plane, report the point of S that is farthest from (or, alternatively, nearest to) the point q among all points to the left of line $$\ell $$ . We present two data structures for this problem. The first data structure uses $$O(n^{1+\varepsilon })$$ space and preprocessing time, and answers queries in $$O(2^{1/\varepsilon }\log n)$$ time, for any $$0< \varepsilon < 1$$ . The second data structure uses $$O(n \log ^3 n)$$ space and polynomial preprocessing time, and answers queries in $$O(\log n)$$ time. These are the first solutions to the problem with $$O(\log n)$$ query time and $$o(n^2)$$ space. The second data structure uses a new representation of nearest- and farthest-point Voronoi diagrams of points in convex position. This representation supports the insertion of new points in clockwise order using only $$O(\log n)$$ amortized pointer changes, in addition to $$O(\log n)$$ -time point-location queries, even though every such update may make $$\Theta (n)$$ combinatorial changes to the Voronoi diagram. This data structure is the first demonstration that deterministically and incrementally constructed Voronoi diagrams can be maintained in o(n) amortized pointer changes per operation while keeping $$O(\log n)$$ -time point-location queries.

Incremental Voronoi Diagrams

We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram (and several variants thereof) of a set S of n sites in the plane as sites are added to the set. To that effect, we define a general update operation for planar graphs that can be used to model the incremental construction of several variants of Voronoi diagrams as well as the incremental construction of an intersection of halfspaces in \(\mathbb {R}^3\). We show that the amortized number of edge insertions and removals needed to add a new site to the Voronoi diagram is \(O(\sqrt{n})\). A matching \(\Omega (\sqrt{n})\) combinatorial lower bound is shown, even in the case where the graph representing the Voronoi diagram is a tree. This contrasts with the \(O(\log {n})\) upper bound of Aronov et al. (LATIN 2006: Theoretical Informatics. Lecture Notes in Computer Science, Springer, Berlin, 2006) for farthest-point Voronoi diagrams in the special case where the points are inserted in clockwise order along their convex hull. We then present a semi-dynamic data structure that maintains the Voronoi diagram of a set S of n sites in convex position. This data structure supports the insertion of a new site p (and hence the addition of its Voronoi cell) and finds the asymptotically minimal number K of edge insertions and removals needed to obtain the diagram of \(S \cup \{p\}\) from the diagram of S, in time \(O(K\,\mathrm {polylog}\ n)\) worst case, which is \(O(\sqrt{n}\;\mathrm {polylog}\ n)\) amortized by the aforementioned combinatorial result. The most distinctive feature of this data structure is that the graph of the Voronoi diagram is maintained explicitly at all times and can be retrieved and traversed in the natural way; this contrasts with other known data structures supporting nearest neighbor queries. Our data structure supports general search operations on the current Voronoi diagram, which can, for example, be used to perform point location queries in the cells of the current Voronoi diagram in \(O(\log n)\) time, or to determine whether two given sites are neighbors in the Delaunay triangulation.

A primal algorithm for the weighted minimum covering ball problem in $$\mathbb {R}^n$$ R n

The nonlinear programming problem of finding the minimum covering ball of a finite set of points in \(\mathbb {R}^n\), with a positive weight corresponding to each point, is solved by a directional search method. At each iteration, the search path is either a ray or the arc of a circle and is determined by bisectors of points. Each step size along the search path is determined explicitly. The primal algorithm is shown to search along the farthest point Voronoi diagram of the given points. We provide computational results that show the efficiency of the algorithm when compared to general convex nonlinear optimization solvers.

Euclidean farthest-point Voronoi diagram of a digital edge

A digital edge is a digitization of a straight segment joining two points of integer coordinates. Such a digital set may be analytically defined by the rational slope of the straight segment. We show in this paper that the convex hull, the Euclidean farthest-point Voronoi diagram as well as the dual farthest-point Delaunay triangulation of a digital edge can be fully described by the continued fraction expansion of its slope.

Minimum enclosing circle of a set of fixed points and a mobile point

Given a set S of n static points and a mobile point p in R2, we study the variations of the smallest circle that encloses S∪{p} when p moves along a straight line ℓ. In this work, a complete characterization of the locus of the center of the minimum enclosing circle (MEC) of S∪{p}, for p∈ℓ, is presented. The locus is a continuous and piecewise differentiable linear function, and each of its differentiable pieces lies either on the edges of the farthest-point Voronoi diagram of S, or on a line segment parallel to the line ℓ. Moreover, the locus has O(n) differentiable pieces, which can be computed in linear time, given the farthest-point Voronoi diagram of S.

A geometric characterisation of the quadratic min-power centre

For a given set of nodes in the plane the min-power centre is a point such that the cost of the star centred at this point and spanning all nodes is minimised. The cost of the star is defined as the sum of the costs of its nodes, where the cost of a node is an increasing function of the length of its longest incident edge. The min-power centre problem provides a model for optimally locating a cluster-head amongst a set of radio transmitters, however, the problem can also be formulated within a bicriteria location model involving the 1-centre and a generalised Fermat-Weber point, making it suitable for a variety of facility location problems. We use farthest point Voronoi diagrams and Delaunay triangulations to provide a complete geometric description of the min-power centre of a finite set of nodes in the Euclidean plane when cost is a quadratic function. This leads to a new linear-time algorithm for its construction when the convex hull of the nodes is given. We also provide an upper bound for the performance of the centroid as an approximation to the quadratic min-power centre. Finally, we briefly describe the relationship between solutions under quadratic cost and solutions under more general cost functions.

Farthest line segment Voronoi diagrams

The farthest line segment Voronoi diagram shows properties different from both the closest-segment Voronoi diagram and the farthest-point Voronoi diagram. Surprisingly, this structure did not receive attention in the computational geometry literature. We analyze its combinatorial and topological properties and outline an O ( n log n ) time construction algorithm that is easy to implement. No restrictions are placed upon the n input line segments; they are allowed to touch or cross.

Computing farthest neighbors on a convex polytope

Let N be a set of n points in convex position in R 3. The farthest point Voronoi diagram of~ N partitions~ R 3 into n convex cells. We consider the intersection~ G( N) of the diagram with the boundary of the convex hull of N. We give an algorithm that computes an implicit representation of G( N) in expected O(n log 2 n) time. More precisely, we compute the combinatorial structure of G( N), the coordinates of its vertices, and the equation of the plane defining each edge of G( N). The algorithm allows us to solve the all-pairs farthest neighbor problem for N in expected time O(n log 2 n) , and to perform farthest-neighbor queries on N in O( log 2 n) time with high probability.

An efficient algorithm for the three-dimensional diameter problem

We explore a new approach for computing the diameter of n points in ?3 that is based on the restriction of the furthest-point Voronoi diagram to the convex hull. We show that the restricted Voronoi diagram has linear complexity. We present a deterministic algorithm with O(n log2n) running time. The algorithm is quite simple and is a good candidate to be implemented in practice. Using our approach the chromatic diameter and all-furthest neighbors in ?3 can be found in the same running time.

Data structures and algorithms to support interactive spatial analysis using dynamic Voronoi diagrams

To support the need for interactive spatial analysis, it is often necessary to rethink the data structures and algorithms underpinning applications. This paper describes the development of an interactive environment in which a number of different Voronoi models of space can be manipulated together in real time to: (1) study their behaviour; (2) select appropriate models for specific analysis tasks; and (3) to examine how choice of one model over another will affect the interpretation of data. The paper studies six specific Voronoi diagram variants: the Ordinary Voronoi Diagram, the Farthest-point Voronoi Diagram, the Order- k Voronoi Diagram, the Ordered Order- k Voronoi Diagram, the kth Nearest-point Voronoi Diagram and the Multiplicatively Weighted Voronoi Diagram, and develops algorithms and data structures to store, rebuild and query these variants. From this, a generalised Voronoi data structure is proposed, from which specific Voronoi variants can be reconstructed dynamically as required. Algorithms for diagram reconstruction and for querying neighbourhood (topology or adjacency relations) of generator points and Voronoi regions are presented. An application program, developed on these ideas, is used to generate example results as proof of concept. It may be downloaded from a supporting website.

A geometrical solution for quadratic bicriteria location models

This paper deals with a single-facility, unweighted, quadratic bicriteria location problem defined in the continuous space. First we describe an analytical method for constructing the set of Pareto-optimal locations for this problem based on the farthest-point Voronoi diagram. Then we present an analytical method for finding the solution to the scalarized location problems based on the Pareto-optimal locations. The analytical results enable planners to carry out some sensitivity analyses. Since both methods are based entirely on geometrical analysis, they can be understood intuitively.

ON THE WIDTH AND ROUNDNESS OF A SET OF POINTS IN THE PLANE

Let S be a set of points in the plane. The width (resp. roundness) of S is defined as the minimum width of any slab (resp. annulus) that contains all points of S. We give a new characterization of the width of a point set. Also, we give a rigorous proof of the fact that either the roundness of S is equal to the width of S, or the center of the minimum-width annulus is a vertex of the closest-point Voronoi diagram of S, the furthest-point Voronoi diagram of S, or an intersection point of these two diagrams. This proof corrects the characterization of roundness used extensively in the literature.

Intersections with random geometric objects

We present a systematic study of the expected complexity of the intersection of geometric objects. We first study the expected size of the intersection between a random Voronoi diagram and a generic geometric object that consists of a finite collection of line segments in the plane. Using this result, we explore the intersection complexity of a random Voronoi diagram with the following target objects which may or may not be random: a line segment, a Voronoi diagram, a minimum spanning tree, a Gabriel graph, a relative neighborhood graph, a Hamiltonian circuit, a furthest point Voronoi diagram, a convex hull, a k-dimensional tree, and a rectangular grid.

Locational optimization problems solved through Voronoi diagrams

This paper reviews a class of continuous locational optimization problems (where an optimal location or an optimal configuration of facilities is found in a continuum on a plane or a network) that can be solved through the Voronoi diagram. Eight types of continuous locational optimization problems are formulated, and these problems are solved through the ordinary Voronoi diagram, the farthest-point Voronoi diagram, the weighted Voronoi diagram, the network Voronoi diagram, the Voronoi diagram with a convex distance function, the line Voronoi diagram, and the area Voronoi diagram.