Order-k Voronoi Diagram Research Articles

Efficient Algorithm for Constructing Order K Voronoi Diagrams in Road Networks

The order k Voronoi diagram (OkVD) is an effective geometric construction to partition the geographical space into a set of Voronoi regions such that all locations within a Voronoi region share the same k nearest points of interest (POIs). Despite the broad applications of OkVD in various geographical analysis, few efficient algorithms have been proposed to construct OkVD in real road networks. This study proposes a novel algorithm consisting of two stages. In the first stage, a new one-to-all k shortest path finding procedure is proposed to efficiently determine the shortest paths to k nearest POIs for each node. In the second stage, a new recursive procedure is introduced to effectively divide boundary links within different Voronoi regions using the hierarchical tessellation property of the OkVD. To demonstrate the applicability of the proposed OkVD construction algorithm, a case study of place-based accessibility evaluation is carried out. Computational experiments are also conducted on five real road networks with different sizes, and results show that the proposed OkVD algorithm performed significantly better than state-of-the-art algorithms.

Backup gateways for IoT mesh network using order-k hops Voronoi diagram

Mesh network is a common topology in deploying Edge/Fog computing in IoT due to its robustness, expandability and reliability. In the Mesh topology, gateways are the key role for the entire networks to communicate with the clouds. In order to ensure network availability in a failover scenario, a router must always have backup gateways to maintain mesh robustness during primary gateway failover. Order-k Voronoi diagram is known for its capability to identify k-nearest facilities and ensure that all objects will always have k-nearest backup facilities. In this paper, we utilize order-k Voronoi diagram with sink tree to produce order-k hops Voronoi diagram to identify k-gateways coverage with minimal hops for all routers as the backup gateways. Our experiment shows that order-k hops Voronoi diagram is more effective in ensuring that all routers have backup gateways with a minimum number of hops than an ordinary order-k network Voronoi diagram, hence reduce network latency for the entire mesh networks and maintain the robustness of the mesh network.

Separation by Convex Pseudo-Circles

Let S be a finite set of n points in the plane in general position. We prove that every inclusion-maximal family of subsets of S separable by convex pseudo-circles has the same cardinal $$\left( {\begin{array}{c}n 0\end{array}}\right) +\left( {\begin{array}{c}n 1\end{array}}\right) +\left( {\begin{array}{c}n 2\end{array}}\right) +\left( {\begin{array}{c}n 3\end{array}}\right) $$ . This number does not depend on the configuration of S and is the same as the number of subsets of S separable by true circles. For a fixed $$k \in \{1,\dots ,n-1\}$$ , we also count the number of elements in a maximal family of k-subsets of S separable by convex pseudo-circles. This time the number depends on the configuration of S, but it is again equal to the number of k-subsets of S separable by true circles. Thus, it is an invariant of S: it does not depend on the choice of the maximal family. To achieve these results, we introduce a graph that generalizes the dual graph of the order-k Voronoi diagram. The vertices of the graph are the elements of a maximal family of k-subsets of S separable by convex pseudo-circles. In order to count the number of these vertices, we show that the graph is realizable as a triangulation.

An Efficient Randomized Algorithm for Higher-Order Abstract Voronoi Diagrams

Given a set of n sites in the plane, the order-k Voronoi diagram is a planar subdivision such that all points in a region share the same k nearest sites. The order-k Voronoi diagram arises for the k-nearest-neighbor problem, and there has been a lot of work for point sites in the Euclidean metric. In this paper, we study order-k Voronoi diagrams defined by an abstract bisecting curve system that satisfies several practical axioms, and thus our study covers many concrete order-k Voronoi diagrams. We propose a randomized incremental construction algorithm that runs in $$O(k(n-k)\log ^2 n +n\log ^3 n)$$ steps, where $$O(k(n-k))$$ is the number of faces in the worst case. This result applies to disjoint line segments in the $$L_p$$ norm, convex polygons of constant size, points in the Karlsruhe metric, and so on. In fact, a running time with a polylog factor to the number of faces was only achieved for point sites in the $$L_1$$ or Euclidean metric before.

A randomized divide and conquer algorithm for higher-order abstract Voronoi diagrams

Given a set of n sites in the plane, their order-k Voronoi diagram partitions the plane into regions such that all points within one region have the same k nearest sites. The order-k abstract Voronoi diagram offers a unifying framework that represents a wide range of concrete order-k Voronoi diagrams. It is defined in terms of bisecting curves satisfying some simple combinatorial properties, rather than the geometric notions of sites and distance.In this paper we develop a randomized divide-and-conquer algorithm to compute the order-k abstract Voronoi diagram in expected O(kn1+ε) operations. For solving small sub-instances in the divide-and-conquer process, we also give two auxiliary algorithms with expected O(k2nlog⁡n) and O(n22α(n)log⁡n) time, respectively, where α(⋅) is the inverse of the Ackermann function. Our approach directly implies an O(kn1+ε)-time algorithm for several concrete order-k instances such as points in any convex distance, disjoint line segments or convex polygons of constant size in the Lp norms, and others. It also provides basic techniques that can enable the application of well-known random sampling techniques to the construction of Voronoi diagrams in the abstract setting and for non-point sites.

Optimal Deterministic Algorithms for 2-d and 3-d Shallow Cuttings

We present optimal deterministic algorithms for constructing shallow cuttings in an arrangement of lines in two dimensions or planes in three dimensions. Our results improve the deterministic polynomial-time algorithm of Matousek (Comput Geom 2(3):169---186, 1992) and the optimal but randomized algorithm of Ramos (Proceedings of the Fifteenth Annual Symposium on Computational Geometry, SoCG'99, 1999). This leads to efficient derandomization of previous algorithms for numerous well-studied problems in computational geometry, including halfspace range reporting in 2-d and 3-d, k nearest neighbors search in 2-d, $$({\le }k)$$(≤k)-levels in 3-d, order-k Voronoi diagrams in 2-d, linear programming with k violations in 2-d, dynamic convex hulls in 3-d, dynamic nearest neighbor search in 2-d, convex layers (onion peeling) in 3-d, $$\varepsilon $$?-nets for halfspace ranges in 3-d, and more. As a side product we also describe an optimal deterministic algorithm for constructing standard (non-shallow) cuttings in two dimensions, which is arguably simpler than the known optimal algorithms by Matousek (Discrete Comput Geom 6(1):385---406, 1991) and Chazelle (Discrete Comput Geom 9(1):145---158, 1993).

On the complexity of higher order abstract Voronoi diagrams

Abstract Voronoi diagrams (AVDs) are based on bisecting curves enjoying simple combinatorial properties, rather than on the geometric notions of sites and circles. They serve as a unifying concept. Once the bisector system of any concrete type of Voronoi diagram is shown to fulfill the AVD properties, structural results and efficient algorithms become available without further effort.In a concrete order-k Voronoi diagram, all points are placed into the same region that have the same k nearest neighbors among the given sites. This paper is the first to study abstract Voronoi diagrams of arbitrary order k. We prove that their complexity in the plane is upper bounded by 2k(n−k). So far, an O(k(n−k)) bound has been shown only for point sites in the Euclidean and Lp planes, and, recently, for line segments, in the Lp metric. These proofs made extensive use of the geometry of the sites.Our result on AVDs implies a 2k(n−k) upper bound for a wide range of cases for which only trivial upper complexity bounds were previously known, and a slightly sharper bound for the known cases. Also, our proof shows that the reasons for this bound are combinatorial properties of certain permutation sequences.

Ordered Reverse k Nearest Neighbor Search via On-demand Broadcast

The Reverse k Nearest Neighbor (RkNN) query is valuable for finding objects influenced by a specific object and is widely used in both scientific and commercial systems. However, the influence level of each object is unknown, information that is critical for some applications (e.g. target marketing). In this paper, we propose a new query type, Ordered Reverse k Nearest Neighbor (ORkNN), and make efforts to adapt it in an on-demand scenario. An Order-k Voronoi diagram based approach is used to answer ORkNN queries. In particular, for different values of k, we pre-construct only one Voronoi diagram. Algorithms on both the server and the clients are presented. We also present experimental results that suggest our proposed algorithms may have practical applications.

Order-k α-hulls and α-shapes

We introduce order-k α-hulls and α-shapes – generalizations of α-hulls and α-shapes. Being also a generalization of k-hull (known in statistics as “k-depth contour”), order-k α-hull provides a link between shape reconstruction and statistical depth. As a generalization of α-hull, order-k α-hull gives a robust shape estimation by ignoring locally up to k outliers in a point set. Order-kα-shape produces an “inner” shape of the set, with the amount of “digging” into the points controlled by k. As a generalization of k-hull, order-k α-hull is capable of determining “deep” points amidst samples from a multimodal distribution: it correctly identifies points which lie outside clusters of samples.The order-k α-hulls and α-shapes are related to order-k Voronoi diagrams in the same way in which α-hulls and α-shapes are related to Voronoi diagrams. This implies that order-k α-hull and α-shape can be readily built from order-k Voronoi diagram, and that the number of different order-kα-shapes for all possible values of α is proportional to the complexity of order-k Voronoi diagram.

Approximating generalized distance functions on weighted triangulated surfaces with applications

Given P, a simple connected, possibly non-convex, polyhedral surface composed of positively weighted triangular faces, we consider paths from generalized sources (points, segments, polygonal chains or polygonal regions) to points on P that stay on P and avoid obstacles (segments, polygonal chains or polygonal regions). The distance function defined by a generalized source is a function that assigns to each point of P the cost of the shortest path from the source to the point. In this paper we present an algorithm for computing approximate generalized distance functions. We also provide an algorithm that computes a discrete representation of the approximate distance function and, as applications, algorithms for computing discrete order-k Voronoi diagrams and for approximately solving facility location problems. Finally, we present experimental results obtained with our implementation of the provided algorithms.

CENTROID TRIANGULATIONS FROM k-SETS

Given a set V of n points in the plane, no three of them being collinear, a convex inclusion chain of V is an ordering of the points of V such that no point belongs to the convex hull of the points preceding it in the ordering. We call k-set of the convex inclusion chain, every k-set of an initial subsequence of at least k points of the ordering. We show that the number of such k-sets (without multiplicity) is an invariant of V, that is, it does not depend on the choice of the convex inclusion chain. Moreover, this number is equal to the number of regions of the order-k Voronoi diagram of V (when no four points are cocircular). The dual of the order-k Voronoi diagram belongs to the set of so-called centroid triangulations that have been originally introduced to generate bivariate simplex spline spaces. We show that the centroids of the k-sets of a convex inclusion chain are the vertices of such a centroid triangulation. This leads to the currently most efficient algorithm to construct particular centroid triangulations of any given point set; it runs in O(n log n + k(n - k) log k) worst case time.

Sorted Order-$K$ Voronoi Diagrams for Model-Independent Source Localization in Wireless Sensor Networks

Localization of an isotropic source using energy measurements from randomly deployed sensors is considered. In particular, an optimization problem that does not require knowledge of the underlaying energy decay model is proposed, and a condition under which the optimal solution can be computed is given. This condition employs a new geometric construct introduced here, called the sorted order-K Voronoi diagram. We give centralized and distributed algorithms for source localization in this setting. Finally, analytical results and simulations are used to verify the performance of the developed algorithms.

Random Sampling, Halfspace Range Reporting, and Construction of \lowercase$(\le k)$-Levels in Three Dimensions

Given n points in three dimensions, we show how to answer halfspace range reporting queries in O(log n+k) expected time for an output size k. Our data structure can be preprocessed in optimal O(n log n) expected time. We apply this result to obtain the first optimal randomized algorithm for the construction of the $(\le k)$-level in an arrangement of n planes in three dimensions. The algorithm runs in O(n log n+nk2 ) expected time. Our techniques are based on random sampling. Applications in two dimensions include an improved data structure for "k nearest neighbors" queries and an algorithm that constructs the order-k Voronoi diagram in O(n log n+nk log k) expected time.

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).

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

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.

Order-k voronoi diagrams of sites with additive weights in the plane

It is shown that the order-k Voronoi diagram of n sites with additive weights in the plane has at most (4kź2)(nźk) vertices, (6kź3)(nźk) edges, and (2kź1)(nźitk) + 1 regions. These bounds are approximately the same as the ones known for unweighted order-k Voronoi diagrams. Furthermore, tight upper bounds on the number of edges and vertices are given for the case that every weighted site has a nonempty region in the order-1 diagram. The proof is based on a new algorithm for the construction of these diagrams which generalizes a plane-sweep algorithm for order-1 diagrams developed by Steven Fortune. The new algorithm has time-complexityO(k2n logn) and space-complexityO(kn). It is the only nontrivial algorithm known for constructing order-kc Voronoi diagrams of sites withadditive weights. It is fairly simple and of practical interest also in the special case of unweighted sites.

A new duality result concerning voronoi diagrams

A new duality between order-k Voronoi diagrams inEd and convex hulls inEd+1 is established. It implies a reasonably simple algorithm for computing the order-k diagram forn points in the plane inO(k2n logn) time and optimalO(k(n?k)) space.

New applications of random sampling in computational geometry

This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry. One new algorithm creates a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer. This algorithm requiresO(sd+?) expected preprocessing time to build a search structure for an arrangement ofs hyperplanes ind dimensions. The expectation, as with all expected times reported here, is with respect to the random behavior of the algorithm, and holds for any input. Given the data structure, and a query pointp, the cell of the arrangement containingp can be found inO(logs) worst-case time. (The bound holds for any fixed ?>0, with the constant factors dependent ond and ?.) Using point-plane duality, the algorithm may be used for answering halfspace range queries. Another algorithm finds random samples of simplices to determine the separation distance of two polytopes. The algorithm uses expectedO(n[d/2]) time, wheren is the total number of vertices of the two polytopes. This matches previous results [10] for the cased = 3 and extends them. Another algorithm samples points in the plane to determine their orderk Voronoi diagram, and requires expectedO(s1+?k) time fors points. (It is assumed that no four of the points are cocircular.) This sharpens the boundO(sk2 logs) for Lee's algorithm [21], andO(s2 logs+k(s?k) log2s) for Chazelle and Edelsbrunner's algorithm [4]. Finally, random sampling is used to show that any set ofs points inE3 hasO(sk2 log8s/(log logs)6) distinctj-sets withj≤k. (ForS ?Ed, a setS? ?S with |S?| =j is aj-set ofS if there is a half-spaceh+ withS? =S ?h+.) This sharpens with respect tok the previous boundO(sk5) [5]. The proof of the bound given here is an instance of a probabilistic method [15].