Most of the emphasis in path planning, a topic of much interest in several domains, has been on finding the optimal path or at most k optimal paths, but in domains such as adversarial planning, one of the agents might deliberately take less optimal paths to confuse the opponent, and by the same token an agent, for inferring opponent’s intent, has to consider all possible paths that the opponent...

Given a protein, it is often necessary to study its geometric and physicochemical properties for studying its structure and predicting funtions of a protein. In this case, a connolly surface of a protein plays important roles for these purpose. A protein consists of a set of amino acids and a set of atoms comprise an amino acide. Since an atom can be represented by a hard 3D sphere in van der W...

We investigate higher-order Voronoi diagrams in the city metric. This metric is induced by quickest paths in the L1 metric in the presence of an accelerating transportation network of axis-parallel line segments. For the structural complexity of k-order city Voronoi diagrams of n point sites, we show an upper bound of O(k(n − k) + kc) and a lower bound of Ω(n+ kc), where c is the complexity of ...

The Voronoi diagram of a point set has been extensively used in various disciplines ever since it was first proposed. Its application realms have been even further extended to estimate the shape of point clouds when Edelsbrunner and Mücke introduced the concept of α-shape based on the Delaunay triangulation of a point set. In this paper, we present the theory of β-shape for a set of three-dimen...

Voronoi diagrams are among the most important data structures in geometric modeling. Among many efficient algorithms for computing 2D Voronoi diagrams, Fortune’s sweepline algorithm (Fortune, 1986 [5]) is popular due to its elegance and simplicity. Dehne and Klein (1987) [8] extended sweepline to sweepcircle and suggested computing a type of transformed Voronoi diagram, which is parallel in nat...

A protein consists of one or more chains where each chain is a linear sequence of amino acids bonded by peptide bonds. Chains in a protein interact with each other and the interaction is known to be one of the most fundamental factors which determine important physiological phenomena in the body. Hence, biologists have been investigating protein interactions since the early days of life science...

Our goal in this paper is to examine the application of Voronoi diagrams, a fundamental concept of computational geometry, to the nearest neighbor algorithm used in machine learning. We consider the question “Given a planar polygonal tessellation T and an integer k, is there a set of k points whose Voronoi diagram contains every edge in T?” We show that this question is NP-hard. We encountered ...

The Network Voronoi diagram and its variants have been extensively used in the context of numerous applications in road networks, particularly to efficiently evaluate various spatial proximity queries such as k nearest neighbor (kNN), reverse kNN, and closest pair. Although the existing approaches successfully utilize the network Voronoi diagram as a way to partition the space for their specifi...

In this work, austenite–martensite transformation in a low alloy steel is modeled using Voronoi algorithm. The evolved fraction of martensite predicted from the geometry diagram constructed, incorporating nucleation kinetics and obtained results are validated against Koistinen–Marburger equation. geometrical properties phase identified characteristics Poisson–Voronoi diagram. Finally, microstru...

In the authors’ recent paper (de Lacy Costello et al., 2010) the authors described the formation of complex tessellations of the plane arising from the various reactions of metal salts with potassium ferricyanide and ferrocyanide loaded gels. In addition to producing colourful tessellations these reactions are naturally computing generalised Voronoi diagrams of the plane. The reactions reported...