This paper presents a dynamic algorithm for the construction of the Euclidean Voronoi diagram of a set of convex objects in the plane. We consider first the Voronoi diagram of smooth convex objects forming pseudocircles set. A pseudo-circles set is a set of bounded objects such that the boundaries of any two objects intersect at most twice. Our algorithm is a randomized dynamic algorithm. It does not use a conflict graph or any sophisticated data structure to perform conflict detection. This feature allows us to handle deletions in a relatively easy way. In the case where objects do not intersect, the randomized complexity of an insertion or deletion can be shown to be respectively O(log 2 n) and O(log 3 n). Our algorithm can easily be adapted to the case of pseudo-circles sets formed by piecewise smooth convex objects. Finally, given any set of convex objects in the plane, we show how to compute the restriction of the Voronoi diagram in the complement of the objects' union.
Abstract. In this paper we present a dynamic algorithm for the construction of the additively weighted Voronoi diagram of a set of weighted points in the plane. The novelty in our approach is that we use the dual of the additively weighted Voronoi diagram to represent it. This permits us to perform both insertions and deletions of sites easily. Given a set B of n sites, among which h sites have a non-empty cell, our algorithm constructs the additively weighted Voronoi diagram of B in O(nT (h) + h log h) expected time, where T (k) is the time to locate the nearest neighbor of a query site within a set of k sites. Deletions can be performed for all sites whether or not their cell is empty. The space requirements for the presented algorithm is O(n). Our algorithm is simple to implement and experimental results suggest an O(n log h) behavior.
We present a new methodology for agent modeling that is scalable and efficient. It is based on the integration of nonlinear dynamical systems and kinetic data structures. The method consists of three layers, which together model 3D agent steering, crowds and flocks among moving and static obstacles. The first layer, the local layer employs nonlinear dynamical systems theory to models low-level behaviors. It is fast and efficient, and it does not depend on the total number of agents in the environment. This dynamical systems-based approach also allows us to establish continuous numerical parameters for modifying each agent's behavior. The second layer, a global environment layer consists of a specifically designed kinetic data structure to track efficiently the immediate environment of each agent and know which obstacles/agents are near or visible to the given agent. This layer reduces the complexity in the local layer. In the third layer, a global planning layer, the problem of target tracking is generalized in a way that allows navigation in maze-like terrains, avoidance of local minima and cooperation between agents. We implement this layer based on two approaches that are suitable for different applications: One approach is to track the closest single moving or static target; the second is to use a pre-specified vector field, which may be generated automatically (with harmonic functions, for example) or based on user input to achieve the desired output. We also discuss how hybrid systems concepts for global planning can capitalize on both our layered approach and the continuous, reactive nature of our agent steering.We demonstrate the power of the approach through a series of experiments simulating single/multiple agents and crowds moving towards moving/static targets in complex environments. r
Real solving of univariate polynomials is a fundamental problem with several important applications. This paper is focused on the comparison of black-box implementations of state-of-the-art algorithms for isolating real roots of univariate polynomials over the integers. We have tested 9 different implementations based on symbolic-numeric methods, Sturm sequences, Continued Fractions and Descartes' rule of sign. The methods under consideration were developed at the GALAAD group at INRIA, the VEGAS group at LO-RIA and the MPI-Saarbrücken. We compared their sensitivity with respect to various aspects such as degree, bitsize or root separation of the input polynomials. Our datasets consist of 5 000 polynomials from many different settings, which have maximum coefficient bitsize up to bits 8 000, and the total running time of the experiments was about 50 hours. Thereby, all implementations of the theoretically exact methods always provided correct results throughout this extensive study. For each scenario we identify the currently most adequate method, and we point to weaknesses in each approach, which should lead to further improvements. Our results indicate that there is no "best method" overall, but one can say that for most instances the solvers based on Continued Fractions are among the best methods. To the best of our knowledge, this is the largest number of tests for univariate real solving up to date.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.