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For a set of n disjoint line segments S in R2, the visibility testing problem (VTP) is to test whether the query point p sees a query segment s∈S. For this configuration, the visibility counting problem (VCP) is to preprocess S such that the number of visible segments in S from any query point p can be computed quickly. In this paper, we solve VTP in expected logarithmic query time using quadratic preprocessing time and space. Moreover, we propose a (1+δ)-approximation algorithm for VCP using at most quadratic preprocessing time and space. The query time of this method is Oε (1/δ 2√n) where Oε (f(n))=O(f(n)nε) and ε>0 is an arbitrary constant number  

In this paper, we consider the problem of computing the visibility polygon (VP) of a query point q (or VP(q)) in a scene consisting of some obstacles with total complexity of n. We show that the combinatorial representation of VP(q) can be computed in logarithmic time by preprocessing the scene in O( n4logn) time and using O( n4) space. We can also report the actual VP(q) in additional O(|VP(q)|) time. As a main result of this paper, we will prove that we can have a tradeoff between the query time and the preprocessing time/space. In other words, we will show that using O(m) space, we can obtain O( n2log(m/n)/m) query time, where m is a parameter satisfying n2≤m≤ n4. For example, when m= n3,... 

This paper presents a kinetic data structure (KDS) for maintenance of the Euclidean minimum spanning tree (EMST) on a set of moving points in 2-dimensional space. For a set of n points moving in the plane we build a KDS of size O(n) in O(nlogn) preprocessing time by which the EMST is maintained efficiently during the motion. This is done by applying the required changes to the combinatorial structure of the EMST which is changed in discrete timestamps. We assume that the motion of the points, i.e. x and y coordinates of the points, are defined by algebraic functions of constant maximum degree. In terms of the KDS performance parameters, our KDS is responsive, local, and compact. The... 

For a set of n disjoint line segments S in R2, the visibility counting problem (VCP) is to preprocess S such that the number of visible segments in S from a query point p can be computed quickly. For this configuration, the visibility testing problem (VTP) is to test whether p sees a fixed segment s. These problems can be solved in logarithmic query time by using O(n4) preprocessing time and space. In this paper, we approximately solve this problem using quadratic preprocessing time and space. Our methods are superior to current approximation algorithms in terms of both approximation factor and preprocessing cost. In this paper, we propose a 2-approximation algorithm for the VCP using at... 

This paper considers the problem of computing the weak visibility polygon (WVP) of any query line segment pq (or WVP(pq)) inside a given simple polygon P. We present an algorithm that preprocesses P and creates a data structure from which WVP(pq) is efficiently reported in an output sensitive manner. Our algorithm needs O(n2log n) time and O(n2) space in the preprocessing phase to report WVP(pq) of any query line segment pq in time O(|WVP(pq)|+log2 n+κlog2 (nκ)), where κ is an input and output sensitive parameter of at most |WVP(pq)|. We improve the preprocessing time and space of current results for this problem [11,6] at the expense of more query time  

A Semi-Separated Pair Decomposition (SSPD), with parameter s>1, of a set SâŠ"Rd is a set {(Ai,Bi)} of pairs of subsets of S such that for each i, there are balls DAi and DBi containing Ai and Bi respectively such that d(DAi,DBi)≥s×min(radius(DAi), radius(DBi)), and for any two points p,qâ̂̂S there is a unique index i such that pâ̂̂Ai and qâ̂̂Bi or vice versa. In this paper, we use the SSPD to obtain the following results: First, we consider the construction of geometric t-spanners in the context of imprecise points and we prove that any set SâŠ"Rd of n imprecise points, modeled as pairwise disjoint balls, admits a t-spanner with O(nlogn/(t-1)d) edges that can be computed in O(nlogn/(t-1)d)... 

We construct a new proximity graph, called the Pie Delaunay graph, on a set of n points which is a super graph of Yao graph and Euclidean minimum spanning tree (EMST). We efficiently maintain the Pie Delaunay graph where the points are moving in the plane. We use the kinetic Pie Delaunay graph to create a kinetic data structure (KDS) for maintenance of the Yao graph and the EMST on a set of n moving points in 2-dimensional space. Assuming x and y coordinates of the points are defined by algebraic functions of at most degree s, the structure uses O(n) space, O(nlogn) preprocessing time, and processes O(n 2 λ 2s∈+∈2(n)β s + 2(n)) events for the Yao graph and O(n 2 λ 2s + 2(n)) events for the... 

This paper presents the first kinetic data structure (KDS) for maintenance of the Euclidean minimum spanning tree (EMST) on a set of n moving points in 2-dimensional space. We build a KDS of size O(n) in O(nlogn) preprocessing time by which their EMST is maintained efficiently during the motion. In terms of the KDS performance parameters, our KDS is responsive, local, and compact  

Given a set S of n disjoint line segments in ℝ2, the visibility counting problem (VCP) is to preprocess S such that the number of segments in S visible from any query point p can be computed quickly. This problem can trivially be solved in logarithmic query time using O(n4) preprocessing time and space. Gudmundsson and Morin proposed a 2-approximation algorithm for this problem with a tradeoff between the space and the query time. They answer any query in Oε(n1−α) with Oε(n2+2α) of preprocessing time and space, where α is a constant 0 ≤ α ≤ 1, ε > 0 is another constant that can be made arbitrarily small, and Oε(f(n)) = O(f(n)nε). In this paper, we propose a randomized approximation algorithm... 

Given a set S of n disjoint line segments in R2, the visibility counting problem (VCP) is to preprocess S such that the number of segments in S visible from any query point p can be computed quickly. This problem can be solved trivially in O(log⁡n) query time using O(n4log⁡n) preprocessing time and O(n4) space. Gudmundsson and Morin (2010) [10] proposed a 2-approximation algorithm for this problem with a tradeoff between the space and the query time. For any constant 0≤α≤1, their algorithm answers any query in Oϵ(m(1−α)/2) time with Oϵ(m1+α) of preprocessing time and space, where ϵ>0 is a constant that can be made arbitrarily small and Oϵ(f(n))=O(f(n)nϵ) and m=O(n2) is a number that depends... 

Computing the visibility polygon, VP, of a point in a polygonal scene, is a classical problem that has been studied extensively. In this paper, we consider the problem of computing VP for any query point efficiently, with some additional preprocessing phase. The scene consists of a set of obstacles, of total complexity O(n). We show for a query point q, VP(q) can be computed in logarithmic time using O(n 4) space and O(n 4 logn) preprocessing time. Furthermore to decrease space usage and preprocessing time, we make a tradeoff between space usage and query time; so by spending O(m) space, we can achieve O(n 2 log √ m/n)/√ m)query time, where n 2≤m≤n 4. These results are also applied to...