Parallel methods for visibility and shortest-path problems in simple polygons

In this paper we give efficient parallel algorithms for solving a number of visibility and shortest-path problems for simple polygons. Our algorithms all run inO(logn) time and are based on the use of a new data structure for implicitly representing all shortest paths in a simple polygonP, which we call thestratified decomposition tree. We use this approach to derive efficient parallel methods for computing the visibility ofP from an edge, constructing the visibility graph of the vertices ofP (using an output-sensitive number of processors), constructing the shortest-path tree from a vertex ofP, and determining all-farthest neighbors for the vertices inP. The computational model we use is the CREW PRAM.This research was announced in preliminary form in theProceedings of the 6th ACM Symposium on Computational Geometry, 1990, pp. 73–82. The research of Michael T. Goodrich was supported by the National Science Foundation under Grants CCR-8810568 and CCR-9003299, and by the NSF and DARPA under Grant CCR-8908092.     

一个改进的简单多边形凸包算法

凸包问题是计算几何的基本问题之一，在许多领域均有应用。该文通过给出反例，证明文献[4]提出的简单多边形凸包的双动线检测算法不能正确求出任意多边形的凸包，并分析了其缺点，提出了一个改进的算法。改进的算法解决了线性算法所不能解决的自交问题，且实现简单。     

VC-dimension of perimeter visibility domains

We obtain an upper bound of 7 for the VC-dimension of Perimeter Visibility Domains in simple polygons.     

A fast algorithm for an envelope construction of a huge set of topologically consistent polygons

This paper describes an efficient algorithm for the determination of an envelope of a set of polygons. The polygons have to be aligned along their borders – the case which, for example, appears in geographic information systems. The edges, which belong to two neighbouring polygons, are called twin edges, and are eliminated first. To accelerate the geometric search, a two-level uniform plane subdivision structure is proposed. The remaining non-twin edges belong to the envelope, and they have to be joined to form the simple polygons at the end. For this task, a new algorithm has been developed. At the end, spatial relationships between resulting polygons are established. The whole algorithm for envelope determination works in expected time O(n log n) using O(n) memory space, where n is the total number of edges belonging to the input polygons. In the last part of the paper, practical results using data from a geographical database are considered.     

Optimal parallel algorithms for point-set and polygon problems

In this paper we give parallel algorithms for a number of problems defined on point sets and polygons. All our algorithms have optimalT(n) * P(n) products, whereT(n) is the time complexity andP(n) is the number of processors used, and are for the EREW PRAM or CREW PRAM models. Our algorithms provide parallel analogues to well-known phenomena from sequential computational geometry, such as the fact that problems for polygons can oftentimes be solved more efficiently than point-set problems, and that nearest-neighbor problems can be solved without explicitly constructing a Voronoi diagram.The research of R. Cole was supported in part by NSF Grants CCR-8702271, CCR-8902221, and CCR-8906949, by ONR Grant N00014-85-K-0046, and by a John Simon Guggenheim Memorial Foundation fellowship. M. T. Goodrich's research was supported by the National Science Foundation under Grant CCR-8810568 and by the National Science Foundation and DARPA under Grant CCR-8908092.     

A fast algorithm for computing sparse visibility graphs

AnO(¦E¦log² n) algorithm is presented to construct the visibility graph for a collection ofn nonintersecting line segments, where ¦E¦ is the number of edges in the visibility graph. This algorithm is much faster than theO(n ²)-time andO(n ²)-space algorithms by Asanoet al., and by Welzl, on sparse visibility graphs. Thus we partially resolve an open problem raised by Welzl. Further, our algorithm uses onlyO(n) working storage.     

On the minimum size of visibility graphs

In this paper we give tight lower bounds on the size of the visibility graph, the contracted visibility graph, and the bar-visibility graph of n disjoint line segments in the plane, according to their vertex-connectivity.     

Maintaining the visibility map of spheres while moving the viewpoint on a circle at infinity

We investigate three-dimensional visibility problems for scenes that consist ofn non-intersecting spheres. The viewing point moves on a flightpath that is part of a circle at infinity given by a planeP and a range of angles {(t)¦t[01]} [02]. At timet, the lines of sight are parallel to the ray inP, which starts in the origin ofP and represents the angle(t) (orthographic views of the scene). We give an algorithm that computes the visibility graph at the start of the flight, all time parameters at which the topology of the scene changes, and the corresponding topology changes. The algorithm has running time0(n + k + p) logn), wheren is the number of spheres in the scene;p is the number of transparent topology changes (the number of different scene topologies visible along the flight path, assuming that all spheres are transparent); andk denotes the number of vertices (conflicts) which are in the (transparent) visibility graph at the start and do not disappear during the flight.The second author was supported by the ESPRIT II Basic Research Actions Program, under Contract No. 3075 (project ALCOM).     

Measure of circularity for parts of digital boundaries and its fast computation

This paper focuses on the design of an effective method that computes the measure of circularity of a part of a digital boundary. An existing circularity measure of a set of discrete points, which is used in computational metrology, is extended to the case of parts of digital boundaries. From a single digital boundary, two sets of points are extracted so that the circularity measure computed from these sets is representative of the circularity of the digital boundary. Therefore, the computation consists of two steps. First, the inner and outer sets of points are extracted from the input part of a digital boundary using digital geometry tools. Next, the circularity measure of these sets is computed using classical tools of computational geometry. It is proved that the algorithm is linear in time in the case of convex parts thanks to the specificity of digital data, and is in O(nlogn) otherwise. Experiments done on synthetic and real images illustrate the interest of the properties of the circularity measure.     

Some chain visibility problems in a simple polygon

In this paper, the notions of convex chain visibility and reflex chain visibility of a simple polygonP are introduced, and some optimal algorithms concerned with convex- and reflex-chain visibility problems are described. For a convex-chain visibility problem, two linear-time algorithms are exhibited for determining whether or notP is visible from a given convex chain; one is the turn-checking approach and the other is the decomposition approach based on checking edge visibilities. We also present a linear-time algorithm for finding, if any, all maximal convex chains of a given polygonP from whichP is visible, where a maximal convex chain is a convex chain which does not properly include any other convex chains. It can be made by showing that there can be at most four visible maximal convex chains inP with an empty kernel. By similar arguments, we show that the same problems for reflex chain visibility can be easily solved in linear time.     

Multi-dimensional dynamic facility location and fast computation at query points

We present O(nlogn) time algorithms for the minimax rectilinear facility location problem in R¹ and R². The algorithms enable, once they terminate, computing the cost of any given query point in O(logn) time. Based on these algorithms, we develop a preprocessing procedure which enables solving the following two problems: Fast computation of the cost of any query point in Rd, and fast solution for the dynamic location problem in R² (namely, in the presence of an additional facility). Finally, we show that the preprocessing always gives a bound on the optimal value, which allows us in many cases to find the optimum fast (for both the traditional and the dynamic location problems in Rd for any d).     

Fast computation of smallest enclosing circle with center on a query line segment

Here we propose an efficient algorithm for computing the smallest enclosing circle whose center is constrained to lie on a query line segment. Our algorithm preprocesses a given set of n points P={p₁,p₂,…,pn} such that for any query line or line segment L, it efficiently locates a point c on L that minimizes the maximum distance among the points in P from c. Roy et al. [S. Roy, A. Karmakar, S. Das, S.C. Nandy, Constrained minimum enclosing circle with center on a query line segment, in: Proc. of the 31st Mathematical Foundation of Computer Science, 2006, pp. 765-776] have proposed an algorithm that solves the query problem in O(log²n) time using O(nlogn) preprocessing time and O(n) space. Our algorithm improves the query time to O(logn); but the preprocessing time and space complexities are both O(n²).     

Constructing the visibility graph for n-line segments in O(n2) time

Given a set S of line segments in the plane, its visibility graph G_S is the undirected graph which has the endpoints of the line segments in S as nodes and in which two nodes (points) are adjacent whenever they ‘see’ each other (the line segments in S are regarded as nontransparent obstacles). It is shown that G_S can be constructed in O(n²) time and space for a set S of n nonintersecting line segments. As an immediate implication, the shortest path between two points in the plane avoiding a set of n nonintersecting line segments can be computed in O(n²) time and space     

Minimum-link watchman tours

We consider the problem of computing a watchman route in a polygon with holes. We show that the problem of finding a minimum-link watchman route is NP-complete, even if the holes are all convex. The proof is based on showing that the related problem of finding a minimum-link tour on a set of points in the plane is NP-complete. We provide a provably good approximation algorithm that achieves an approximation factor of O(logn).     

Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons

Given a triangulation of a simple polygonP, we present linear-time algorithms for solving a collection of problems concerning shortest paths and visibility withinP. These problems include calculation of the collection of all shortest paths insideP from a given source vertexS to all the other vertices ofP, calculation of the subpolygon ofP consisting of points that are visible from a given segment withinP, preprocessingP for fast "ray shooting" queries, and several related problems.Work on this paper by this author has been supported by Office of Naval Research Grant N00014-82-K-0381, National Science Foundation Grant No. NSF-DCR-83-20085, and by grants from the Digital Equipment Corporation, the IBM Corporation, and from the U.S.-Israel Binational Science Foundation.Work on this paper by this author has been supported by National Science Foundation Grant DCR-86-05962.     

Applying the improved Chen and Han’s algorithm to different versions of shortest path problems on a polyhedral surface

The computation of shortest paths on a polyhedral surface is a common operation in many computer graphics applications. There are two best known exact algorithms for the “single source, any destination” shortest path problem. One is proposed by Mitchell et al. (1987) [1]. The other is by Chen and Han (1990) [11]. Recently, Xin and Wang (2009) [9] improved the CH algorithm by exploiting a filtering theorem and achieved a practical method that outperforms both the CH algorithm and the MMP algorithm whether in time or in space.In this paper, we apply the improved CH algorithm to different versions of shortest path problems. The contributions of this paper include: (1) For a surface point p∈△v₁v₂v₃, we present an unfolding technique for estimating the distance value at p using the distances at v₁,v₂ and v₃. (2) We show that the improved CH algorithm can be naturally extended to the “multiple sources, any destination” version. Also, introducing a well-chosen heuristic factor into the improved CH algorithm will induce an exact solution to the “single source, single destination” version. (3) At the conclusion of multi-source shortest path algorithms, we can use the distance values at vertices to approximately compute the geodesic-distance-based offsets, the Voronoi diagram and the Delaunay triangulation in O(n) time. (4) By importing a precision parameter λ, we obtain a precision controlled approximant which varies from the improved CH algorithm to Dijkstra’s algorithm as λ increases from 0 to 1. Thus, an interesting relationship between them can be naturally established.     

Characterizing and recognizing the visibility graph of a funnel-shaped polygon

A funnel, which is notable for its fundamental role in visibility algorithms, is defined as a polygon that has exactly three convex vertices, two of which are connected by a boundary edge. In this paper we investigate the visibility graph of a funnel which we call an F-graph.We first present two characterizations of an F-graph, one of whose sufficiency proof itself is a linear time Real RAM algorithm for drawing a funnel on the plane that corresponds to an F-graph. We next give a linear-time algorithm for recognizing an F-graph. When the algorithm recognizes an F-graph, it also reports one of the Hamiltonian cycles defining the boundary of its corresponding funnel. This recognition algorithm takes linear time even on a RAM.We finally show that an F-graph is weakly triangulated and therefore perfect, which agrees with the fact that perfect graphs are related to geometric structures.This work was supported in part by the Korea Science and Engineering Foundation under Grant 91-01-01.     

A new algorithm for the largest empty rectangle problem

A rectangleA and a setS ofn points inA are given. We present a new simple algorithm for the so-called largest empty rectangle problem, i.e., the problem of finding a maximum area rectangle contained inA and not containing any point ofS in its interior. The computational complexity of the presented algorithm isO(n logn + s), where s is the number of possible restricted rectangles considered. Moreover, the expected performance isO(n · logn).     

A linear-time algorithm for constructing a circular visibility diagram

To computer circular visibility inside a simple polygon, circular arcs that emanate from a given interior point are classified with respect to the edges of the polygon they first intersect. Representing these sets of circular arcs by their centers results in a planar partition called the circular visibility diagram. AnO(n) algorithm is given for constructing the circular visibility diagram for a simple polygon withn vertices.