Results 1  10
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20
Computing Minimum Length Paths of a Given Homotopy Class
 Comput. Geom. Theory Appl
, 1991
"... In this paper, we show that the universal covering space of a surface can be used to unify previous results on computing paths in a simple polygon. We optimize a given path among obstacles in the plane under the Euclidean and link metrics and under polygonal convex distance functions. Besides reveal ..."
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Cited by 74 (7 self)
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In this paper, we show that the universal covering space of a surface can be used to unify previous results on computing paths in a simple polygon. We optimize a given path among obstacles in the plane under the Euclidean and link metrics and under polygonal convex distance functions. Besides revealing connections between the minimum paths under these three distance functions, the framework provided by the universal cover leads to simplified lineartime algorithms for shortest path trees, for minimumlink paths in simple polygons, and for paths restricted to c given orientations. 1 Introduction If a wire, a pipe, or a robot must traverse a path among obstacles in the plane, then one might ask what is the best route to take. For the wire, perhaps the shortest distance is best; for the pipe, perhaps the fewest straightline segments. For the robot, either might be best depending on the relative costs of turning and moving. In this paper, we find shortest paths and shortest closed curve...
Efficient Algorithms for Approximating Polygonal Chains
"... We consider the problem of approximating a polygonal chain C by another polygonal chain C ′ whose vertices are constrained to be a subset of the set of vertices of C. The goal is to minimize the number of vertices needed in the approximation C ′. Based on a framework introduced by Imai and Iri [25 ..."
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Cited by 39 (2 self)
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We consider the problem of approximating a polygonal chain C by another polygonal chain C ′ whose vertices are constrained to be a subset of the set of vertices of C. The goal is to minimize the number of vertices needed in the approximation C ′. Based on a framework introduced by Imai and Iri [25], we define an error criterion for measuring the quality of an approximation. We consider two problems. (1) Given a polygonal chain C and a parameter ε ≥ 0, compute an approximation of C, among all approximations whose error is at most ε, that has the smallest number of vertices. We present an O(n 4/3+δ)time algorithm to solve this problem, for any δ>0; the constant of proportionality in the running time depends on δ. (2) Given a polygonal chain C and an integer k, compute an approximation of C with at most k vertices whose error is the smallest among all approximations with at most k vertices. We present a simple randomized algorithm, with expected running time O(n 4/3+δ), to solve this problem.
On The Complexity Of Polyhedral Separability
 DISCRETE COMPUTATIONAL GEOMETRY
, 1988
"... It is NPcomplete to recognize whether two sets of points in general space can be separated by two hyperplanes. It is NPcomplete to recognize whether two sets of points in the plane can be separated with k lines. For every fixed k in any fixed dimension, it takes polynomial time to recognize whethe ..."
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Cited by 39 (2 self)
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It is NPcomplete to recognize whether two sets of points in general space can be separated by two hyperplanes. It is NPcomplete to recognize whether two sets of points in the plane can be separated with k lines. For every fixed k in any fixed dimension, it takes polynomial time to recognize whether two sets of points can be separated with k hyperplanes.
Efficient PiecewiseLinear Function Approximation Using the Uniform Metric
 Discrete & Computational Geometry
, 1994
"... We give an O(n log n)time method for finding a best klink piecewiselinear function approximating an npoint planar data set using the wellknown uniform metric to measure the error, ffl 0, of the approximation. Our method is based upon new characterizations of such functions, which we exploit to ..."
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Cited by 38 (0 self)
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We give an O(n log n)time method for finding a best klink piecewiselinear function approximating an npoint planar data set using the wellknown uniform metric to measure the error, ffl 0, of the approximation. Our method is based upon new characterizations of such functions, which we exploit to design an efficient algorithm using a plane sweep in "ffl space" followed by several applications of the parametric searching technique. The previous best running time for this problem was O(n 2 ). 1 Introduction Approximating a set S = f(x 1 ; y 1 ); (x 2 ; y 2 ); : : : ; (x n ; y n )g of points in the plane by a function is a classic problem in applied mathematics. The general goals in this area of research are to find a function F belonging to a class of functions F such that each F 2 F is simple to describe, represent, and compute and such that the chosen F approximates S well. For example, one may desire that F be the class of linear or piecewiselinear functions, and, for any parti...
Sweeping simple polygons with a chain of guards
 In Proceedings of the 11th ACMSIAM Symposium on Discrete Algorithms (SODA
, 2000
"... Abstract We consider the problem of locating a continuouslymoving target using a group of guardsmoving inside a simple polygon. Our guards always form a simple polygonal chain within the polygon such that consecutive guards along the chain are mutually visible. We developalgorithms that sweep such ..."
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Cited by 36 (2 self)
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Abstract We consider the problem of locating a continuouslymoving target using a group of guardsmoving inside a simple polygon. Our guards always form a simple polygonal chain within the polygon such that consecutive guards along the chain are mutually visible. We developalgorithms that sweep such a chain of guards through a polygon to locate the target. Our two main results are the following: 1. an algorithm to compute the minimum number r * of guards needed to sweep an nvertexpolygon that runs in O(n3) time and uses O(n2) working space, and 2. a faster algorithm, using O(n log n) time and O(n) space, to compute an integer r suchthat max( r 16, 2) < = r * < = r and P can be swept with a chain of r guards. We develop two other techniques to approximate r*. Using O(n2) time and space, we show howto sweep the polygon using at most r * + 2 guards. We also show that any polygon can be sweptby a number of guards equal to two more than the link radius of the polygon. As a key component of our exact algorithm, we introduce the notion of the link diagramof a polygon, which encodes the link distance between all pairs of points on the boundary of the polygon. We prove that the link diagram has size \Theta (n3) and can be constructed in \Theta (n3)time. We also show link diagram provides a data structure for optimal twopoint linkdistance queries, matching an earlier result of Arkin et al.As a key component of our O(n log n)time approximation algorithm, we introduce the notionof the "link width " of a polygon, which may have independent interest, as it captures important
Maintaining Approximate Extent Measures of Moving Points
 In Proc. 12th ACMSIAM Sympos. Discrete Algorithms
, 2001
"... We present approximation algorithms for maintaining various descriptors of the extent of moving points in R d . We first describe a data structure for maintaining the smallest orthogonal rectangle containing the point set. We then use this data structure to maintain the approximate diameter, smal ..."
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Cited by 31 (4 self)
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We present approximation algorithms for maintaining various descriptors of the extent of moving points in R d . We first describe a data structure for maintaining the smallest orthogonal rectangle containing the point set. We then use this data structure to maintain the approximate diameter, smallest enclosing disk, width, and smallest area or perimeter bounding rectangle of a set of moving points in R 2 so that the number of events is only a constant. This contrasts with\Omega\Gamma n 2 ) events that data structures for the maintenance of those exact properties have to handle. 1 Introduction With the rapid advances in positioning systems, e.g., GPS, adhoc networks, and wireless communication, it is becoming increasingly feasible to track and record the changing position of continuously moving objects. These developments have raised a wide range of challenging geometric problems involving moving objects, including efficient data structures for answering proximity queries, fo...
Separation and Approximation of Polyhedral Objects
, 1993
"... Given a family of disjoint polygons P1, P2, : ::, Pk in the plane, and an integer parameter m, it is NPcomplete to decide if the Pi's can be pairwise separated by a polygonal family with at most m edges, that is, if there exist polygons R1; R2; : ::; Rk with pairwisedisjoint boundaries such that P ..."
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Cited by 30 (3 self)
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Given a family of disjoint polygons P1, P2, : ::, Pk in the plane, and an integer parameter m, it is NPcomplete to decide if the Pi's can be pairwise separated by a polygonal family with at most m edges, that is, if there exist polygons R1; R2; : ::; Rk with pairwisedisjoint boundaries such that Pi Ri andP jRij m. In three dimensions, the problem is NPcomplete even for two nested convex polyhedra. Many other extensions and generalizations of the polyhedral separation problem, either to families of polyhedra or to higher dimensions, are also intractable. In this paper, we present e cient approximation algorithms for constructing separating families of nearoptimal size. Our main results are as follows. In two dimensions, we give an O(n log n) time algorithm for constructing a separating family whose size is within a constant factor of an optimal separating family; n is the number of edges in the input family of polygons. In three dimensions, we show how to separate a convex polyhedron from a nonconvex polyhedron with a polyhedral surface whose facetcomplexity is O(log n) times the optimal, where n = jPj+jQj is the complexity of the input polyhedra. Our algorithm runs in O(n4) time, but improves to O(n3) time if the two polyhedra are nested and convex. Our algorithm for separating a convex polyhedron from a nonconvex polyhedron extends to higher dimensions. In d dimensions, for d 4, the facetcomplexity of the approximation polyhedron is O(d log n) times the optimal, and the algorithm runs in O(nd+1) time. Finally, we also obtain results on separating sets of points, a family of convex polyhedra, and separation by nonpolyhedral surfaces, such as spherical patches.
Guarding Art Galleries  Methods for Mobile Guards; doctoral thesis
, 1995
"... Typeset in LATEX Pictures where done using xfig and inserted as Postscript1 figures 1Postscript is a registered trademark of Adobe Systems Incorporated. Abstract Many problems that arise in everyday situations involve guarding or illuminating objects or regions. Some examples are: placing TVcameras ..."
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Cited by 25 (5 self)
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Typeset in LATEX Pictures where done using xfig and inserted as Postscript1 figures 1Postscript is a registered trademark of Adobe Systems Incorporated. Abstract Many problems that arise in everyday situations involve guarding or illuminating objects or regions. Some examples are: placing TVcameras in the local supermarket, positioning radar observation stations in a mountainous terrain, or arranging the lighting in one's living room. The art gallery problem is a famous problem in computational geometry that formally models these types of applications. The problem is that of placing the minimum number of guards in an art gallery so that the whole gallery is seen by the guards. The art gallery is represented geometrically by a polygon in the plane, where the edges correspond to the walls of the gallery. The geometric interpretation of a guard in this original setting is to be a point in the art gallery representing a static sensor system that can see things along straight lines. As a first step of this work, we generalize the problem to placing either mobile or static guards in a gallery subject to an optimization criterion, and possibly some locality restrictions. In this way, we can model many new practical problems such as guiding robots through a burning building in search of victims, or having robots survey a nuclear reactor. Unfortunately, the original art gallery problem can be shown to be intractable, which also implies that our generalized version is intractable. Hence, we have to content ourselves with solving restricted cases of the problem. This work is divided into two parts where we investigate two variations of the problem. The first part addresses the following specific problem: given an art gallery and the route followed by a robot guarding the gallery, what is the minimum number of points on the route where the robot must engage its sensor system? We show that, in general, this problem is also intractable, but we solve it for some interesting special classes of art galleries.
Approximation algorithms for geometric separation problems
 Department of
, 1993
"... In computer graphics and solid modeling, one is interested in representing complex geometric objects with combinatorially simpler ones. It turns out that via a “fattening ” transformation, one obtains a formulation of the approximation problem in terms of separation: Find a minimumcomplexity surface ..."
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Cited by 14 (4 self)
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In computer graphics and solid modeling, one is interested in representing complex geometric objects with combinatorially simpler ones. It turns out that via a “fattening ” transformation, one obtains a formulation of the approximation problem in terms of separation: Find a minimumcomplexity surface that separates two sets. In this paper, we provide approximation algorithms for several geometric separation problems, including: • Given a set of triangles T and a set S of points that lie within the union of the triangles, find a minimumcardinality set, T ′ , of pairwisedisjoint triangles, each contained within some triangle of T, that cover the point set S. • Given finite sets of “red ” and “blue ” points in the plane, determine a simple polygon of fewest edges that separates the red points from the blue points. More generally, given finite sets of points of many color classes, determine a planar “separating ” subdivision of minimum combinatorial complexity, which has the property that each face of the subdivision contains points of at most one color class; • Given two polyhedral terrains, P and Q, over a common support set (e.g., the unit square), with P lying above Q, compute a nested polyhedral terrain R that lies between P and Q such that R has a minimum number of facets. Exact solution of the above problems in polynomial time is highly unlikely: The decision versions of all three problems are known to be NPhard. We provide polynomialtime algorithms that are guaranteed to produce an answer within a logarithmic factor (O(log n), where n is the complexity of the input problem instance) of optimal. (The error factor is constant in the orthogonal case — coverage by disjoint aligned rectangles, or separation of orthohedral terrains.) We also discuss extensions to higher dimensions. 1
On the Complexity of Optimization Problems for 3Dimensional Convex Polyhedra and Decision Trees
 Comput. Geom. Theory Appl
, 1995
"... We show that several wellknown optimization problems involving 3dimensional convex polyhedra and decision trees are NPhard or NPcomplete. One of the techniques we employ is a lineartime method for realizing a planar 3connected triangulation as a convex polyhedron, which may be of independent i ..."
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Cited by 14 (0 self)
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We show that several wellknown optimization problems involving 3dimensional convex polyhedra and decision trees are NPhard or NPcomplete. One of the techniques we employ is a lineartime method for realizing a planar 3connected triangulation as a convex polyhedron, which may be of independent interest. Key words: Convex polyhedra, approximation, Steinitz's theorem, planar graphs, art gallery theorems, decision trees. 1 Introduction Convex polyhedra are fundamental geometric structures (e.g., see [20]). They are the product of convex hull algorithms, and are key components for problems in robot motion planning and computeraided geometric design. Moreover, due to a beautiful theorem of Steinitz [20, 38], they provide a strong link between computational geometry and graph theory, for Steinitz shows that a graph forms the edge structure of a convex polyhedra if and only if it is planar and 3connected. Unfortunately, algorithmic problems dealing with 3dimensional convex polyhedra ...