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Geometric Shortest Paths and Network Optimization
 Handbook of Computational Geometry
, 1998
"... Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of t ..."
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Cited by 146 (12 self)
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Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal
Efficient algorithms for geometric optimization
 ACM Comput. Surv
, 1998
"... We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear progra ..."
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Cited by 92 (12 self)
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We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other querytype problems.
Exact and Approximate Distances in Graphs  a survey
 In ESA
, 2001
"... We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems. ..."
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Cited by 54 (0 self)
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We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems.
An Efficient Algorithm for Euclidean Shortest Paths Among Polygonal Obstacles in the Plane
, 1988
"... We give an algorithm to compute a (Euclidean) shortest path in a polygon with h holes and a total of n vertices. The algorithm uses O(n) space and requires O(n + h² log n) time. ..."
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Cited by 34 (1 self)
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We give an algorithm to compute a (Euclidean) shortest path in a polygon with h holes and a total of n vertices. The algorithm uses O(n) space and requires O(n + h² log n) time.
Approximate Distance Oracles for Geometric Graphs
, 2002
"... Given a geometric tspanner graph G in E d with n points and m edges, with edge lengths that lie within a polynomial (in n) factor of each other. Then, after O(m+n log n) preprocessing, we present an approximation scheme to answer (1+") approximate shortest path queries in O(1) time. The data str ..."
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Cited by 34 (10 self)
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Given a geometric tspanner graph G in E d with n points and m edges, with edge lengths that lie within a polynomial (in n) factor of each other. Then, after O(m+n log n) preprocessing, we present an approximation scheme to answer (1+") approximate shortest path queries in O(1) time. The data structure uses O(n log n) space.
The VisibilityVoronoi complex and its applications
 In Proc. 21st Annu. ACM Sympos. Comput. Geom. (SCG
, 2005
"... We introduce a new type of diagram called the VV (c)diagram (the Visibility–Voronoi diagram for clearance c), which is a hybrid between the visibility graph and the Voronoi diagram of polygons in the plane. It evolves from the visibility graph to the Voronoi diagram as the parameter c grows from 0 ..."
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Cited by 25 (3 self)
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We introduce a new type of diagram called the VV (c)diagram (the Visibility–Voronoi diagram for clearance c), which is a hybrid between the visibility graph and the Voronoi diagram of polygons in the plane. It evolves from the visibility graph to the Voronoi diagram as the parameter c grows from 0 to ∞. This diagram can be used for planning naturallooking paths for a robot translating amidst polygonal obstacles in the plane. A naturallooking path is short, smooth, and keeps — where possible — an amount of clearance c from the obstacles. The VV (c)diagram contains such paths. We also propose an algorithm that is capable of preprocessing a scene of configurationspace polygonal obstacles and constructs a data structure called the VVcomplex. The VVcomplex can be used to efficiently plan motion paths for any start and goal configuration and any clearance value c, without having to explicitly construct the VV (c)diagram for that cvalue. The preprocessing time is O(n 2 log n), where n is the total number of obstacle vertices, and the data structure can be queried directly for any cvalue by merely performing a Dijkstra search. We have implemented a Cgalbased software package for computing the VV (c)diagram in an exact manner for a given clearance value, and used it to plan naturallooking paths in various applications.
New Results on Shortest Paths in Three Dimensions
 Proc. 20th Annual ACM Symposium on Computational Geometry
, 2004
"... We revisit the problem of computing shortest obstacleavoiding paths among obstacles in three dimensions. We prove new hardness results, showing, e.g., that computing Euclidean shortest paths among sets of “stacked ” axisaligned rectangles is NPcomplete, and that computing L1shortest paths among ..."
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Cited by 21 (0 self)
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We revisit the problem of computing shortest obstacleavoiding paths among obstacles in three dimensions. We prove new hardness results, showing, e.g., that computing Euclidean shortest paths among sets of “stacked ” axisaligned rectangles is NPcomplete, and that computing L1shortest paths among disjoint balls is NPcomplete. On the positive side, we present an efficient algorithm for computing an L1shortest path between two given points that lies on or above a given polyhedral terrain. We also give polynomialtime algorithms for some versions of stacked polygonal obstacles that are “terrainlike ” and analyze the complexity of shortest path maps in the presence of parallel halfplane “walls.”
ResourceConstrained Geometric Network Optimization (Extended Abstract)
"... We study a variety of geometric network optimization problems on a set of points, in which we are given a resource bound, B, on the total length of the network, and our objective is to maximize the number of points visited (or the total "value" of points visited). In particular, we resolve the wel ..."
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Cited by 21 (1 self)
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We study a variety of geometric network optimization problems on a set of points, in which we are given a resource bound, B, on the total length of the network, and our objective is to maximize the number of points visited (or the total "value" of points visited). In particular, we resolve the wellpublicized open problem on the approximability of the rooted "orienteering problem" for the case in which the sites are given as points in the plane and the network required is a cycle. We obtain a 2approximation for this problem. We also obtain approximation algorithms for variants of this problem in which the network required is a tree (3approximation) or a path (2approximation). No prior approximation bounds were known for any of these problems. We also obtain improved approximation algorithms for geometric instances of the unrooted orienteering problem, where we obtain a 2approximation for both the cycle and tree versions of the problem on points in the plane, as well as a ...
TwoPoint Euclidean Shortest Path Queries in the Plane (Extended Abstract)
, 1999
"... ) To appear in Proc. Tenth Annual ACMSIAM Symposium on Discrete Algorithms (SODA '99), January 1719, 1999 YiJen Chiang Joseph S. B. Mitchell y Abstract We consider the twopoint query version of the fundamental geometric shortest path problem: Given a set h of polygonal obstacles in the pla ..."
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Cited by 18 (2 self)
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) To appear in Proc. Tenth Annual ACMSIAM Symposium on Discrete Algorithms (SODA '99), January 1719, 1999 YiJen Chiang Joseph S. B. Mitchell y Abstract We consider the twopoint query version of the fundamental geometric shortest path problem: Given a set h of polygonal obstacles in the plane, having a total of n vertices, build a data structure such that for any two query points s and t we can efficiently determine the length, d(s; t), of an Euclidean shortest obstacleavoiding path, ß(s; t), from s to t. Additionally, our data structure should allow one to report the path ß(s; t), in time proportional to its (combinatorial) size. We present various methods for solving this twopoint query problem, including algorithms with o(n), O(log n+h), O(h log n), O(log 2 n) or optimal O(log n) query times, using polynomialspace data structures, with various tradeoffs between space and query time. While several results have been known for approximate twopoint Euclidean shortest p...