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39
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 ..."
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Cited by 187 (15 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
Robust Geometric Computation
, 1997
"... Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section... ..."
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Cited by 80 (14 self)
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Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section...
Approximate convex decomposition of polyhedra
 In Proc. of ACM Symposium on Solid and Physical Modeling
, 2005
"... Decomposition is a technique commonly used to partition complex models into simpler components. While decomposition into convex components results in pieces that are easy to process, such decompositions can be costly to construct and can result in representations with an unmanageable number of compo ..."
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Cited by 51 (3 self)
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Decomposition is a technique commonly used to partition complex models into simpler components. While decomposition into convex components results in pieces that are easy to process, such decompositions can be costly to construct and can result in representations with an unmanageable number of components. In this paper, we explore an alternative partitioning strategy that decomposes a given model into “approximately convex ” pieces that may provide similar benefits as convex components, while the resulting decomposition is both significantly smaller (typically by orders of magnitude) and can be computed more efficiently. Indeed, for many applications, an approximate convex decomposition (ACD) can more accurately represent the important structural features of the model by providing a mechanism for ignoring less significant features, such as surface texture. We describe a technique for computing ACDs of threedimensional polyhedral solids and surfaces of arbitrary genus. We provide results illustrating that our approach results in high quality decompositions with very few components and applications showing that comparable or better results can be obtained using ACD decompositions in place of exact convex decompositions (ECD) that are several orders of magnitude larger. 1 ECD Figure 1: The approximate convex decompositions (ACD) of the Armadillo and the David models consist of a small number of nearly convex components that characterize the important features of the models better than the exact convex decompositions (ECD) that have orders of magnitude more components. The Armadillo (500K edges, 12.1MB) has a solid ACD with 98 components (14.2MB) that was computed in 232 seconds while the solid “ECD ” has more than 726,240 components (20+ GB) and could not be completed because disk space was exhausted after nearly 4 hours of computation. The David (750K edges, 18MB) has a surface ACD with 66 components (18.1MB) while the surface ECD has 85,132 components (20.1MB). 1
Approximating Shortest Paths on Weighted Polyhedral Surfaces
"... Shortest path problems are among the... In this paper we propose several simple and practical algorithms (schemes) to compute an approximated weighted shortest path Π'(s, t) points s and t on the surface of a polyhedron P. ..."
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Cited by 38 (8 self)
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Shortest path problems are among the... In this paper we propose several simple and practical algorithms (schemes) to compute an approximated weighted shortest path &Pi;'(s, t) points s and t on the surface of a polyhedron P.
Touring a sequence of polygons
 Proc. 35th ACM Sympos. Theory Comput
, 2003
"... All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. ..."
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Cited by 36 (5 self)
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All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.
Constructing Approximate Shortest Path Maps in Three Dimensions
 SIAM J. Comput
, 1999
"... We present a new technique for constructing a datastructure that approximates shortest path maps in lH'. By applying tbia technique, we get the following two results on approximate ahorteet path maps in lRB. (i) Given a polyhedral surface or a convex polytope P with n edges in lRa, a source p ..."
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Cited by 32 (6 self)
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We present a new technique for constructing a datastructure that approximates shortest path maps in lH'. By applying tbia technique, we get the following two results on approximate ahorteet path maps in lRB. (i) Given a polyhedral surface or a convex polytope P with n edges in lRa, a source point s on P, and a real parameter 0 < c 5 1, we present an algorithm that computes a subdivision of P of size O((n/e)log(l/e)) 17hich can be used to answer efficiently approximate shortest path queries, Namely, given any point t on P, one can compute, in O(log(n/s)) time, a distance Appls(t), such that dp+(t) 5 A?,,(t) 5 (1 + e)d~,#(t), where d~,~(t) is the length of a shortest path between s and t on P. The map can be computed in O(ns log n+(n/a) log (l/c) time, for the case of a polyhedral surface, and in log(l/s) + (n/e"") log (l/s) log n) time if P is a convex polytope. (ii) Given a set of polyhedral obstacles 0 with a total of n edges in lRs, a source point s in lH3 \ int UO~,, 0, and a renl pnrameter 0 < e 5 1, we present an algonthm that computes a subdivision of lEtB, which can be used to answer efllciently approximate shortest path queries. That is, for any point t E lIta, one can compute, in O(log (n/e)) time, a distance A+(t) that sapproximates the length of a shorteat path from s to t that avoids the interiors of the obsta;ple,e This subdivision can be computed in roughly O(n4/e6) .
Approximate ShortestPath and Geodesic Diameter on Convex Polytopes in Three Dimensions
, 1996
"... Given a convex polytope P with n edges in R³, we present a relatively simple algorithm that preprocesses P in O(n) time, such that, given any two points s; t 2 @P , and a parameter 0 ! " 1, it computes, in O((logn)=" 1:5 + 1=" 3 ) time, a distance \Delta P (s; t), such that d ..."
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Cited by 29 (3 self)
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Given a convex polytope P with n edges in R³, we present a relatively simple algorithm that preprocesses P in O(n) time, such that, given any two points s; t 2 @P , and a parameter 0 ! " 1, it computes, in O((logn)=" 1:5 + 1=" 3 ) time, a distance \Delta P (s; t), such that d P (s; t) \Delta P (s; t) (1 + ")d P (s; t), where d P (s; t) is the length of the shortest path between s and t on @P . The algorithm also produces a polygonal path with O(1=" 1:5 ) segments that avoids the interior of P and has length \Delta P (s; t). Our second related result is: Given a convex polytope P with n edges in R³, and a parameter 0 ! " 1, we present an O(n + 1=" 6 )time algorithm that computes two points s; t 2 @P such that d P (s; t) (1 \Gamma ")D P , where D P = max s;t2@P d P (s; t) is the geodesic diameter of P .
PrecisionSensitive Euclidean Shortest Path in 3Space
 11TH ACM SYMP. ON COMP. GEOM
, 1995
"... This paper introduces the concept of precisionsensitive algorithms, in analogy to the wellknown outputsensitive algorithms. We exploit this idea in studying the complexity of the 3dimensional Euclidean shortest path problem. Specifically, we analyze an incremental approximation approach based o ..."
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Cited by 25 (6 self)
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This paper introduces the concept of precisionsensitive algorithms, in analogy to the wellknown outputsensitive algorithms. We exploit this idea in studying the complexity of the 3dimensional Euclidean shortest path problem. Specifically, we analyze an incremental approximation approach based on ideas in [CSY], and show that this approach yields an asymptotic improvement of running time. By using an optimization technique to improve paths on fixed edge sequences, we modify this algorithm to guarantee a relative error of O(2 \Gammar ) in a time polynomial in r and 1=ffi, where ffi denotes the relative difference in path length between the shortest and the second shortest path. Our result is the best possible in some sense: if we have a strongly precisionsensitive algorithm then we can show that USAT (unambiguous SAT) is in polynomial time, which is widely conjectured to be unlikely. Finally, we discuss the practicability of this approach. Experimental results are provided.
Polytime Algorithm for the Shortest Path in a Homotopy Class amidst SemiAlgebraic Obstacles in the Plane
, 1998
"... Given a set of semialgebraic obstacles in the plane and two points in the same connected component of the complement, the problem is to construct the shortest path between these points in a given homotopy class. This path is unique and has some canonical form. We use the representation of homoto ..."
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Cited by 18 (1 self)
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Given a set of semialgebraic obstacles in the plane and two points in the same connected component of the complement, the problem is to construct the shortest path between these points in a given homotopy class. This path is unique and has some canonical form. We use the representation of homotopy classes in a way that is as general as the classical one. It consists in representing generators of a free group which describes the classes of homotopy by disjoint cuts [GS97] homeomorphic to rays. We show that given such a system of generators and a word representing a homotopy class, one can contruct the shortest path of this class in time polynomial in the size of the word and in the size of the representation of the obstacles and the cuts. The homotopy class may also be represented by a path, then the polynomial complexity will depend on the size of the representation of this path. As a technical notion we introduce one particular system of cuts, which we call an extremity ba...