There is a planar graph almost as good as the complete graph
SCG '86 Proceedings of the second annual symposium on Computational geometry
On shortest paths amidst convex polyhedra
SCG '86 Proceedings of the second annual symposium on Computational geometry
On shortest paths in polyhedral spaces
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
Visibility-polygon search and euclidean shortest paths
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
Geometric applications of Davenport-Schinzel sequences
SFCS '86 Proceedings of the 27th Annual Symposium on Foundations of Computer Science
Computing geographic nearest neighbors using monotone matrix searching (preliminary version)
CSC '90 Proceedings of the 1990 ACM annual conference on Cooperation
Approximate Euclidean shortest path in 3-space
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Shortest paths in the plane with polygonal obstacles
Journal of the ACM (JACM)
A single-exponential upper bound for finding shortest paths in three dimensions
Journal of the ACM (JACM)
Approximating shortest paths on a convex polytope in three dimensions
Proceedings of the twelfth annual symposium on Computational geometry
A new algorithm for computing shortest paths in weighted planar subdivisions (extended abstract)
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Approximating weighted shortest paths on polyhedral surfaces
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Approximate shortest paths and geodesic diameters on convex polytopes in three dimensions
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Approximating shortest paths on a convex polytope in three dimensions
Journal of the ACM (JACM)
Efficient algorithms for constructing fault-tolerant geometric spanners
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Constructing approximate shortest path maps in three dimensions
Proceedings of the fourteenth annual symposium on Computational geometry
On the all-pairs Euclidean short path problem
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
Practical methods for approximating shortest paths on a convex polytope in R3
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
Approximation algorithms for curvature-constrained shortest paths
Proceedings of the seventh annual ACM-SIAM symposium on Discrete algorithms
Two-point Euclidean shortest path queries in the plane
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Approximation algorithms for geometric shortest path problems
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Approximate distance oracles for geometric graphs
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Linear Time Euclidean Distance Algorithms
IEEE Transactions on Pattern Analysis and Machine Intelligence
New Results of Fault Tolerant Geometric Spanners
WADS '99 Proceedings of the 6th International Workshop on Algorithms and Data Structures
I/O-Efficient Batched Range Counting and Its Applications to Proximity Problems
FST TCS '01 Proceedings of the 21st Conference on Foundations of Software Technology and Theoretical Computer Science
Touring a sequence of polygons
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
New results on shortest paths in three dimensions
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Approximating shortest path for the skew lines problem in time doubly logarithmic in 1/epsilon
Theoretical Computer Science - Algebraic and numerical algorithm
On the symmetric angle-restricted nearest neighbor problem
Information Processing Letters
Geometric spanners with applications in wireless networks
Computational Geometry: Theory and Applications
Region-fault tolerant geometric spanners
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Local properties of geometric graphs
Computational Geometry: Theory and Applications
Efficiently determining a locally exact shortest path on polyhedral surfaces
Computer-Aided Design
Flying over a polyhedral terrain
Information Processing Letters
Approximate distance oracles for geometric spanners
ACM Transactions on Algorithms (TALG)
I/O-efficient algorithms for computing planar geometric spanners
Computational Geometry: Theory and Applications
Geometric Spanner of Objects under L1 Distance
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
Approximate Euclidean shortest paths amid convex obstacles
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Proceedings of the twenty-fifth annual symposium on Computational geometry
A near linear time approximation scheme for Steiner tree among obstacles in the plane
Computational Geometry: Theory and Applications
A Genetic Algorithm for Shortest Path Motion Problem in Three Dimensions
ICIC '07 Proceedings of the 3rd International Conference on Intelligent Computing: Advanced Intelligent Computing Theories and Applications. With Aspects of Artificial Intelligence
Fréchet Distance Problems in Weighted Regions
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
On the symmetric angle-restricted nearest neighbor problem
Information Processing Letters
Experimental study of geometric t-spanners: a running time comparison
WEA'07 Proceedings of the 6th international conference on Experimental algorithms
Compact oracles for approximate distances around obstacles in the plane
ESA'07 Proceedings of the 15th annual European conference on Algorithms
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Connections between theta-graphs, delaunay triangulations, and orthogonal surfaces
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
Property testing
Property testing
Approximate shortest paths in simple polyhedra
DGCI'11 Proceedings of the 16th IAPR international conference on Discrete geometry for computer imagery
The euclidean bottleneck steiner path problem
Proceedings of the twenty-seventh annual symposium on Computational geometry
A nearly optimal algorithm for finding L1shortest paths among polygonal obstacles in the plane
ESA'11 Proceedings of the 19th European conference on Algorithms
Experimental study of geometric t-spanners
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Constant-time all-pairs geodesic distance query on triangle meshes
I3D '12 Proceedings of the ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games
Querying two boundary points for shortest paths in a polygonal domain
Computational Geometry: Theory and Applications
Fast edge-routing for large graphs
GD'09 Proceedings of the 17th international conference on Graph Drawing
Algorithms for computing Best Coverage Path in the presence of obstacles in a sensor field
Journal of Discrete Algorithms
On plane constrained bounded-degree spanners
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
A near linear time approximation scheme for steiner tree among obstacles in the plane
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
Direct multiple shooting method for solving approximate shortest path problems
Journal of Computational and Applied Mathematics
On the stretch factor of the theta-4 graph
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
On the spanning ratio of theta-graphs
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
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This paper gives approximation algorithms of solving the following motion planning problem: Given a set of polyhedral obstacles and points s and t, find a shortest path from s to t that avoids the obstacles. The paths found by the algorithms are piecewise linear, and the length of a path is the sum of the lengths of the line segments making up the path. Approximation algorithms will be given for versions of this problem in the plane and in three-dimensional space. The algorithms return an &egr;-short path, that is, a path with length within (1 + &egr;) of shortest. Let n be the total number of faces of the polyhedral obstacles, and &egr; a given value satisfying &Ogr; &egr; ≤ &pgr;. The algorithm for the planar case requires &Ogr;(n log n)/&egr; time to build a data structure of size &Ogr;(n/&egr;). Given points s and t, and &egr;-short path from s to t can be found with the use of the data structure in time &Ogr;(n/&egr; + n log n). The data structure is associated with a new variety of Voronoi diagram. Given obstacles S ⊂ &Egr;3 and points s, t &egr; E3, an &egr;-short path between s and t can be found in &Ogr;(n2&lgr;(n) log(n/&egr;)/&egr;4 + n2 lognp log(n logp)) time, where p is the ratio of the length of the longest obstacle edge to the distance between s to t. The function &lgr;(n) = &agr;(n)&Ogr;(&agr;(n)&Ogr;(1)), where the &agr;(n) is a form of inverse of Ackermann's function. For log(1/&egr;) and log p that are &Ogr;(log n), this bound is &Ogr;(log n2(n)&lgr;(n)/&egr;4).