Matrix analysis
The hierarchical representation of objects: the Delaunay tree
SCG '86 Proceedings of the second annual symposium on Computational geometry
Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
A fast planar partition algorithm, I
Journal of Symbolic Computation
A fast planar partition algorithm, II
Journal of the ACM (JACM)
Computational Geometry: Theory and Applications
Small-dimensional linear programming and convex hulls made easy
Discrete & Computational Geometry
Dynamic maintenance of geometric structures made easy
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
Applications of random sampling to on-line algorithms in computational geometry
Discrete & Computational Geometry
On the randomized construction of the Delaunay tree
Theoretical Computer Science
Fully dynamic Delaunay triangulation in logarithmic expected time per operation
Computational Geometry: Theory and Applications
Four results on randomized incremental constructions
Computational Geometry: Theory and Applications
Tail estimates for the efficiency of randomized incremental algorithms for line segment intersection
Computational Geometry: Theory and Applications
SIAM Journal on Discrete Mathematics
Separators for sphere-packings and nearest neighbor graphs
Journal of the ACM (JACM)
Proceedings of the 24th annual conference on Computer graphics and interactive techniques
Algorithmic geometry
Realistic image synthesis using photon mapping
Realistic image synthesis using photon mapping
SIAM Journal on Computing
Self-Organizing Data Structures with Dependent Accesses
ICALP '96 Proceedings of the 23rd International Colloquium on Automata, Languages and Programming
Remembering Conflicts in History Yields Dynamic Algorithms
ISAAC '93 Proceedings of the 4th International Symposium on Algorithms and Computation
Markov Localization for Reliable Robot Navigation and People Detection
Selected Papers from the International Workshop on Sensor Based Intelligent Robots
Incremental constructions con BRIO
Proceedings of the nineteenth annual symposium on Computational geometry
Building Voronoi Diagrams for Convex Polygons in Linear Expected Time
Building Voronoi Diagrams for Convex Polygons in Linear Expected Time
Navigation in 3D Game by Markov Model Based Head Pose Estimating
ICIG '04 Proceedings of the Third International Conference on Image and Graphics
A Generative Model of Terrain for Autonomous Navigation in Vegetation
International Journal of Robotics Research
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
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A classic result asserts that many geometric structures can be constructed optimally by successively inserting their constituent parts in random order. These randomized incremental constructions (RICs) still work with imperfect randomness: the dynamic operations need only be "locally" random. Much attention has been given recently to inputs generated by Markov sources. These are particularly interesting to study in the framework of RICs, because Markov chains provide highly nonlocal randomness, which incapacitates virtually all known RIC technology. We generalize Mulmuley's theory of Θ-series and prove that Markov incremental constructions with bounded spectral gap are optimal within polylog factors for trapezoidal maps, segment intersections,and convex hulls in any fixed dimension. The main contribution of this work is threefold: (i)extending the theory of abstract configuration spaces to the Markov setting; (ii)proving Clarkson-Shor type bounds for this new model; (iii)applying the results to classical geometric problems. We hope that this work will pioneer a new approach to average-case analysis in computational geometry.