ACM SIGACT News
ACM Transactions on Graphics (TOG)
Computing the betti numbers of arrangements
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Three dimensional euclidean Voronoi diagrams of lines with a fixed number of orientations
Proceedings of the eighteenth annual symposium on Computational geometry
Polyhedral Voronoi diagrams of polyhedra in three dimensions
Proceedings of the eighteenth annual symposium on Computational geometry
Improved construction of vertical decompositions of three-dimensional arrangements
Proceedings of the eighteenth annual symposium on Computational geometry
On the overlay of envelopes in four dimensions
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
New constructions of weak epsilon-nets
Proceedings of the nineteenth annual symposium on Computational geometry
Proceedings of the nineteenth annual symposium on Computational geometry
Cutting triangular cycles of lines in space
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Computing the Betti numbers of arrangements via spectral sequences
Journal of Computer and System Sciences - STOC 2002
Ray shooting and stone throwing with near-linear storage
Computational Geometry: Theory and Applications
The interface between computational and combinatorial geometry
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Ray shooting amid balls, farthest point from a line, and range emptiness searching
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Crossing patterns of semi-algebraic sets
Journal of Combinatorial Theory Series A
Ray shooting and stone throwing with near-linear storage
Computational Geometry: Theory and Applications
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We show that the complexity of the vertical decomposition of an arrangement of n fixed-degree algebraic surfaces or surface patches in four dimensions is 0(n^{4 + \varepsilon } ), for any \varepsilon 0. This improves the best previously known upper bound for this problem by a near-linear factor, and settles a major problem in the theory of arrangements of surfaces, open since 1989. The new bound can be extended to higher dimensions, yielding the bound 0(n^{2d - 4 + \varepsilon } ) for any \varepsilon 0 on the complexity of vertical decompositions in dimensions d \geqslant 4. We also describe the immediate algorithmic applications of these results, which include improved algorithms for point location, range searching, ray shooting, robot motion planning, and some geometric optimization problems.