A randomized algorithm for closest-point queries
SIAM Journal on Computing
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
An algorithm for generalized point location and its applications
Journal of Symbolic Computation
Applications of parametric searching in geometric optimization
SODA selected papers from the third annual ACM-SIAM symposium on Discrete algorithms
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Ray Shooting Amidst Spheres in Three Dimensions and Related Problems
SIAM Journal on Computing
Computing Envelopes in Four Dimensions with Applications
SIAM Journal on Computing
Handbook of discrete and computational geometry
Handbook of discrete and computational geometry
Vertical Decomposition of Shallow Levels in 3-Dimensional Arrangements and Its Applications
SIAM Journal on Computing
Ray Shooting, Depth Orders and Hidden Surface Removal
Ray Shooting, Depth Orders and Hidden Surface Removal
A Singly-Expenential Stratification Scheme for Real Semi-Algebraic Varieties and Its Applications
ICALP '89 Proceedings of the 16th International Colloquium on Automata, Languages and Programming
Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition
Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages
Sharp Bounds for Vertical Decompositions of Linear Arrangements in Four Dimensions
Discrete & Computational Geometry
Pianos are not flat: rigid motion planning in three dimensions
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
More algorithms for all-pairs shortest paths in weighted graphs
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Smoothed analysis of probabilistic roadmaps
Computational Geometry: Theory and Applications
Arrangements in geometry: recent advances and challenges
ESA'07 Proceedings of the 15th annual European conference on Algorithms
Optimal triangulation with Steiner points
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Geometric optimization and sums of algebraic functions
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
More Algorithms for All-Pairs Shortest Paths in Weighted Graphs
SIAM Journal on Computing
Semialgebraic Range Reporting and Emptiness Searching with Applications
SIAM Journal on Computing
Cuttings for disks and axis-aligned rectangles
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
Geometric optimization and sums of algebraic functions
ACM Transactions on Algorithms (TALG)
| Hi-index | 0.00 |
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 O(n4+ϵ), for any ϵ 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 O(n2d−4+ϵ), for any ϵ 0, on the complexity of vertical decompositions in dimensions d ≥ 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.