On the overlay of envelopes in four dimensions
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Proceedings of the nineteenth annual symposium on Computational geometry
Lenses in arrangements of pseudo-circles and their applications
Journal of the ACM (JACM)
Almost tight upper bounds for vertical decompositions in four dimensions
Journal of the ACM (JACM)
Algorithms for the placement of diagrams on maps
Proceedings of the 12th annual ACM international workshop on Geographic information systems
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
On approximating the depth and related problems
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Minimum-cost load-balancing partitions
Proceedings of the twenty-second annual symposium on Computational geometry
Boundary labeling: Models and efficient algorithms for rectangular maps
Computational Geometry: Theory and Applications
Approximation algorithms for spreading points
Journal of Algorithms
On bounded leg shortest paths problems
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Near-linear approximation algorithms for geometric hitting sets
Proceedings of the twenty-fifth annual symposium on Computational geometry
A general approach for cache-oblivious range reporting and approximate range counting
Proceedings of the twenty-fifth annual symposium on Computational geometry
Proceedings of the twenty-sixth annual symposium on Computational geometry
The 2-center problem in three dimensions
Proceedings of the twenty-sixth annual symposium on Computational geometry
A general approach for cache-oblivious range reporting and approximate range counting
Computational Geometry: Theory and Applications
Simple wriggling is hard unless you are a fat hippo
FUN'10 Proceedings of the 5th international conference on Fun with algorithms
Optimal cover of points by disks in a simple polygon
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Algorithms for the transportation problem in geometric settings
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
A near-linear time ε-approximation algorithm for geometric bipartite matching
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Bottleneck non-crossing matching in the plane
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
The 2-center problem in three dimensions
Computational Geometry: Theory and Applications
Nearest neighbor searching under uncertainty II
Proceedings of the 32nd symposium on Principles of database systems
A sub-quadratic algorithm for bipartite matching of planar points with bounded integer coordinates
Proceedings of the twenty-ninth annual symposium on Computational geometry
Two-Sided boundary labeling with adjacent sides
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
Bottleneck non-crossing matching in the plane
Computational Geometry: Theory and Applications
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Let ${\cal F}$ be a collection of n bivariate algebraic functions of constant maximum degree. We show that the combinatorial complexity of the vertical decomposition of the $({\le}k)$-level of the arrangement $\A({\cal F})$ is $O(k^{3+\varepsilon}\psi({n/k}))$ for any $\varepsilon0$, where $\psi (r)$ is the maximum complexity of the lower envelope of a subset of at most $r$ functions of ${\cal F}$. This bound is nearly optimal in the worst case and implies the existence of shallow cuttings, in the sense of [J. Matousek, Comput. Geom., 2 (1992), pp. 169--186], of small size in arrangements of bivariate algebraic functions. We also present numerous applications of these results, including (i) data structures for several generalized 3-dimensional range-searching problems; (ii) dynamic data structures for planar nearest- and farthest-neighbor searching under various fairly general distance functions; (iii) an improved (near-quadratic) algorithm for minimum-weight bipartite Euclidean matching in the plane; and (iv) efficient algorithms for certain geometric optimization problems in static and dynamic settings.