Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications
Published
Journal Article
Let 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 (≤k)-level of the arrangement A(F) is O(k3+ε ψ(n/k)) for any ε > 0, where ψ(r) is the maximum complexity of the lower envelope of a subset of at most r functions of 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.
Full Text
Duke Authors
Cited Authors
- Agarwal, PK; Efrat, A; Sharir, M
Published Date
- January 1, 1999
Published In
Volume / Issue
- 29 / 3
Start / End Page
- 912 - 953
International Standard Serial Number (ISSN)
- 0097-5397
Digital Object Identifier (DOI)
- 10.1137/s0097539795295936
Citation Source
- Scopus