Abstract
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+∈ψ(n/k)), for any ∈ > 0, where ψ(r) is the maximum complexity of the lower envelope of a subset of at most r functions of F. This result implies the existence of shallow cuttings, in the sense of [3, 31], of small size in arrangements of bivariate algebraic functions. We also present numerous applications of these results, including: (i) data structures for several generalized three-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.
| Original language | English (US) |
|---|---|
| Title of host publication | Proceedings of the 11th Annual Symposium on Computational Geometry, SCG 1995 |
| Publisher | Association for Computing Machinery |
| Pages | 39-50 |
| Number of pages | 12 |
| Volume | Part F129372 |
| ISBN (Electronic) | 0897917243 |
| DOIs | |
| State | Published - Sep 1 1995 |
| Externally published | Yes |
| Event | 11th Annual Symposium on Computational Geometry, SCG 1995 - Vancouver, Canada Duration: Jun 5 1995 → Jun 7 1995 |
Other
| Other | 11th Annual Symposium on Computational Geometry, SCG 1995 |
|---|---|
| Country | Canada |
| City | Vancouver |
| Period | 6/5/95 → 6/7/95 |
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ASJC Scopus subject areas
- Theoretical Computer Science
- Geometry and Topology
- Computational Mathematics
Cite this
Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. / Agarwal, Pankaj K.; Efrat, Alon; Sharir, Micha.
Proceedings of the 11th Annual Symposium on Computational Geometry, SCG 1995. Vol. Part F129372 Association for Computing Machinery, 1995. p. 39-50.Research output: Chapter in Book/Report/Conference proceeding › Conference contribution
}
TY - GEN
T1 - Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications
AU - Agarwal, Pankaj K.
AU - Efrat, Alon
AU - Sharir, Micha
PY - 1995/9/1
Y1 - 1995/9/1
N2 - 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+∈ψ(n/k)), for any ∈ > 0, where ψ(r) is the maximum complexity of the lower envelope of a subset of at most r functions of F. This result implies the existence of shallow cuttings, in the sense of [3, 31], of small size in arrangements of bivariate algebraic functions. We also present numerous applications of these results, including: (i) data structures for several generalized three-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.
AB - 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+∈ψ(n/k)), for any ∈ > 0, where ψ(r) is the maximum complexity of the lower envelope of a subset of at most r functions of F. This result implies the existence of shallow cuttings, in the sense of [3, 31], of small size in arrangements of bivariate algebraic functions. We also present numerous applications of these results, including: (i) data structures for several generalized three-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.
UR - http://www.scopus.com/inward/record.url?scp=84958016763&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84958016763&partnerID=8YFLogxK
U2 - 10.1145/220279.220284
DO - 10.1145/220279.220284
M3 - Conference contribution
AN - SCOPUS:84958016763
VL - Part F129372
SP - 39
EP - 50
BT - Proceedings of the 11th Annual Symposium on Computational Geometry, SCG 1995
PB - Association for Computing Machinery
ER -