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

N1 - Funding Information:
* Work on this paper by the first author has been supported by National Science Foundation Grant CCR-93-01259, an NYI award, and by matching funds from Xerox Corp. Work on this paper by the third author has been supported by NSF Grants CCR-91-22103 and CCR-93-11127, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Academy of Sciences, and the G. I. F., the German-Israeli Foundation for Scientific Research and Development. t Department of Computer Science, Box 1029, Duke University, Durham, NC 27708-0129, USA 1 School of Mathematical Sciences, Tel Aviv 69978, Israel $ School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel, and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
Publisher Copyright:
© 1995 ACM.

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

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 39

EP - 50

BT - Proceedings of the 11th Annual Symposium on Computational Geometry, SCG 1995

PB - Association for Computing Machinery

T2 - 11th Annual Symposium on Computational Geometry, SCG 1995

Y2 - 5 June 1995 through 7 June 1995

ER -