Construction of ɛ-nets
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LetS be a set ofn points in the plane and letɛ be a real number, 0<ɛ<1. We give a deterministic algorithm, which in timeO(nɛ−2 log(1/ɛ)+ɛ−8) (resp.O(nɛ−2 log(1/ɛ)+ɛ−10) constructs anɛ-netN⊂S of sizeO((1/ɛ) (log(1/ɛ))2) for intersections ofS with double wedges (resp. triangles); this means that any double wedge (resp. triangle) containing more thatɛn points ofS contains a point ofN. This givesO(n logn) deterministic preprocessing for the simplex range-counting algorithm of Haussler and Welzl [HW] (in the plane).
We also prove that given a setL ofn lines in the plane, we can cut the plane intoO(ɛ−2) triangles in such a way that no triangle is intersected by more thanɛn lines ofL. We give a deterministic algorithm for this with running timeO(nɛ−2 log(1/ɛ)). This has numerous applications in various computational geometry problems.
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- [A1]P. K. Agarwal: A deterministic algorithm for partitioning arrangements of lines and its applications,Proc. 5th Ann. ACM Symposium on Computational Geometry (1989), pp. 11–21 (full version to apear inDiscrete Comput. Geom.).Google Scholar
- [A2]P. K. Agarwal: Ray shooting and other applications of spanning trees with low stabbing number,Proc. 5th Ann. ACM Symposium on Computational Geometry (1989), pp. 315–325 (full version to appear inDiscrete Comput. Geom.).Google Scholar
- [AKS]M. Ajtai, J. Komlós, E. Szemerédi: AnO(n logn) sorting network,Proc. 15th ACM Symp. on Theory on Computing (1983), pp. 1–9.Google Scholar
- [CF]B. Chazelle, J. Friedman: A deterministic view of random sampling and its use in geometry, Report CS-TR-181-88 (extended abstract inProc. 29th Ann. IEEE Symposium on Foundations of Computer Science (1988), pp. 539–549).Google Scholar
- [CEG*]K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir, E. Welzl: Combinatorial complexity for arrangements of curves and surfaces,Proc. 29th Ann. IEEE Symposium on Foundations of Computer Science (1988), pp. 568–579.Google Scholar
- [Co]R. Cole: Slowing down sorting networks to obtain faster sorting algorithms,J. Assoc. Comput. Mach. 31 (1984), 200–208.Google Scholar
- [M1]J. Matoušek:Approximate Halfplanar Range Counting, KAM Series 59-87, Charles University, Prague, 1987.Google Scholar
- [M2]J. Matoušek: Cutting hyperplane arrangements, 6th ACM Symposium on Computational Geometry, 1990.Google Scholar
- [M3]J. Matoušek: Spanning trees with low crossing number, to appear inInform. Theoret. Applic. Google Scholar
- [PSS]J. Pach, W. Steiger, E. Szemerédi: An upper bound for the number of planark-sets,Proc. 30th Ann. IEEE Symposium on Foundations of Computer Science (1989), pp. 72–81.Google Scholar
- [W1]E. Welzl: Partition trees for triangle counting and other range searching problems,Proc. 4th ACM Symposium on Computational Geometry (1988), pp. 23–33.Google Scholar