@article{4072,
abstract = {We show that the total number of edges ofm faces of an arrangement ofn lines in the plane isO(m 2/3– n 2/3+2 +n) for any>0. The proof takes an algorithmic approach, that is, we describe an algorithm for the calculation of thesem faces and derive the upper bound from the analysis of the algorithm. The algorithm uses randomization and its expected time complexity isO(m 2/3– n 2/3+2 logn+n logn logm). If instead of lines we have an arrangement ofn line segments, then the maximum number of edges ofm faces isO(m 2/3– n 2/3+2 +n (n) logm) for any>0, where(n) is the functional inverse of Ackermann's function. We give a (randomized) algorithm that produces these faces and takes expected timeO(m 2/3– n 2/3+2 log+n(n) log2 n logm).},
author = {Herbert Edelsbrunner and Guibas, Leonidas J and Sharir, Micha},
journal = {Discrete & Computational Geometry},
number = {1},
pages = {161 -- 196},
publisher = {Springer},
title = {{The complexity and construction of many faces in arrangements of lines and of segments}},
doi = { 10.1007/BF02187784},
volume = {5},
year = {1990},
}
@inproceedings{4073,
abstract = {A number of rendering algorithms in computer graphics sort three-dimensional objects by depth and assume that there is no cycle that makes the sorting impossible. One way to resolve the problem caused by cycles is to cut the objects into smaller pieces. The problem of estimating how many such cuts are always sufficient is addressed. A few related algorithmic and combinatorial geometry problems are considered},
author = {Chazelle, Bernard and Herbert Edelsbrunner and Guibas, Leonidas J and Pollack, Richard and Seidel, Raimund and Sharir, Micha and Snoeyink, Jack},
pages = {242 -- 251},
publisher = {IEEE},
title = {{Counting and cutting cycles of lines and rods in space}},
doi = {10.1109/FSCS.1990.89543},
year = {1990},
}
@article{4074,
author = {Clarkson, Kenneth L and Herbert Edelsbrunner and Guibas, Leonidas J and Sharir, Micha and Welzl, Emo},
journal = {Discrete & Computational Geometry},
number = {1},
pages = {99 -- 160},
publisher = {Springer},
title = {{Combinatorial complexity bounds for arrangements of curves and spheres}},
doi = {10.1007/BF02187783},
volume = {5},
year = {1990},
}
@article{4075,
abstract = {A key problem in computational geometry is the identification of subsets of a point set having particular properties. We study this problem for the properties of convexity and emptiness. We show that finding empty triangles is related to the problem of determining pairs of vertices that see each other in a star-shaped polygon. A linear-time algorithm for this problem which is of independent interest yields an optimal algorithm for finding all empty triangles. This result is then extended to an algorithm for finding empty convex r-gons (r> 3) and for determining a largest empty convex subset. Finally, extensions to higher dimensions are mentioned.},
author = {Dobkin, David P and Herbert Edelsbrunner and Overmars, Mark H},
journal = {Algorithmica},
number = {4},
pages = {561 -- 571},
publisher = {Springer},
title = {{Searching for empty convex polygons}},
doi = {10.1007/BF01840404},
volume = {5},
year = {1990},
}
@inproceedings{4076,
abstract = {We present an algorithm to compute a Euclidean minimum spanning tree of a given set S of n points in Ed in time O(Td(N, N) logd N), where Td(n, m) is the time required to compute a bichromatic closest pair among n red and m blue points in Ed. If Td(N, N) = Ω(N1+ε), for some fixed ε > 0, then the running time improves to O(Td(N, N)). Furthermore, we describe a randomized algorithm to compute a bichromatic closets pair in expected time O((nm log n log m)2/3+m log2 n + n log2 m) in E3, which yields an O(N4/3log4/3 N) expected time algorithm for computing a Euclidean minimum spanning tree of N points in E3.},
author = {Agarwal, Pankaj K and Herbert Edelsbrunner and Schwarzkopf, Otfried and Welzl, Emo},
pages = {203 -- 210},
publisher = {ACM},
title = {{ Euclidean minimum spanning trees and bichromatic closest pairs}},
doi = {10.1145/98524.98567},
year = {1990},
}