TY - JOUR
T1 - The 2-center problem with obstacles
AU - Halperin, Dan
AU - Sharir, Micha
AU - Goldberg, Ken
N1 - Funding Information:
Given a set S of n points in the plane and a set O of pairwise disjoint simple polygons with a total of m edges, we wish to find two congruent disks of smallest radius whose union covers S and whose centers lie outside the polygons in O (referred to as locational constraints in facility location theory). We present an algorithm to solve this problem in randomized expected time O m log2 mn + mn log2 n log mn . We also present an efficient approximation scheme that constructs, for a given ε > 0, two disks as above of radius at most 1 + ε r∗, where r∗ is the optimal radius, in time O 1/ε log 1/ε m log2 m + n log2 n or in randomized expected time O 1/ε log 1/ε m + n log n log mn . 2002 Elsevier Science 1Work on this paper by Ken Goldberg and Dan Halperin has been supported by a grant from U.S.–Israeli Binational Science Foundation. Work by Dan Halperin and Micha Sharir has been supported by The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities (Center for Geometric Computing and its Applications), by a Franco–Israeli research grant “factory of the future” (monitored by AFIRST/France and The Israeli Ministry of Science), and by the Hermann Minkowski–Minerva Center for Geometry at Tel Aviv University. A preliminay version of this paper appeared in Proceedings of the 16th ACM Symposium on Computational Geometry, Hong Kong, 2000, pp. 80–90.
PY - 2002/1
Y1 - 2002/1
N2 - Given a set S of n points in the plane and a set O of pairwise disjoint simple polygons with a total of m edges, we wish to find two congruent disks of smallest radius whose union covers S and whose centers lie outside the polygons in O (referred to as locational constraints in facility location theory). We present an algorithm to solve this problem in randomized expected time O(m log2(mn) + mn log2 n log(mn)). We also present an efficient approximation scheme that constructs, for a given ε > 0, two disks as above of radius at most (1 + ε)r*, where r* is the optimal radius, in time O(1/ε log(1/ε)(m log2 m + n log2 n)) or in randomized expected time O(1/ε log(1/ε)(m + n log n) log(mn)).
AB - Given a set S of n points in the plane and a set O of pairwise disjoint simple polygons with a total of m edges, we wish to find two congruent disks of smallest radius whose union covers S and whose centers lie outside the polygons in O (referred to as locational constraints in facility location theory). We present an algorithm to solve this problem in randomized expected time O(m log2(mn) + mn log2 n log(mn)). We also present an efficient approximation scheme that constructs, for a given ε > 0, two disks as above of radius at most (1 + ε)r*, where r* is the optimal radius, in time O(1/ε log(1/ε)(m log2 m + n log2 n)) or in randomized expected time O(1/ε log(1/ε)(m + n log n) log(mn)).
UR - https://www.scopus.com/pages/publications/0036462083
U2 - 10.1006/jagm.2001.1194
DO - 10.1006/jagm.2001.1194
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AN - SCOPUS:0036462083
SN - 0196-6774
VL - 42
SP - 109
EP - 134
JO - Journal of Algorithms
JF - Journal of Algorithms
IS - 1
ER -