TY - GEN
T1 - Computing the smallest k-enclosing circle and related problems
AU - Efrat, Alon
AU - Sharir, Micha
AU - Ziv, Alon
N1 - Funding Information:
Much attention has recently been given to problems of the form: "Given a set S of n objects and a parameter k < n, find a k-subset (namely, a subset of cardinality k) of the objects that optimizes some cost function, among all possible k-subsets." This problem was studied for a variety of cost functions. Aggarwal et. al. \[4\] solve this problem when the parameter to be optimized is the diameter of the k-subset (in time O(k2"~n log k + n log n)), the variance of the k-subset (in time O(k2n + n log n)), the size of an axis-parallel enclosing square, or the perimeter of an axis-parallel enclosing rectangle (both in time O(nk 2 log n); the solution to the first problem has recently been improved to O(n log 2 n); see \[6\]).I n \[9\], t email: alone@math, tau. ac. il * Work on this paper by the second author has been supported by NSF Grant CCR-91-22103, and by grants from the U.S.-Israeli Binational Science Foundation, the G.I.F., the German-Israeli Foundation for Scientific Research and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences.
Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 1993.
PY - 1993
Y1 - 1993
N2 - We present an efficient algorithm for solving the “smallest kenclosing circle” (kSC) problem: Given a set of n points in the plane and an integer k ≤ n, find the smallest disk containing k of the points. We present several algorithms that run in O(nk logc n) time, where the constant c depends on the storage that the algorithm is allowed. When using O(nk) storage, the problem can be solved in time O(nklog2 n). When only O(n log n) storage is allowed, the running time is (formula presentd). The method we describe can be easily extended to obtain efficient solutions of several related problems (with similar time and storage bounds). These related problems include: finding the smallest homothetic copy of a given convex polygon P, which contains k points from a given planar set, and finding the smallest hypodrome of a given length and orientation (formally defined in Section 4) containing k points from a given planar set.
AB - We present an efficient algorithm for solving the “smallest kenclosing circle” (kSC) problem: Given a set of n points in the plane and an integer k ≤ n, find the smallest disk containing k of the points. We present several algorithms that run in O(nk logc n) time, where the constant c depends on the storage that the algorithm is allowed. When using O(nk) storage, the problem can be solved in time O(nklog2 n). When only O(n log n) storage is allowed, the running time is (formula presentd). The method we describe can be easily extended to obtain efficient solutions of several related problems (with similar time and storage bounds). These related problems include: finding the smallest homothetic copy of a given convex polygon P, which contains k points from a given planar set, and finding the smallest hypodrome of a given length and orientation (formally defined in Section 4) containing k points from a given planar set.
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U2 - 10.1007/3-540-57155-8_259
DO - 10.1007/3-540-57155-8_259
M3 - Conference contribution
AN - SCOPUS:85029531691
SN - 9783540571551
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 325
EP - 336
BT - Algorithms and Data Structures - 3rd Workshop, WADS 1993, Proceedings
A2 - Dehne, Frank
A2 - Sack, Jorg-Rudiger
A2 - Santoro, Nicola
A2 - Whitesides, Sue
PB - Springer-Verlag
T2 - 3rd Workshop on Algorithms and Data Structures, WADS 1993
Y2 - 11 August 1993 through 13 August 1993
ER -