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 -