TY - JOUR

T1 - Computing the smallest k-enclosing circle and related problems

AU - Efrat, Alon

AU - Sharir, Micha

AU - Ziv, Alon

N1 - Funding Information:
*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. * Corresponding author.

PY - 1994/7

Y1 - 1994/7

N2 - We present an efficient algorithm for solving the "smallest k-enclosing 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 two solutions. When using O(nk) storage, the problem can be solved in time O(nk log2 n). When only O(n log n) storage is allowed, the running time is O(nk log2 n log n/k). We also extend our technique 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 disk intersecting k segments from a given planar set of non-intersecting segments.

AB - We present an efficient algorithm for solving the "smallest k-enclosing 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 two solutions. When using O(nk) storage, the problem can be solved in time O(nk log2 n). When only O(n log n) storage is allowed, the running time is O(nk log2 n log n/k). We also extend our technique 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 disk intersecting k segments from a given planar set of non-intersecting segments.

KW - Geometric optimization

KW - Smallest enclosing circle

UR - http://www.scopus.com/inward/record.url?scp=38149145977&partnerID=8YFLogxK

U2 - 10.1016/0925-7721(94)90003-5

DO - 10.1016/0925-7721(94)90003-5

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AN - SCOPUS:38149145977

SN - 0925-7721

VL - 4

SP - 119

EP - 136

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

IS - 3

ER -