Abstract
The planar 3-center problem for a set S of points given in the plane asks for three congruent circular disks with the minimum radius, whose union can cover all points of S completely. In this paper, we present an O(n2 log3n) time algorithm for a restricted planar 3-center problem in which the given points are in the convex positions , i.e. The given points are the vertices of a convex polygon exactly.
Highlights
Let S denote a set of n points given in the plane
We know that the problem can be solved in O(n) time[3] when k = 1, i.e. the 1-center problem, and some efficient algorithms are known for k = 2
Eppstein and Chan made some further refinement of Sharir’s work, and Eppstein presented a randomized algorithm with O(nlog2n) expected time[5], Chan gave out an O(nlog2n) time randomized algorithm with high probability, and an O(n(lognloglogn)2) time deterministic algorithm[6]
Summary
The planar k-center problem asks for k congruent closed disks of the minimum radius, whose union covers S completely. Tan et al improved the work of Kim, and presented an O(nlog2n) time algorithm which uses the binary search and the known algorithms only, instead of using the complicated parametric search which is the base for most planar 2center algorithms, for computing the smallest enclosing disk of a point set to solve the planar 2-center problem[9]. We gave an O(n2log3n) time algorithm for solving a restricted planar 3-center problem in which the given points are in convex position, i.e. these points are the vertices of a minimum convex polygon which surrounds all of the given points
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