Linear time algorithms for Euclidean 1-center in ℜd with non-linear convex constraints

Article Type

Research Article

Publication Title

Discrete Applied Mathematics


In this paper, we first present a linear-time algorithm to find the smallest circle enclosing n given points in ℜ2 with the constraint that the center of the smallest enclosing circle lies inside a given disk. We extend this result to ℜ3 by computing constrained smallest enclosing sphere centered on a given sphere. We generalize the result for the case of points in ℜd where the center of the minimum enclosing ball lies inside a given ball. We show that similar problem of computing minimum intersecting/stabbing ball for a set of hyper planes in ℜd can also be solved using similar techniques. We also show how the minimum intersecting disk with the center constrained on a given disk can be computed to intersect a set of convex polygons. Lastly, we show that this technique is applicable when the center of minimum enclosing/intersecting ball lies in a convex region bounded by constant number of non-linear constraints with computability assumptions. We solve each of these problems in linear time in input size for fixed dimension.



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