Linear Time Algorithms for Euclidean 1-Center in $\mathfrak {R}^d$ with Non-linear Convex Constraints

Abstract

In this paper, we first present a linear-time algorithm to find the smallest circle enclosing n given points in $\mathfrak {R}^2$ with the constraint that the center of the smallest enclosing circle lies inside a given disk. We extend this result to $\mathfrak {R}^3$ by computing constrained smallest enclosing sphere centered on a given sphere. We generalize the result for the case of points in $\mathfrak {R}^d$ where center of the minimum enclosing ball lies inside a given ball. We show that similar problem of minimum intersecting/stabbing ball for set of hyper planes in $\mathfrak {R}^d$ can also be solved using similar techniques. We also show how minimum intersecting disk with 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 complexity for fixed dimension.