Comuting Core-Sets and Approximate Smallest Enclosing HyperSpheres in High Dimensions
نویسندگان
چکیده
We study the minimum enclosing ball (MEB) problem for sets of points or balls in high dimensions. Using techniques of second-order cone programming and “core-sets”, we have developed (1 + )approximation algorithms that perform well in practice, especially for very high dimensions, in addition to having provable guarantees. We prove the existence of core-sets of size O(1/ ) , improving the previous bound of O(1/ 2) , and we study empirically how the core-set size grows with dimension. We show that our algorithm, which is simple to implement, results in fast computation of nearly optimal solutions for point sets in much higher dimension than previously computable using exact techniques.
منابع مشابه
Computing Core-Sets and Approximate Smallest Enclosing HyperSpheres in High Dimensions∗
We study the minimum enclosing ball (MEB) problem for sets of points or balls in high dimensions. Using techniques of second-order cone programming and “coresets”, we have developed (1 + )-approximation algorithms that perform well in practice, especially for very high dimensions, in addition to having provable guarantees. We prove the existence of core-sets of size O(1/ ), improving the previo...
متن کاملOptimal core-sets for balls
Given a set of points P ⊂ R and value > 0, an core-set S ⊂ P has the property that the smallest ball containing S is within of the smallest ball containing P . This paper shows that any point set has an -core-set of size d1/ e, and this bound is tight in the worst case. A faster algorithm given here finds an core-set of size at most 2/ . These results imply the existence of small core-sets for ...
متن کاملFast Smallest-Enclosing-Ball Computation in High Dimensions
We develop a simple combinatorial algorithm for computing the smallest enclosing ball of a set of points in high dimensional Euclidean space. The resulting code is in most cases faster (sometimes significantly) than recent dedicated methods that only deliver approximate results, and it beats off-the-shelf solutions, based e.g. on quadratic programming solvers. The algorithm resembles the simple...
متن کاملSmallest Enclosing Disks (balls and Ellipsoids)
A simple randomized algorithm is developed which computes the smallest enclosing disk of a nite set of points in the plane in expected linear time. The algorithm is based on Seidel's recent Linear Programming algorithm, and it can be generalized to computing smallest enclosing balls or ellipsoids of point sets in higher dimensions in a straightforward way. Experimental results of an implementat...
متن کاملSmallest Enclosing Ball for a Point Set with Strictly Convex Level Sets
Let the center point be the point that minimizes the maximum distance from a point of a given point set to the center point. Finding this center point is referred to as the smallest enclosing ball problem. In case of points with Euclidean distance functions, the smallest enclosing ball is actually the center of a geometrical ball. We consider point sets with points that have distance functions ...
متن کامل