Optimal core-sets for balls

Mihai Badoiu, Kenneth L. Clarkson

Abstract: Given a set of points P⊂Rd<math><mi is="true">P</mi><mo is="true">⊂</mo><msup is="true"><mi mathvariant="double-struck" is="true">R</mi><mi is="true">d</mi></msup></math> and value ε>0<math><mi is="true">ε</mi><mo is="true">&gt;</mo><mn is="true">0</mn></math>, an ε-core-set S⊂P<math><mi is="true">S</mi><mo is="true">⊂</mo><mi is="true">P</mi></math> has the property that the smallest ball containing S has radius within 1+ε<math><mn is="true">1</mn><mo is="true">+</mo><mi is="true">ε</mi></math> of the radius of the smallest ball containing P. This paper shows that any point set has an ε-core-set of size ⌈1/ε⌉<math><mo stretchy="false" is="true">⌈</mo><mn is="true">1</mn><mo stretchy="false" is="true">/</mo><mi is="true">ε</mi><mo stretchy="false" is="true">⌉</mo></math>, and this bound is tight in the worst case. Some experimental results are also given, comparing this algorithm with a previous one, and with a more powerful, but slower one.