1 research outputs found
Helly-Type Theorems in Property Testing
Helly's theorem is a fundamental result in discrete geometry, describing the
ways in which convex sets intersect with each other. If is a set of
points in , we say that is -clusterable if it can be
partitioned into clusters (subsets) such that each cluster can be contained
in a translated copy of a geometric object . In this paper, as an
application of Helly's theorem, by taking a constant size sample from , we
present a testing algorithm for -clustering, i.e., to distinguish
between two cases: when is -clusterable, and when it is
-far from being -clusterable. A set is -far
from being -clusterable if at least
points need to be removed from to make it -clusterable. We solve
this problem for and when is a symmetric convex object. For , we
solve a weaker version of this problem. Finally, as an application of our
testing result, in clustering with outliers, we show that one can find the
approximate clusters by querying a constant size sample, with high probability