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 $S$ is a set of $n$ points in $R^d$, we say that $S$ is $(k,G)$-clusterable if it can be partitioned into $k$ clusters (subsets) such that each cluster can be con...
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Zusammenfassung: | Helly's theorem is a fundamental result in discrete geometry, describing the
ways in which convex sets intersect with each other. If $S$ is a set of $n$
points in $R^d$, we say that $S$ is $(k,G)$-clusterable if it can be
partitioned into $k$ clusters (subsets) such that each cluster can be contained
in a translated copy of a geometric object $G$. In this paper, as an
application of Helly's theorem, by taking a constant size sample from $S$, we
present a testing algorithm for $(k,G)$-clustering, i.e., to distinguish
between two cases: when $S$ is $(k,G)$-clusterable, and when it is
$\epsilon$-far from being $(k,G)$-clusterable. A set $S$ is $\epsilon$-far
$(01$, 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. |
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DOI: | 10.48550/arxiv.1307.8268 |