Given a planar point set X, we study the convex shells and the convex layers . We prove that when X consists of points independently and uniformly sampled inside a convex polygon with k vertices , the expected number of vertices on the first t convex shells is O(kt\log{n\left(\right) and the expected number of vertices on the first t convex layers is O\left(kt^{3} \log{\frac{n \right). We also show a lower bound of \Omega(t\log n) for both quantities in the special cases where k=3,4. The implications of those results in the average-case analysis of two computational geometry algorithms are then discussed.
We study the Tukey layers and convex layers of a planar point set, which consists of n points independently and uniformly sampled from a convex polygon with k vertices . We show that the expected number of vertices on the first t Tukey layers is O\left(kt\log(n/k)\right) and the expected number of vertices on the first t convex layers is O\left(kt^{3} \log(n/(kt^2)) \right). We also show a lower bound of \Omega(t\log n) for both quantities in the special cases where k=3,4. The implications of those results in the average-case analysis of two computational geometry algorithms are then discussed.
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Given a planar point set X, we study the convex shells and the convex layers . We prove that when X consists of
We study the Tukey layers and convex layers of a planar point set, which consists of n
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inside
from
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,
. We show that
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convex shells is O(kt\log{n
Tukey layers is O
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kt\log(n/k)
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\log{\frac{n
\log(n/(kt^2))