diff --git a/main.tex b/main.tex index 456147f..4471013 100644 --- a/main.tex +++ b/main.tex @@ -37,7 +37,12 @@ The last line is greater than $(\sqrt{d}/5)^d$ for large enough $d$. \end{proof} \begin{exercise} -Compute clustering radius +Compute clustering radius. +Let $C$ and $P$ be two given set of points such that $k=|C|$ and $n=|P|$. Define the covering radius of $P$ by $C$ as $r=\max_{p\in P} \min_{c\in C} \norm{p-c}$. +\begin{enumerate} +\item find an $O(n+k\log n)$ expected time alg that outputs $\alpha$ such that $\alpha \leq r \leq 10\alpha$. +\item for prescribed $\varepsilon>0$, find an $O(n+k\varepsilon^{-2}\log n)$ expected time alg that outputs $\alpha$ s.t. $\alpha