ex1.2 draft...

main.tex

@@ -1,4 +1,4 @@
-\documentclass[12pt]{article}
+\documentclass[11pt]{article}
 % \usepackage{chao}
 \usepackage[sans]{xenotes}
 % \usepackage{natbib}
@@ -16,7 +16,7 @@
 
 For errata and more stuff, see \url{https://sarielhp.org/book/}
 
-Note that unless specifically stated, we always consider the RAM model.
+% Note that unless specifically stated, we always consider the RAM model.
 
 \section{Grid}
 \begin{exercise}\label{ex1.1}
@@ -44,10 +44,21 @@ The last line is greater than $(\sqrt{d}/5)^d$ for large enough $d$.
 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 find an $O(n+k\log n)$ expected time alg that outputs $\alpha$ such that $r \leq \alpha \leq 10r$.
 \item for prescribed $\varepsilon>0$, find an $O(n+k\varepsilon^{-2}\log n)$ expected time alg that outputs $\alpha$ s.t. $\alpha<r<(1+\epsilon)\alpha$.
 \end{enumerate}
 \end{exercise}
+% a Las Vegas approximation...
+We repeatedly build grid for $C$ with different side length and insert points in $P$ into the grid.
+$\log n$ rebuilds, each takes $O(k)$ time. each insertion takes $O(1)$ for points in $P$... 
+
+but how can i get the approximation...
+
+\begin{exercise}
+Given a set $P$ of $n$ points in the plane and
+parameter $k$, present a (simple) randomized algorithm that computes, in expected $O(n(n/k))$
+time, a circle $D$ that contains $k$ points of $P$ and $\mathrm{radius}(D) ≤2r_{\mathrm{opt}}(P,k)$.
+\end{exercise}
 
 
 \section*{Not in the book}