z

main.tex

@@ -1,7 +1,8 @@
 \documentclass[11pt]{article}
-% \usepackage{chao}
-\usepackage[sans]{xenotes}
-% \usepackage{natbib}
+\usepackage{chao}
+\newaliascnt{exercise}{theorem}
+\newtheorem{exercise}{Ex}[section]
+
 
 \title{Exercises in Sariel Har-Peled's \\ \textit{Geometric Approximation Algorithms}}
 \author{}
@@ -23,12 +24,12 @@ For errata and more stuff, see \url{https://sarielhp.org/book/}
 Let $P$ be a max cardinality point set contained in the $d$-dimensional unit hypercube such that the smallest distance of point pairs in $P$ is 1. Prove that 
 \[\left( \floor{\sqrt{d}}+1 \right)^d \leq |P|\leq \left( \ceil{\sqrt{d}}+1 \right)^d. \]
 \end{exercise}
-hmm... the first exercise in this book is wrong. See \url{https://sarielhp.org/book/errata.pdf}.
+hmm... the first exercise in this book is wrong.
 The stated lowerbound is actually an upperbound.
 \begin{proof}
 We evenly partition the $[0,1]$ interval into $m=\left( \floor{\sqrt{d}}+1 \right)$ small segments for each of the $d$ axes. The unit hypercube is partitioned into $m^d$ cells. The length of each cell's diagonal is $\sqrt{\frac{d}{m^2} }< 1$. Thus there is at most one point of $P$ in each cell and there are $\left( \floor{\sqrt{d}}+1 \right)^d$ cells.
 
-For lowerbound, one can construct a solution of size $2^d$ by selecting vertices of the hypercube. For sufficient large $d$ one can find a solution of size $(\sqrt{d}/5)^d$.\footnote{Exercise 1.1 (C) in \url{https://sarielhp.org/book/chapters/min_disk.pdf}}
+For lowerbound, one can construct a solution of size $2^d$ by selecting vertices of the hypercube. For sufficient large $d$ one can find a solution of size $(\sqrt{d}/5)^d$.\footnote{See exercise 1.1 (C) in \url{https://sarielhp.org/book/chapters/min_disk.pdf}}
 Let point set $P$ be the optimal solution and let $n=|P|$. We place a $d$-dimensional unit sphere around each point of $P$. These $n$ spheres must cover the unit hypercube since otherwise we can add more points into $P$. Thus one has $n\vol(1b^d)\geq 1$.
 \begin{equation*}
 \begin{aligned}
@@ -57,7 +58,7 @@ 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 $\mathrm{radius}(D) ≤2r_{\mathrm{opt}}(P,k)$.
+time, a circle $D$ that contains $k$ points of $P$ and $\mathrm{radius}(D)\leq 2r_{\mathrm{opt}}(P,k)$.
 \end{exercise}
 
 \section*{Not in the book}