45 lines
2.0 KiB
TeX
45 lines
2.0 KiB
TeX
\documentclass[12pt]{article}
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% \usepackage{chao}
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\usepackage[sans]{xenotes}
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% \usepackage{natbib}
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\title{Exercises in Sariel Har-Peled's \\ \textit{Geometric Approximation Algorithms}}
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\author{}
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\date{}
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\begin{document}
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\maketitle
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% \tableofcontents
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% \newpage
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For errata and more stuff, see \url{https://sarielhp.org/book/}
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\section{Grid}
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\begin{oneshot}{1.1}
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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
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\[\left( \floor{\sqrt{d}}+1 \right)^d \leq |P|\leq \left( \ceil{\sqrt{d}}+1 \right)^d. \]
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\end{oneshot}
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hmm... the first exercise in this book is wrong. See \url{https://sarielhp.org/book/errata.pdf}.
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The stated lowerbound is actually an upperbound.
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\begin{proof}
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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.
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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}}
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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 $n\vol(1b^d)\geq 1$.
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\begin{equation*}
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\begin{aligned}
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n &\geq 1/\vol(1b^d)\\
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&= \frac{\Gamma(d/2+1)}{\pi^{d/2}}\\
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&\geq \sqrt{2\pi/(d/2+1)} (\frac{\sqrt{d}}{\sqrt{2e\pi}})^{d}
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\end{aligned}
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\end{equation*}
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The last line is greater than $(\sqrt{d}/5)^d$ for large enough $d$.
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\end{proof}
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\begin{oneshot}{1.2}
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Compute clustering radius
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\end{oneshot}
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\end{document}
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