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81 lines
3.3 KiB
TeX
81 lines
3.3 KiB
TeX
\newpage
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\subsubsection{Convex hull}
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\begin{definition}[]{Convex hull}
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Let $S \subseteq \R^d, d \in \N$. The \textit{convex hull}, $\text{conv}(S)$ of $S$ is the cut of all convex sets that contains $S$, i.e.
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\begin{align*}
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\text{conv}(S) := \bigcap_{S\subseteq C \subseteq \R^d} C
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\end{align*}
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where $C$ is convex
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\end{definition}
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A convex hull is always convex again.
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\setcounter{all}{37}
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\begin{theorem}[]{JarvisWrap}
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The \verb|JarvisWrap| algorithm can find the convex hull in \tco{nh}, where $h$ is the number of corners of $P$ and $P$ is a set of $n$ points in any position in $\R^2$.
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\end{theorem}
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\begin{algorithm}
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\caption{JarvisWrap}
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\begin{algorithmic}[1]
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\Procedure{FindNext}{$q$}
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\State Choose $p_0 \in P\backslash\{q\}$ arbitrarily
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\State $q_{next} \gets p_0$
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\For{\textbf{all} $p \in P\backslash \{q, p_0\}$}
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\If{$p$ is to right of $q q_{next}$}
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$q_{next} \gets p$ \Comment{$qq_{next}$ is vector from $q$ to $q_{next}$}
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\EndIf
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\EndFor
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\State \Return $q_{next}$
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\EndProcedure
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\Procedure{JarvisWrap}{$P$}
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\State $h \gets 0$
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\State $p_{now} \gets$ point in $P$ with lowest $x$-coordinate
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\While{$p_{now} \neq q_0$}
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\State $q_h \gets p_{now}$
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\State $p_now \gets$ \Call{FindNext}{$q_h$}
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\State $h \gets h + 1$
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\EndWhile
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\State \Return $(q_0, q_1, \ldots, q_{h - 1})$
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\EndProcedure
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\end{algorithmic}
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\end{algorithm}
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\newpage
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\fhlc{Cyan}{LocalRepair}
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\begin{algorithm}
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\caption{LocalRepair}
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\begin{algorithmic}[1]
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\Procedure{LocalRepair}{$p_1, \ldots, p_n$}
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\State $q_0 \gets q_1$ \Comment{input sorted according to $x$-coordinate}
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\State $h \gets 0$
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\For{$i \in \{2, \ldots, n\}$} \Comment{Lower convex hull, left to right}
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\While{$h > 0$ and $q_h$ is to the left of $q_{h - 1}p_i$} \Comment{$q_{h - 1}p_i$ is a vector}
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\State $h \gets h - 1$
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\EndWhile
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\State $h \gets h + 1$
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\State $q_h \gets p_i$ \Comment{$(q_0, \ldots, q_h)$ is lower convex hull of $\{p_1, \ldots, p_i\}$}
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\EndFor
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\State $h' \gets h$
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\For{$i \in \{n - 1, \ldots, 1\}$} \Comment{Upper convex hull, right to left}
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\While{$h > h'$ and $q_h$ is to the left of $q_{h - 1}p_i$} \Comment{$q_{h - 1}p_i$ is a vector}
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\State $h \gets h - 1$
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\EndWhile
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\State $h \gets h + 1$
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\State $q_h \gets p_i$
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\EndFor
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\State \Return $(q_0, \ldots, q_{h - 1})$ \Comment{The corners of the convex hull, counterclockwise}
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\EndProcedure
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\end{algorithmic}
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\end{algorithm}
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\setcounter{all}{42}
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\begin{theorem}[]{LocalRepair}
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The \verb|LocalRepair| algorithm can find the convex hull of a (in respect to the $x$-coordinate of each point) sorted set $P$ of $n$ points in $\R^2$ in \tco{n}.
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\end{theorem}
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The idea of the \verb|LocalRepair| algorithm is to repeatedly correct mistakes in the originally randomly chosen polygon.
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The improvement steps add edges to the polygon such that eventually all vertices lay withing the smallest enclosing circle of the polygon.
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