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$a$ und $b$: $y$ kontrollflussabhängig von $v$ $\Leftrightarrow$ mindestens ein Pad von $a$ zum Exit-Knoten ohne $y$ und jeder Pfad von $b$ zum Exit-Knoten über $y$ \end{itemize} \subsection{Dominanz} \begin{itemize} \item Knoten $x$ dominiert $y$ ($x \geq \geq y$), wenn jeder Pfad von Wurzel zu $y$ durch $x$ laufen muss \item Strikte Dominanz $x >> y$, falls zusätzlich $x \neq y$ gilt \item $\mathrm{ImmDom[}y\mathrm{]}$ ist strikter Dominator von $y$, der $y$ am Nächsten ist - \item Bestimmung des ImmDomms: TODO \item Dominatorbaum enthält jeden Knoten als Kind seines ImmDomms $\rightarrow$ Pfad zwischen $x$ und $z$ in Dominatorbaum $\Leftrightarrow$ $x >> z$ \end{itemize} +\subsubsection{Berechnung der Dominatoren $D(n)$ eines Knoten $n$} + +\paragraph{Iterativer Fixpunkt-Algorithmus (Lengauer)} +\begin{itemize} + \item mit $\mathcal{O}(\vert E\vert \vert N\vert ^2)$ + \item Zunächst Überapproximation der Dominatorenmenge + \item Initialisierung aller $D(n) \in N$ mit $N$ außer Startknoten $S$ mit $D(S) = S$ + \item Bis Fixpunkt erreicht ist: alle $D(n)$ zu $D'(n) = \lbrace n\rbrace \cup \bigcap_{(p, n) \in E} D(p)$ + \item $n$ am besten in Tiefensuchereihenfolge durchlaufen +\end{itemize} + +\paragraph{Verfahren mit Spannendem Tiefenbaum $T$} +\begin{itemize} + \item Besuch des KFG in Tiefensuchereihenfolge mit zugehöriger Nummerierung: \begin{itemize} + \item \enquote{Spannende} Kanten gehen zu frisch nummerierten Knoten + \item Rückschreitende Kanten gehen zu Vorgänger (kleinere DFS-Nummer) in $T$ + \item Fortschreitende Kanten gehen zu Nachfolger (größere DFS-Nummer) in $T$ + \item Kreuzkanten führen in früher besuchten Ast in $T$ + \end{itemize} + \item Dominatoren $D(n)$ liegen auf jeden Fall \enquote{über} $n$ in $T$ + \item Berechnung der Semidominatoren $\mathrm{SemDom}\lbrack w\rbrack$ in Reihenfolge fallender DFS-Nummern: \begin{itemize} + \item Direkte Vorgänger auf $T$ sind Kandidaten + \item $\min_{u\in \mathrm{Pred}(w)} \mathrm{SemDom}\lbrack u\rbrack$ ist Kandidat + \item Minimum der Kandidaten ist $\mathrm{SemDom}\lbrack w\rbrack$ + \end{itemize} + \item Berechnung von $\mathrm{ImmDom}\lbrack w\rbrack$ durch Durchlaufen in Tiefenordnung von $\mathrm{SemDom}\lbrack w\rbrack$ nach $w$:\begin{itemize} + \item Jeweils alle Vorgänger $u$ untersuchen und $u$ mit kleinstem $\mathrm{SemDom}\lbrack u\rbrack$ finden + \item \begin{equation*} + \mathrm{ImmDom}\lbrack w\rbrack = \begin{cases} + \mathrm{SemDom}\lbrack u\rbrack & \mathrm{falls}\, \mathrm{SemDom}\lbrack w\rbrack = \mathrm{SemDom}\lbrack u \rbrack\\ + \mathrm{ImmDom}\lbrack u\rbrack & \mathrm{sonst} + \end{cases} + \end{equation*} + \end{itemize} +\end{itemize} + +\subsubsection{Dominanzgrenze} +\begin{itemize} + \item Dominanzgrenze $DG[x]$ enthält Knoten $y$, die einen von $x$ dominierten Vorgänger besitzen, aber nicht von $x$ streng dominiert werden + \item Berechnung der $DG[x]$: \begin{align*} + DG[x] &= DG_\text{local}[x] \cup \bigcup_{z \in N, \mathrm{ImmDom}[z] = x} DG_\text{up}[x, z] \\ + DG_\text{local}[x] &= \lbrace y \in \mathrm{Succ}(x) \mid \mathrm{ImmDom}[y] \neq x\rbrace\\ + DG_\text{up}[x, z] &= \lbrace y \in DG[z] \mid \mathrm{ImmDom}[y] \neq x\rbrace + \end{align*} + \item Invertierung der Dominanzgrenzen liefert Kontrollflussabhängigkeiten +\end{itemize} + \subsection{Schleifenerkennung} \begin{itemize} \item Region: \begin{itemize} @@ -86,7 +133,6 @@ \end{itemize} \end{itemize} \end{itemize} -\subsection{Kontrollflussabhängigkeit} \section{Datenflussanalyse} \subsection{Datenabhängigkeiten}