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\end{cases*} + \end{equation*} + \item Richtungsvektor: Zusammenfassung aller Abhängigkeitsrichtungen einer Schleife:\begin{itemize} + \item \enquote{übliche} Zusammenfassungen (z.B. $=$ und $<$ zu $\leq$) + \item keine Aussage $*$ bei $<$ und $>$ + \end{itemize} + \item Die Abhängigkeit tragende Schleife $p$ folgt aus der Struktur der Abhängigkeitsdistanzen bei Inkrementschleifen: + \begin{equation*} + (\underbrace{d_1, d_2, \textellipsis}_{0}, \underbrace{d_p}_{>0}, \underbrace{\textellipsis, d_n}_{\mathrm{beliebig}}) + \end{equation*} + \item Parallele Ausführbarkeit aller Iterationen einer Schleife $p$ gegeben, wenn \begin{itemize} + \item diese keine Abhängigkeit trägt (alle Abhängigkeitsdistanzen haben $d_p = 0$) oder + \item jede schleifengetragene Abhängigkeit von einer $p$ umgebenden Schleife getragen wird (für jede Abhängigkeitsdistanz $\exists q < p: d_q > 0$) + \end{itemize} + \item Parallele Ausführung nicht immer gewinnbringend, da auf Cache-Ebene Abhängigkeiten auftreten können (\enquote{False Sharing}) + \item Optimierungsmöglichkeiten:\begin{itemize} + \item Skalare Ersetzung von Array-Ausdrücken (\enquote{register pipelining}), wenn: \begin{itemize} + \item Abhängigkeit wird von innerster Schleife getragen + \item Konstante Abhängigkeitsdistanz $d$ + \item $d+1$ freie Register + \end{itemize}\begin{enumerate} + \item Vor Schleife Initialisierung der Register und \enquote{Abschälen} der ersten $d$ Iterationen + \item In Schleife Benutzung der Register bei Lesezugriff und Speicherung der Registerinhalte bei Speicherzugriff, sowie am Anfang des Rumpfs \enquote{Aufrücken} der Register + \end{enumerate} + \item Skalar-Vervielfachung: Einführung eines Arrays zur Elimination von unnötigen Abhängigkeiten einer Temporärvariable + \item Verbesserung der zeitlichen Lokalität durch optimierte Registerverwendung + \item Verbesserung der räumlichen Lokalität durch optimierte Cache-Verwendung + \item Parallelisierung/Vektorisierung von Schleifen + \item Daten-/Prozessplatzierung zur Kommunikationsoptimierung + \end{itemize} +\end{itemize} + \subsection{Schleifentransformationen}