From 000ef23a5a8749891dfa34e3b125d6e03eb5e325 Mon Sep 17 00:00:00 2001 From: Eva Dengler Date: Wed, 29 Mar 2017 13:28:17 +0200 Subject: [PATCH] added zusammenfassungen erstes Semester --- horschtey_zusammenfassungen/.gitignore | 5 + horschtey_zusammenfassungen/aud/baum.png | Bin 0 -> 29534 bytes horschtey_zusammenfassungen/aud/baumref.png | Bin 0 -> 46287 bytes horschtey_zusammenfassungen/aud/document.tex | 930 +++++++++++++ horschtey_zusammenfassungen/aud/dowhile.png | Bin 0 -> 9640 bytes .../aud/exceptionhierachy.png | Bin 0 -> 41175 bytes horschtey_zusammenfassungen/aud/ifelse.png | Bin 0 -> 19706 bytes .../aud/instanzklasse.PNG | Bin 0 -> 124434 bytes .../aud/rotationavl.PNG | Bin 0 -> 272932 bytes horschtey_zusammenfassungen/aud/sortier.PNG | Bin 0 -> 67624 bytes horschtey_zusammenfassungen/aud/uml.png | Bin 0 -> 42919 bytes horschtey_zusammenfassungen/aud/while.png | Bin 0 -> 9548 bytes .../gti/1erkomplement.png | Bin 0 -> 7644 bytes .../gti/2erkomplement.png | Bin 0 -> 7541 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a=\{viele viele bunte Werte\} oder int[\ ] a=new int[10] + ACHTUNG: Indizierung beginnt bei 0. + + \subsection{Operatoren und Ausdrücke} + \begin{tabular}{@{}c|c|c c c|c|c} + $+$&$x+y$& Addition &\ \ \ \ \ \ \ \ \ &$>$&$x>y$& x größer y\\ + $-$&$x-y$& Subtraktion &&$>=$&$x\geq y$& x größer gleich y\\ + $*$&$x+y$& Multiplikation &\ &$<$&$x>$ right shift sign. Beachtet Vorzeichen, von links altes Vorzeichenbit einschieben.\\ + $>>>$ right shift no-sign. ignoriert Vorzeichen, von links neue Nullen einschieben.\\ + \\ + Auswertungsreihenfolge, von unten nach oben:\\ + \begin{tabular}{@{}ll} + Postfix-Operatoren & [] . (params) expr++ expr--\\ + unäre Operatoren & ++expr --expr +expr -expr ! ~\\ + Erzeugung oder Typumwandlung & new (type) expr\\ + Multiplikationsoperatoren &* / \%\\ + Additionsoperatoren & + -\\ + Verschiebeoperatoren & $<<$ $>>$ $>>>$\\ + Vergleichsoperatoren & $<$ $>$ $<=$ $>=$ instanceof\\ + Gleichheitsoperatoren & == !=\\ + Bitoperator Und & \&\\ + Bitoperator exklusives Oder & \textasciicircum\\ + Bitoperator inklusives Oder & $|$ \\ + logisches Und & \&\&\\ + logisches Oder &$||$\\ + Fragezeichenoperator& ?\\ + Zuweisungsoperatoren& = += -= *= /= \%= $>>$= $<<$= $>>>$= \&= \textasciicircum= $|$=\\ + \end{tabular} + \subsection{Typkonvertierung} + implizite Typkonvertierung: Automatische Konvertierung durch Compiler zu größeren Datentypen\\ + explizite Typkonvertierung: Konvertierung zu kleinerem Datentypen, muss von Programmierer explizit gefordert werden.\\ + Typkonvertierungsoperator: (Zieldatentyp) Wert\\ + \newpage + \section{Ablaufstrukturen und Methoden} + \begin{tabular}{@{}lll} + if-else:&& switch:\\ + if ($<$Bedingung$>$) \{&& switch ($<$Ausdruck$>$) \{\\ + $<$Anweisungen$>$&& case $<$konstanter Ausdruck$>$: $<$Anweisung$>$: break;\\ + \}else \{&& ...\\ + $<$Anweisungen$>$&& default: $<$Anweisung$>$;\\ + \}&\ \ \ \ \ \ \ \ \ \ \ \ &\}\\ + &&\\ + &&\\ + while ($<$Bedingung$>$) \{ && do \{\\ + $<$Anweisungen$>$&&$<$Anweisungen$>$\\ + \}&& \} while ($<$Bedingung$>$); + \end{tabular}\\ + \\ + while-Schleifen, wenn die Anzahl an Iterationen unbekannt ist. \\ + \\ + for:\\ + for ($<$Initialausdruck$>$; $<$Bedingung$>$; $<$Inkrementausdruck$>$) \{\\ + $<$Anweisungen$>$\\ + \} \\ + for-Schleifen, wenn die Anzahl an Iterationen bekannt ist.\\ + \\ + bedingter Ausdruck:\\ + allgemein: $<$Bedingung$>$ ? $<$Ausdruck1$>$ : $<$Ausdruck2$>$\\ + \begin{tabular}{@{}lll} + statt&\ \ \ \ \ \ \ & könnte man auch schreiben:\\ + int max;&& int max = $(i > j)$ ? i : j;\\ + if $(i>j)$ {max= i;} else {max=j;}&&\\ + \end{tabular} + \subsection{Ablaufdiagramme} + \includegraphics[width=0.58\linewidth]{ifelse} + \includegraphics[width=0.21\linewidth]{dowhile} + \includegraphics[width=0.21\linewidth]{while} + \subsection{Methoden} + Deklaration:\\ + $<$Sichtbarkeitsmodifikator$>$ $<$Rückgabetyp$>$ $<$Methodenname$>$ (Param 1, ..., Param n)\{\\ + Anweisungen;\\ + \}\\\\ + Methodenaufruf: $<$Objektname$>$.$<$Methodenname$>$($<$Eingabewerte$>$) + \newpage + \section{Rekursion} + \begin{defi}[Rekursion] + Eine Funktion heißt rekursiv, wenn zur Berechnung von f diese Funktion f wieder benutzt wird.\\ + Eine Methodendeklaration f(x) \{ ...\} heißt rekursive Methodendefinition, gdw. f im Block \{...\} aufgerufen wird. + Funktion ruft sich selbst auf $\Rightarrow$ Rekursionsschritt\\ + \\ + Voraussetzungen: Selbstähnlichkeit, Monotonieeigenschaft, Beschränktheit\\ + \\ + Eine Funktion heißt linear rekursiv, wenn sie in jedem Zweig einer Fallunterscheidung ihres Rumpfes höchstens einmal aufgerufen wird.\\ + Eine Methode f heiß endrekursiv, wenn der rekursive Methodenaufruf stets die letzte Aktion des Zweigs zur Berechnung von f ist. Kann unmittelbar entrekursiviert werden: rekursiv besser zum Aufschreiben, iterativ schneller in der Ausführung.\\ + Eine Methode f heißt kaskadenartig rekursiv, wenn in einem Zweig einer Fallunterscheidung im Rumpf von f zwei oder mehr Aufrufe von f auftreten.\\ + Zwei Methoden f und g heißen verschränkt (auch: wechselseitig) rekursiv, wenn die Methodendeklaration von f einen Aufruf der Methode g enthält und die Methodendeklaration von g einen Aufruf der Methode f enthält.\\ + Eine Methode f heißt verschachtelt rekursiv, wenn man zur Bestimmung der Parameter des rekursiven Aufrufs einen rekursiven Aufruf der Funktion ausführen muss. + \end{defi} + Terminierungsbeweis:\\ + finde ganzzahlige Terminierungsfunktion und beweise, dass t bei jedem Rekursionsschritt streng monoton fällt und nach unten beschränkt ist.\\ + \subsection{Entrekursivierung} + \begin{tabular}{@{}lll} + $f(x)$& $= g(x)$ falls $praed(x)$& (nicht rekursiver Basisfall)\\ + & $=f(r(x))$ sonst& (rekursiver Aufruf) + \end{tabular}\\ + \\ + \\ + arg = x; // Hilfsvariable\\ + while (!$praed(arg)$) \{\\ + \hspace*{10mm}arg = $r(arg)$; // modifiziere Parameter\\ + \}\\ + return $g(arg)$; // nicht-rekursiver Basisfall + \section{Rekursion im Einsatz} + 1. Durchreichen von Zwischenergebnissen\\ + 2. Dynamisches Programmieren: \\ + Mithilfe von einer Liste werden alle schon berechneten Werte gespeichert(erhöhter Speicherbedarf)\\ + Vorgehen:\\ + $\rightarrow$ rekursive Version implementieren\\ + $\rightarrow$ Tabelle zum Zwischenspeichern der Werte erzeugen und initialisieren\\ + $\rightarrow$ rekursive Version um Lookup/Memoization erweitern\\ + \hspace*{5mm}$\rightarrow$ Tabelle als zusätzliche Eingabe\\ + \hspace*{5mm}$\rightarrow$ zu berechnenden Wert in der Tabelle nachschlagen (Lookup)\\ + \hspace*{5mm}$\rightarrow$ falls bereits berechnet: Wert direkt zurückgeben\\ + \hspace*{5mm}$\rightarrow$ ansonsten:\\ + \hspace*{8mm}$\rightarrow$ Wert rekursiv berechnen\\ + \hspace*{8mm}$\rightarrow$ Wert in der Tabelle zwischenspeichern (Memoization)\\ + \hspace*{8mm}$\rightarrow$ Wert zurückgeben\\ + Backtracking, Rücksetzverfahren:\\ + systematisches Durchsuchen des Suchraums, dabei Anwendung von trial-and-error\\ + Grundgerüst:\\ + static int[\ ][\ ] backtrack(int[\ ][\ ] state) { //z.B. Sudoku\\ + \hspace*{5mm}if (isFinal(state)) \{\\ + \hspace*{10mm} return state;\\ + \hspace*{5mm} \} else \{\\ + \hspace*{10mm} // moegliche Erweiterungen aufzaehlen\\ + \hspace*{10mm} int[] candidates = getExtensions(state);\\ + \hspace*{10mm} for (int i = 0; i < candidates.length; i++) \{\\ + \hspace*{15mm} int c = candidates[i];\\ + \hspace*{15mm} state = apply(state, c); //z.B. Ziffer setzen\\ + \hspace*{15mm} if (backtrack(state) != null) \{ // Rekursion\\ + \hspace*{20mm} return state;\\ + \hspace*{15mm} \}\\ + \hspace*{15mm} state = revert(state, c); //z.B. Ziffer loeschen\\ + \hspace*{10mm} \}\\ + \hspace*{10mm} return null;\\ + \hspace*{5mm} \} + \section{Asymptotische Aufwandsanalyse} + Qualitätskriterien für Algorithmen:\\ + Korrektheit, hohe Verständlichkeit, einfache Implementierbarkeit, geringstmöglicher Bedarf an Ressourcen\\ + \\ + Der exakte Aufwand eines Algorithmus (z.B. als Anzahl Rechenschritte bis zur Terminierung oder als verbrauchte Zeit) kann nur schwer als Funktion von Umfang und Eingabewerten allgemein angegeben werden. Wichtiger ist die ungefähre Größenordnung, mit der der Aufwand in Abhängigkeit vom Umfang der Eingabe wächst.\\ + Eine Elementaroperation entspricht genau einer Instruktion des zugrunde liegenden Von-Neumann-Rechners und hat das Kostenmaß 1. + \begin{defi}[O-Notation (asympt. obere Schranke), Groß-O] + $f(n) \in O(g(n)):$ $f(n)$ wächst höchstens so schnell wie $g(n)$\\ + $O(g(n))=\{f:\mathbb{N}\rightarrow\mathbb{R}^+ | \exists n_0 \in \mathbb{N}, c\in \mathbb{R}, c>0:\forall n\geq n_0 \ f(n)\leq c\cdot g(n)\}$ + \end{defi} + \begin{defi}[$\Omega$ - Notation] + $f(n) \in \Omega (g(n)):$ $f(n)$ wächst mind. so schnell wie $g(n)$\\ + $\Omega(g(n))=\{h(n)|\exists c>0\ \exists n_0 >0\ \forall n\geq n_0: 0\leq c\cdot g(n)\leq h(n)\}$ + \end{defi} + \begin{defi}[$\Theta$ - Notation] + $f(n) \in \Theta (g(n)):$ $f(n)$ wächst ebenso so schnell wie $g(n)$\\ + $\Theta(g(n))=\{h(n)|\exists c_1>0\ \exists c_2>0\ \exists n_0 >0\ \forall n\geq n_0: 0\leq c_1\cdot g(n)\leq h(n)\leq c_2\cdot g(n)\}$ + \end{defi} + \begin{defi}[Rechenregeln für die O-Notation] + Seien $f(n)\in O(r(n)), g(n)\in O(s(n))$ und $c>0$ konstant. Dann gilt:\\ + a) $f(n)+g(n)\in O(r(n)+s(n))=O(max(r(n), s(n)))$\\ + b) $f(n)\cdot g(n)\in O(r(n)\cdot s(n))$\\ + c) $c\cdot f(n)\in O(r(n))$\\ + d) $f(n) \pm c \in O(r(n))$ + \end{defi} + + \section{Objektorientierte Programmierung: Klassen und Objekte} + Datentyp bisher: Daten gleicher Sorte, z.B. ganze Zahlen, Wahrheitswerte, Zeichenketten.\\ + Jetzt: Konzentration auf das Klassen- und Objektkonzept, das Grundlage der objektorientierten Programmierphilosophie ist. + \begin{defi}[Klasse] + Klasse (auch Objekttyp): Konzept der objektorientierten Modellierung und Programmierung, mit dem neue Datentypen definiert werden können.\\ + Die Deklaration einer Klasse stellt damit eine Art Bauanleitung oder Schablone für sog. Objekte (auch: Exemplare, Instanzen) dieser Klasse dar. \\ + Klassennamen sollen selbsterklärend sein und die Lesbarkeit des Programms fördern sowie Substantive sein und mit einem Großbuchstaben beginnen.\\ + \\ + Konkret wird in der Klassendeklaration festgelegt:\\ + - aus welchen Datenkomponenten (sog. Attributen) ein Objekt besteht\\ + - welche Methoden, z.B. zur Manipulation der zugehörigen Attributwerte, zur Verfügung stehen und\\ + - welche Konstruktoren zur Erzeugung von Objekten (auch Instanziierung) verwendet werden können + \end{defi} + \begin{defi}[Attribute] + Attribute sind Datenkomponenten, aus denen Objekte einer Klasse bestehen. Deklaration und Benennung erfolgen analog zu Variablen. Erlaubt sind Attribute primitiven und zusammengesetzten Typs. + \end{defi} + \begin{defi}[Konstruktoren] + Konstruktoren sind Operationen, die es ermöglichen, Objekte einer Klasse zu instanziieren, d.h. anzulegen oder zu erzeugen. Alle Konstruktoren einer Klasse haben denselben Namen, nämlich den der Klasse, zusätzlich können sie Parameter haben. Ein Rückgabetyp wird nicht angegeben.\\ + \\ + 1. explizite Konstruktoren\\ + Sie werden bei der Klassendeklaration angegeben und i.d.R. dazu genutzt, die Attributwerte des zu erzeugenden Objekts auf für den Anwendungszweck sinnvolle Startwerte zu setzen. Explizite Konstruktoren können Parameter haben. Eine Klasse kann mehrere explizite Konstruktoren haben, die aber unterschiedliche Parameterlisten haben müssen.\\ + 2. impliziter Konstruktor (auch Standard-Konstruktor, Default-Konstruktor)\\ + Wird kein expliziter Konstruktor angegeben, so legt der Übersetzer automatisch einen impliziten Konstruktor an. Ein impliziter Konstruktor hat immer die Form $<$Klassenname$>$()\{\ \}. + \end{defi} + Ein objektorientiertes Programm in Java besteht aus Folge von Klassendeklarationen, die jeweils Attribut-, Konstruktor- und Methodendeklarationen enthalten.\\ + In einer der Klassen muss die Methode main deklariert sein, mit der die Ausführung des Gesamtprogramms startet.\\ + Klasse, die die Methode main deklariert, nimmt i.d.R. Sonderrolle ein, nämlich die der Spezifikation des Hauptprogramms und nicht die der Deklaration des Datentyps einer Klasse von Objekten. In der Regel beginnt dort die Erzeugung der für die Problemlösung relevanten Objekte anhand der Klassendeklarationen. Diese Objekte verwalten eigene Daten (Werte der Attribute) und sie reagieren auf Nachrichten (Aufrufe von Methoden, die in der Klasse des Objekts deklariert wurden).\\ + Problemlösung durch schrittweise Objekterzeugung, - verknüpfung und - kommunikation. + \begin{defi}[Objekt] + Objekt ist Datenstruktur, deren Datentyp (Struktur, Verhalten) durch eine Klasse festgelegt ist. Objekt ist konkrete Ausprägung einer Klasse, deshalb auch Instanz genannt.\\ + Klassendeklaration legt fest, welche Attribute und Methoden für ein Objekt der Klasse zur Verfügung stehen und in welcher Form eine Instanziierung stattfinden kann.\\ + Während eine Klasse die Attribute i.d.R. nur deklariert, werden bei einem Objekt die Attribute durch den verwendeten Konstruktor oder durch Methodenaufrufe mit Attributwerten belegt. + \end{defi} + \begin{defi}[Objektvariablen] + Attribute von primitivem Typ sind Behälter im Objekt. Sie verhalten sich wie die Ihnen bekannten Variablen.\\ + Attribute vom Typ einer Klasse (sog. Objektvariablen) können einen Verweis/eine Referenz auf ein Objekt enthalten.\\ + Bei der Deklaration einer Objektvariable wird eine Variable angelegt, die einen Verweis auf ein Objekt der zugehörigen Klasse aufnehmen kann (deshalb auch Referenzvariable). \\ + Bei der Deklaration wird noch keine Objektinstanz erzeugt, der Wert der neu angelegten Variable ist eine sog. Null-Referenz. + \end{defi} + \begin{defi}[Instanziierung] + Instanziierung bezeichnet die Erstellung eines neuen Objekts einer Klasse. Das Objekt wird erzeugt, d.h. der entsprechende Platz im Speicher reserviert. Verweis auf dieses Objekt bzw. auf den Speicherbereich wird zurückgegeben. \\ + Form: Schlüsselwort new gefolgt von Konstruktoraufruf. + \end{defi} + \begin{defi}[Klassenattribut/Klassenmethode] + Klassenattribut (auch: statisches Attribut) ist Sonderfall, bei dem sich alle Objekte der Klasse denselben Attributwert teilen. Schreiboperationen in einem Objekt wirken sich damit auf alle anderen Objekte aus.\\ + Schlüsselwort static.\\ + Zugriff auf Klassenattribute von außen hat die Form $<$klassenname$>$.$<$attributname$>$ oder $<$objektname$>$.$<$attributname$>$, von innen $<$attributname$>$\\ + \\ + Klassenmethoden sind auch ohne instanziiertes Objekt verwendbar, da sie einer Klasse und nicht einem Objekt zugeordnet sind und bieten damit die Möglichkeit, Methoden direkt auf einer Klasse selbst auszuführen. \\ + Man verwendet sie als Getter und Setter für Klassenattribute oder wenn für die Methode kein Objekt der Klasse benötigt wird oder generell zu einer Klasse keine Objekte instanziiert werden sollen. So sparsam wie möglich zu verwenden, da eig. nicht der objektorientierten Philosophie entsprechend. + \end{defi} + \begin{defi}[Sichtbarkeit von Information] + Oft ist nur wichtig, was ein Objekt leistet, nicht aber, wie es realisiert ist. Es werden Programmiertechniken benötigt, mit denen man den Zugriff auf Attribute und Methoden von Objekten einschränken kann.\\ + Sichtbarkeits-Modifizierer:\\ + keiner (paketsichtbar): aus den Klassen desselben Paktes\\ + public: aus allen Klassen heraus\\ + private: nur aus definierender Klasse heraus\\ + protected: aus definierender Klasse, allen Unterklassen und Klassen desselben Pakets + \end{defi} + \begin{defi}[Datenkapselung] + Bezeichnet das Verbergen von Programmierdetails vor dem direkten Zugriff von Außen. Direkter Zugriff auf interne Datenstruktur wird unterbunden und erfolgt stattdessen über wohl definierte Schnittstellen (Black-Box-Modell). + \end{defi} + \begin{defi}[Wrapper-Klassen] + Viele Methoden erwarten Parameter von Typ Object. Um diesen Methoden Werte eines primitiven Datentyps zu übergeben, stell Java für jeden Datentyp eine sog. Wrapper-Klasse zur Verfügung, die diesen Wert kapselt. + \end{defi} + \subsection{UML - Unified Modelling Language} + Assoziation: setzt Objekte von genau 2 Klassen zueinander in Beziehung\\ + Einfache Linie zw. den Klassen, Name an der Mitte dieser notiert, Anzahlangaben (mit wie vielen Objekten der gegenüberliegenden Assoziationsseite ist je ein Objekt mind./höchst. verbunden)\\ + \\ + Multiplizitäten: geben an, wie viele Objekte der an der Assoziation beteiligten Klassen jeweils miteinander in Beziehung stehen können.\\ + 0..* wird mit Liste angegeben\\ + \\ + Aggregation: Spezialform der Assoziation, modelliert eine "hat-Beziehung"\\ + UML-Darstellung: unausgefüllte Raute am Beziehungsende auf der Seite des Ganzen\\ + \\ + Komposition: Spezialform der Aggregation, bei der ein Bestandteil genau zu einem Ganzen gehört und ohne dieses nicht existieren kann.\\ + UML-Darstellung: ausgefüllte Raute am Beziehungsende auf der Seite des Ganzen + \subsubsection{Klassenkarten} + \includegraphics[width=0.6\linewidth]{uml}\\ + Abstrakte Klassen: über dem Klassennamen $<<$abstract$>>$\\ + Interdaces: über dem Klassennamen $<<$interface$>>$\\ + statische Methoden und Attribute durch Unterstreichen kennzeichnen + + \section{OOP: Klassenbeziehungen, Polymorphie, Module} + \textbf{1. objektorientierte Analyse}\\ + Ziel: allgemeines Systemmodell erstellen ohne Implementierungsdetails. Beschreibung, was das System machen soll und Erstellung eines OOA-Modells\\ + \textbf{2. objektorientierter Entwurf, obj. orient. Design}\\ + Planung und Beschreibung, wie das geplante System die Anforderungen realisiert\\ + \textbf{3. objektorientierte Programmierung}\\ + Umsetzung in Quelltest der verwendeten objektorientierten Programmiersprache + \subsection{Vererbung} + Mechanismus, der es ermöglichst, sog. Unterklassen aus einer gegebenen Klasse abzuleiten.\\ + Die Unterklasse erbt dann von der (Ober-)Klasse die Attribute und Methoden, kann aber auf private Methoden und Attribute nicht zugreifen. Es können weitere Attribute und Methoden hinzukommen, die der Spezialisierung dienen.\\ + Ein Objekt der Unterklasse ist auch ein Objekt der Oberklasse.\\ + UML: Linie von Unter- zur Oberklasse mit unausgefülltem Dreieck als Pfeilspitze auf Seite der Oberklasse.\\ + Schlüsselwort super ermöglicht den expliziten Bezug auf Attribute oder Methoden der direkten Oberklasse der Klasse, in der es verwendet wird. + \subsection{Polymorphismus} + Polymorphie = Fähigkeit, verschiedene Gestalt anzunehmen\\ + Beispiel: polymorphe Methoden: haben gleichen Namen, tun aber etwas Unterschiedliches.\\ + \\ + Überladen:\\ + denselben Namen, aber unterschiedlicher Signatur (unterschiedlicher Rückgabetyp reich nicht)\\ + \\ + Überschreiben:\\ + liegt vor, wenn es zu einer Methode, die in einer Klasse deklariert ist, eine Methode in einer Unterklasse gibt, die denselben Namen, diesselbe Parameterliste und denselben Resultattyp hat.\\ + Anwendung: Obj. der Oberklasse: Methode d. Oberkl., Obj. der Unterklasse: Methode d. Unterkl.\\ + Auf eine überschriebene Instanzmethode der Oberklasse kann im Inneren der überschriebenen Klasse per super zugegriffen werden. Klassenmethoden können nicht überschrieben werden.\\ + \\ + Methodenauswahl in Java:\\ + 1. Finden der anwendbaren und zugreifbaren Methoden\\ + 2. Auswahl der spezifischsten Methode + \subsubsection{Polymorphe Variablen} + \textbf{Typ:} Eigenschaft von Variablen und Ausdrücken, definiert Mindestforderung bzgl. anwendbarer Operationen, rein syntaktisch festgelegt.\\ + \\ + \textbf{Klasse:} stellt Konstruktor für Objekte bereit, definiert Signatur oder Implementierung der Operationen. Objekt gehört zu der Klasse, mit deren Konstruktor es konstruiert wurde. Mit der Klassendefinition wird ein gleichnamiger Typ eingeführt.\\ + \\ + Polymorphe Variablen in typsicheren Sprachen:\\ + Typsicherheit:\\ + Es dürfen nur Methoden aufgerufen werden, die schon beim statischen Typ von m verfügbar sind, da der dynamische Typ erst zur Laufzeit feststeht, der Compiler jedoch zur Übersetzungszeit garantieren muss, dass die entsprechende Methode bzw. das entsprechende Attribut tatsächlich existiert.\\ + Eine polymorphe Variable kann im Laufe der Ausführung eines Programms Referenzen auf Objekte verschiedener Klassen haben. Sie hat einen\\ + - statischen Typ: wird durch Angabe der Klasse bei der Deklaration angegeben und kann bei der Übersetzung \\ + \hspace*{2mm}überprüft werden. Ändert sich nicht.\\ + - dynamischen Typ: wird durch die Klasse des Objekts angegeben, auf den die Variable zur Laufzeit zeigt.\\ + \hspace*{2mm} Zur Laufzeit bekannt, kann sich ändern.\\ + \\ + dynamische Bindung bei Instanzmethoden: zur Laufzeit wird in Abhängigkeit von der Tatsache, ob das Objekt eine Instanz der Ober- oder einer Unterklasse ist, die jeweilige Version der Methode aufgerufen.\\ + statische Bindung bei Klassenmethoden: es wird immer die Methode des Typs genutzt als der die Variable deklariert ist.\\ + \\ + Verdecken von (Klassen-)Attributen und Klassenmethoden:\\ + bedeutet, dass ein in der Unterklasse deklariertes Attribut denselben Namen aufweist wie ein entsprechendes Attribut der Oberklasse, oder dass eine in der Unterklasse deklarierte Klassenmethode diessebe Signatur aufweist, wie eine entsprechende Klassenmethode der Oberklasse.\\ + Hierbei wird innerhalb der Unterklasse das Attribut der Oberklasse durch das namensgleiche Attribut der Unterklasse verdeckt, oder die Klassenmethode der Oberklasse durch die signaturgleiche Klassenmethode der Unterklasse verdeckt.\\ + Regel: beim Zugriff auf (Klassen-)Attribute und Klassenmethoden ist immer der statische Typ an der Zugriffsstelle der relevante. + \newpage + \includegraphics[width=1\linewidth]{instanzklasse} + + \subsection{Schnittstellen} + Mit einer Schnittstelle (Schlüsselwort interface) wird in Java definiert, welche Methoden eine diese Schnittstelle implementierende Klasse (Schlüsselwort implements) mindestens haben muss.\\ + In Interface können Konstanten, Signaturen von Methoden deklariert werden, alles andere ist nicht zulässig.\\ + Konstantendeklarationen haben automatisch die Modifizierer static, final und public, Methodendeklarationen abstract und public. Deshalb brauchen diese nicht angegeben werden.\\ + \\ + Auflösung potentieller Mehrfachvererbung bei Schnittstellen:\\ + Methoden: Werden 2 gleich benannte Methoden über versch. Pfade ererbt, so wird bei identischen Signaturen eben diese Signatur übernommen, und bei unterschiedl. Signaturen beide übernommen und dann als überladene Methoden behandelt.\\ + Konstanten: werden 2 gleich benannte Konstanten über versch. Pfade geerbt, meldet der Übersetzer Fehler, nur wenn die Konstante verwendet wird. Ein Interface kann eine Konstante deklarieren, die eine/mehrere geerbte Konstanten gleichen Namens verdeckt.\\ + \\ + Eine abstrakte Klasse ist eine Art Oberklasse mit nur partieller Implementierung, restliche Implementierung muss in einer Unterklasse nachgeholt werden. + \subsection{Pakete und Klassenbibliotheken} + Paket: Zusammenstellung von als zusammengehörig betrachteten Java-Klassen und Interfaces.\\ + Paket besitzt einen Namen, der - wie von Dateiverzeichnissen her bekannt - hierarchisch aufgebaut sein kann. Der Paketname definiert Namensraum für die im Paket vorkommenden Klassendeklarationen und sollte nur aus Kleinbuchstaben bestehen.\\ + \\ + Um außerhalb eines Pakets eine Klasse oder Schnittstelle zu verwenden, gibt es zwei Möglichkeiten:\\ + 1. Zugriff auf Klasse in einem Paket über den qualifizierten Namen\\ + 2. Paketname mittels import bekannt machen. + \section{Robustes Programmieren} + \subsection{Fehlerquellen} + Strategien zum Umgang mit Fehlern bei der Programmentwicklung:\\ + 1. Identifizierung möglicher Fehlerquellen mit Checklisten\\ + 2. Absicherung von Quellcode gegen fehlerhafte Verwendung\\ + 3. Testen von Programmen\\ + 4. Formale Verifikation von Programmteilen\\ + 5. Toleranz gegen verbleibende Fehler\\ + \\ + Fehlervermeidung durch gemeinsame Programmentwicklung oder gemeinsames "Walkthrough" + \subsection{Fehlerbehandlung} + Ausgabe von Fehlermeldungen: System.err als Ausgabe an das Konsolenfenster\\ + \\ + Eine Ausnahme (engl. exception) ist ein Ereignis, durch das die normale Ausführungsreihenfolge unterbrochen wird.\\ + Man sagt: eine Exception wird geworfen (throw), wenn sie ausgelöst wird, und wird gefangen, wenn sie behandelt wird.\\ + Jeder Ausnahme wird ein Ausnahmetyp zugeordnet. Ausnahmetypen erben alle von der vordefinierten Klasse java.lang.Throwable mit genau zwei direkten Unterklassen: \\ + java.lang.Error (= serious problem) zeigt Probleme bei der Programmausführung an, die das Programm i.d.R. nicht selbst beheben kann/sollte. \\ + java.lang.Exception ist die Oberklasse, unter der alle Ausnahmetypen zusammengefasst werden, die das Programm selbst behandeln möchte/sollte.\\ + \\ + Alle Exceptions ohne RuntimeExceptions sind checked exceptions, alles andere aus Throwable sind unchecked exceptions. Checked Exceptions müssen behandelt oder weitergegeben werden.\\ + \\ + \includegraphics[width=\linewidth]{exceptionhierachy}\\ + \\ + \\ + Eigene Ausnahmeklassen erstellen: \\ + die Klasse sollte eine Unterklasse von java.lang.Exception sein.\\ + Ein Ausnahmeobjekt sollte den Fehlerzustand aufnehmen können, also muss man entsprechende Attribute vorsehen.\\ + \\ + Ausnahmen behandeln mit try-catch:\\ + try-Block umfasst Code, der Ausnahmen auslösen kann.\\ + Wird in einem try-Block eine Ausnahme geworfen, wird die Ausführung des try-Blocks sofort abgebrochen. Dann wird der Typ der Ausnahmeobjekts mit den Typangaben der nachfolgenden catch-Klauseln verglichen (von oben nach unten).\\ + Der erste catch-Block, dei dem der Ausnahmetyp bzgl. instanceof zur Typmenge passt, behandelt die Ausnahme und gibt dem Ausnahmeobjekt einen Namen.\\ + Passt keiner der catch-Blöcke oder löst der catch-Block selbst eine Ausnahme aus, so wird die Ausführung abgebrochen und die letzte Ausnahme an den Aufrufer übergeben.\\ + Falls keine Ausnahme ausgelöst wurde oder nach Bearbeitung der passenden catch-Klausel wird mit den nachfolgenden Anweisungen fortgesetzt.\\ + Manchmal soll im Anschluss an den Code im try-Block ein Stück Code auf jeden Fall ausgeführt werden. Solcher Code kann am Ende in einem sog. finally-Block angegeben werden.\\ + \\ + Ausnahmen weitergeben:\\ + Möglichkeit 1: Ausnahme in der verwendenden Methode mit try-catch abfangen\\ + Möglichkeit 2: Ausnahme an aufrufende Methode weiterleiten:\\ + $<$ModifierList$>$ $<$MethodName$>$($<$ParameterList$>$) throws $<$ExceptionClass$^1$$>$, ..., $<$ExceptionClass$^n$$>$ + \subsubsection{Stacktrace} + Ein Stacktrace zeigt die Methodenschachteln auf dem Stack. \\ + Hilfreich für die Rückverfolgung einer Ausnahme: in welcher Methode wurde die Ausnahme geworfen? von wo aus wurde diese Methode aufgerufen?\\ + Falls Exception in Java nicht gefangen wird, bricht Programm ab und Stacktrace wird ausgegeben. \\ + Kann auch manuell ausgegeben werden: printStackTrace() + \subsection{Zusicherungen} + Während der Programmentwicklung ist es möglich, Bedingungen zu formulieren, die am Beginn oder am Ende einer Methode oder während der Ausführung einer Schleife + gelten müssen, damit die Methode korrekt arbeitet (sog. Vor- bzw. Nachbedingung, Schleifeninvariante).\\ + Solche Bedingungen können als Behauptungen während der + Programmentwicklung (genauer: in der Testphase) automatisiert + mittels sog. Zusicherungen (engl. assertions) geprüft werden.\\ + Bei der Ausführung beim Anwender sollten sie nicht eingesetzt und korrekter Programmablauf sollte dort anders sichergestellt werden.\\ + \textbf{assert Expression1 : Expression2}\\ + Expression1 ist boolescher Ausdruck und Expression2 ein Fehlermeldungsausdruck. + \subsection{wp-Kalkül} + Zur Analyse eines Algorithmus A:\\ + Formulierung einer Nachbedingung Q: \\ + - logischer Ausdruck über den Variablen des Algorithmus\\ + - Anforderung an Methode, soll nach Abarbeitung von A erfüllt sein.\\ + Bestimmung der Vorbedingung P:\\ + - logischer Ausdruck über den Eingabewerten von A\\ + - Welche Eigenschaften müssen die Eingabewerte haben, damit nach der Ausführung von A die Nachbe-\\ + \hspace*{3mm}dingung Q auch tatsächlich erfüllt ist?\\ + A ist korekt, falls alle Eingabewerte, die tatsächlich auftreten können, P erfüllen.\\ + wp = weakest precondition, schwächste Vorbedingung\\ + Seien Algorithmus A und Nachbedingung Q gegeben. Das schwächste Prädikat P (also das, das bei möglichst vielen Eingabedaten x wahr ergibt), für das gilt P\{A\}Q gilt, heißt wp(A,Q):=P\\ + Das wp-Kalkül ist ein sog. Verifikationskalkül (verwenden rein syntaktische Regelwerke zur Ableitung zu verifizierender Aussagen, Regeln sind rein mechanisch anwendbar)\\ + \\ + Formale Definition:\\ + E: Menge der möglichen Eingabedaten\\ + O: Menge der möglichen Ausgabedaten\\ + Vorbedingung P: Prädikat auf E\\ + Nachbedingung Q: Prädikat auf E $\times$ O\\ + A: Algorithmus, der Eingabedaten in Ausgabedaten überführt.\\ + $\rightarrow$ ist also eine Abbildung A: E $\rightarrow$ O\\ + $\rightarrow$ A(x) bezeichnet die Ausgabedaten zu den konkreten Eingabedaten x\\ + Falls gilt $\forall x\in E: P(x) \rightarrow Q(x, A(x))$, dann schreibt man $\{P\} A\{Q\}$\\ + Das schwächste Prädikat P mit $\{P\} A\{Q\}$ heißt wp(A, Q).\\ + \\ + Allgemeines Vorgehen:\\ + Rückwärtsanalyse: A schrittweise von hinten abarbeiten, also in jedem Schritt letzte Anweisung X in A verarbeiten. Dazu Auswirkungen von X auf Nachbedingung bestimmen (was muss vor X gelten, damit nach X Nachbedingung erfüllt ist?). Ergebnis wird neue Nachbedingung für restliche Anweisungen in A.\\ + \\ + \begin{tabular}{@{}ll} + Zuweisungen:& in Q werden alle Vorkommen durch neuen Wert ersetzt.\\ + In-/Dekrement-Operatoren:& nach/vor der eigentlichen Substitution behandeln.\\ + Fallunterscheidungen: & Bedingung wahr: b erfüllt und X führt zur Nachbedingung\\ + & Bedingung falsch: b nicht erfüllt und Y führt zur Nachbedingung\\ + Datentypen:& Vereinfachung in Datentyp der Variable, explizit hinschreiben + \end{tabular} + \subsubsection{Korrektheit von Schleifen} + 2 Werkzeuge zum Korrektheitsbeweis von Schleifen:\\ + 1. Schleifeninvariante: muss vor/nach jedem Durchlauf wahr sein, beschreibt Verhalten der Schleife\\ + 2. Schleifenvariante: Funktion über Programm-Variablen, Beweis der Terminierung\\ + \\ + Schleifeninvariante:\\ + Sei eine Schleife der Form while (b) \{A;\} gegeben. Ein Prädikat I ist eine gültige Schleifeninvariante, falls gilt $\{I\wedge b\}\rightarrow wp(A,I)$\\ + Falls vor der Ausführung des Rumpfes die Invariante I und die Schleifen-Bedingung b erfüllt sind (d.h. die Schleife wird auch tatsächlich ausgeführt), dann muss nach Ausführung des Rumpfes zumindest die Invariante erfüllt sein.\\ + Eine geeignete Schleifeninvariante I hat zusätzlich folgende Eigenschaften:\\ + - das Prädikat I ist vor der Schleife erfüllt\\ + - die Nachbedingung Q lässt sich nach der Schleife aus I implizieren\\ + \\ + Schleifenvariante (auch: Terminierungsfunktion)\\ + Schleife muss nach endlich vielen Durchläufen abbrechen\\ + 1. V ist eine geeignete Funktion auf den in der Schleife verfügbaren Variablen\\ + 2. V ist ganzzahlig\\ + 3. V ist streng monoton fallend\\ + 4. V ist nach unten beschränkt ($\{I\wedge b\wedge V\}A\{c\leq V < z\}$ mit $c, z \in \mathbb{Z}$) + \section{Grundlegende Datentypen} + \subsection{Spezifikation von Datentypen} + Abstrakter Datentyp (ADT):\\ + Festlegung der auf den Typ anwendbaren Operationen (Schnittstelle)\\ + Festlegung der Wirkung der Operationen\\ + \\ + Man legt fest:\\ + - adt: Namen des ADTs\\ + - sorts: Liste verwendeter Datentypen\\ + - ops: Schnittstellen der Operationen: Name, Parameter-Typen und Ergebnis-Typ\\ + - axs: Axiome zur Beschreibung der Semantik: so allgemein wie möglich, so ausführlich wie nötig\\ + \\ + Dies geschieht in der Signatur (Bsp.):\\ + \begin{tabular}{@{}llll} + adt& IntSet&&\\ + sorts& IntSet, Int, Boolean&&\\ + ops& create:&& $\rightarrow$ IntSet\\ + & insert:& IntSet $\times$ Int& $\rightarrow$ IntSet\\ + & delete:& IntSet $\times$ Int& $\rightarrow$ IntSet\\ + & contains:& IntSet $\times$ Int& $\rightarrow$ Boolean\\ + & isEmpty: & IntSet & $\rightarrow$ Boolean\\ + axs& isEmpty(create) & = true&\\ + & isEmpty(insert(x,i)) & =false&\\ + & insert(insert (x,i),i) & = insert(x,i)&\\ + & contains(create, i) & = false&\\ + & contains(insert(x,j),i)& = \begin{tabular}{@{}ll} + true & falls i=j\\ + contains (x,i) & sonst\\ + \end{tabular}&\\ + end & IntSet &&\\ + \end{tabular}\\ + \\ + Vorteile:\\ + Man kann Datentyp genau soweit festlegen, wie gewünscht, ohne Implementierungsdetails vorwegzunehmen. Die Sprache, in der Axiome beschrieben werden, folgt formalen Regeln. Daher sind Entwurfswerkzeuge konstruierbar, die Spezifikation prüfen oder Prototyp erzeugen können.\\ + \\ + Nachteile:\\ + Bei komplexen Anwendungen wird die Zahl der Gesetze sehr groß. Es ist nicht immer leicht, anhand der Gesetze die intuitive Bedeutung eines Datentyps zu erkennen. Insbesondere ist es schwierig zu prüfen, ob die Menge der Gesetze vollständig ist\\ + \\ + Umsetzung in Java-Code:\\ + ops: Schnittstellen der Java-Methoden\\ + axs: Umsetzung der Java-Methoden\\ + \\ + \begin{tabular}{@{}ll} + Konstruktoren:& mit create und insert lassen sich alle mögl. Objekte erzeugen. Zwingend erforderlich.\\ + Hilfskonstrukt.:& erzeugen auch Datenobjekte des Typs, aber nicht zwingend erforderlich.\\ + Normalform:& konstruiert durch minimale Zahl von Konstruktoraufrufen. + \end{tabular} + \subsection{Generische/Parametrisierte Klassen} + Generische bzw. parametrisierte Klassen bieten sich an, wenn man feststellt, dass man mehrere Klassen für verschiedene Typen, aber mit ansonsten identischem Code benötigt.\\ + \\ + Vorteile: Code muss nur einmal geschrieben werden. Platzhalter kann durch einen beliebigen Typ ersetzt werden, der so verwendet werden kann, wie der Platzhalter im Code der generischen Klasse.\\ + \\ + Durch Angabe des Typplatzhalters T in spitzen Klammern wird Klasse zur generischen bzw. parametrisierten Klasse. Mehrere Typparameter werden durch Kommata getrennt angegeben. Typparameter können innerhalb der Klasse als Typen von Variablen oder Rückgabewerten oder als Parameter von Methoden genutzt werden.\\ + \\ + Mögliche Typen für Typparameter können eingeschränkt werden: nur Klassen die von bestimmter Klasse erben oder von bestimmtem Interface implementieren.\\ + Einschränkung mittels extends\\ + \\ + Typparameter dürfen nicht in statischen Elementen verwendet werden, da verschiedene Instanzen einer Klasse mit unterschiedlichen generischen Typen dennoch diesselben statischen Elemente verwenden. + \subsection{Elementare Listen} + Listen enthalten endliche Folgen von Objekten eines gegebenen Grundtyps in einer definierten Ordnung. Jedes Element hat Vorgänger und Nachfolger (außer 1. und letztes Element). Listen dürfen Duplikate enthalten. Man unterscheidet unsortierte und sortierte Listen.\\ + Listeneintrags-Objekte haben folgende Struktur:\\ + Verweis auf den Listeneintrag + Verweis auf den Nachfolger des Listeneintrags\\ + Liste besteht aus verketteten Listeneintrags-Objekten, die jeweils vorne angefügt werden\\ + \\ + \textbf{Stapel/Keller (Stacks)}\\ + Ein Stapel/Keller (engl. Stack) ist ein Spezialfall der elementaren Liste.\\ + Stapelprinzip: man kann nur oben etwas drauflegen oder herunternehmen. \\ + Nur oberstes Element ist sichtbar.\\ + Zuletzt eingefügtes Element wird als erstes entnommen: Last-In-First-Out\\ + \textbf{(Warte-) Schlangen (Queues)}\\ + Schlangen sind spezielle Listen, bei denen Elemente nur an einem Ende (vorne) entnommen und nur am anderen Ende (hinten) angehängt werden.\\ + Schlangen sind sog. First-In-First-Out Datenstrukturen. + \section{Verkettete Listen, dynamische Arrays, Mengen} + Einfach verkettete Liste: jedes Element kennt seinen direkten Nachfolger.\\ + Zweifach verkettete Liste: jedes Element kennt zusätzlich seinen direkten Vorgänger. + \subsection{Innere, geschachtelte und lokale Klassen} + 1. innere Klasse:\\ + nicht-statische Klasse innerhalb einer anderen Klasse\\ + Methoden der inneren Klasse können auf alle Elemente der äußeren Klasse zugreifen.\\ + Ein Objekt der inneren Klasse ist immer abhängig von einem Objekt der äußeren Klasse, d.h. es muss ein Objekt der äußeren Klasse existieren, um eines der inneren instanziieren zu können. Deshalb können innere Klasse keine statischen Elemente besitzen.\\ + \\ + 2. geschachtelte Klasse:\\ + statische Klasse innerhalb einer anderen Klasse\\ + wie statische Methoden können geschachtelte Klassen auf statische Elemente der sie enthaltenden Klasse zugreifen, nicht aber auf deren Instanzvariablen.\\ + Objekte sind nicht von einem Objekt der äußeren Klasse abhängig und können statische Elemente besitzen.\\ + \\ + 3. lokale Klasse:\\ + nicht-statische Klasse innerhalb einer Methode + \subsection{Einfach verkettete Listen} + Liste besteht aus verketteten Listeneintrags-Objekten mit sog. Wächter-Objekt. next der Wächterelements zeigt auf den vordersten Listeneintrag. Letzter Listeneintrag verweist auf Wächter $\rightarrow$ Zyklus\\ + \\ + Durchlaufen einer verketteten Liste:\\ + 1. per for-Schleife\\ + \hspace*{5mm}Medium elem;\\ + \hspace*{5mm}for (int i = 0; i $<$ medList.size(); i++) \{\\ + \hspace*{10mm} elem = medList.get(i);\\ + \hspace*{10mm} elem.print();\\ + \hspace*{5mm}\}\\ + 2. per for-each-Schleife\\ + \hspace*{5mm}for (BibMitglied bm : bmList) \{\\ + \hspace*{10mm} bm.print();\\ + \hspace*{5mm}\}\\ + 3. Listen-Iterator:\\ + \hspace*{5mm}Iterator$<$Medium$>$ iter = medList.iterator();\\ + \hspace*{5mm}Medium elem;\\ + \hspace*{5mm}while (iter.hasNext()) \{\\ + \hspace*{10mm} elem = iter.next();\\ + \hspace*{10mm} elem.print(); // mache etwas mit dem Listenelement\\ + \hspace*{5mm}\} + \subsection{Dynamische Arrays} + \begin{tabular}{l||l|l} + & einfach verkettete Listen & Standard-Arrays\\\hline\hline + Speicherbedarf:& ändert sich dynamisch mit der Anzahl & wird einmal festgelegt und ist dann \\ + & der enthaltenen Objekte & nicht veränderbar\\ + & immer nur so viel wie nötig & ggfs. Speicherverschwendung bei nicht \\ + & Speicherbedarf für Zeiger & voll belegtem Array\\ + Element suchen: & lineare Suche & binäre Suche (wenn sortiert)\\ + Zugriff i-tes Element:& Liste von vorne bis zur & direkter Zugriff möglich\\ + & betreffenden Stelle durchlaufen & + \end{tabular}\\ + \\ + Kombination der Vorteile ergibt sog. dynamisches Array + \subsection{Mengen} + Ziel:\\ + Mengen so darstellen, dass die klassischen Operationen Hinzufügen und Entnahme von Elementen, Vereinigung, Durchschnitt und Differenz effizient ausgeführt werden können. \\ + Mengen enthalten keine Duplikate von Elementen.\\ + Elemente haben keine feste Reihenfolge.\\ + \\ + 1. Implementierung mit einfach verketteter Liste ohne Sortierung der Elemente\\ + Mengenelemente werden in nicht-sortierter, einfach verketteter Liste gehalten.\\ + Neue Elemente werden vorne an der zugrundeliegenden Liste angefügt, sofern sie nicht bereits enthalten sind (Test erfordert Durchlaufen der Liste per linearer Suche).\\ + \\ + 2. Implementierung mit einfach verketteter Liste mit Sortierung der Elemente\\ + wenn die Elemente in der Menge aufsteigend sortiert sind, dann muss man nicht die ganze verkettete Liste durchlaufen, um nach Duplikaten zu suchen und die Mengenoperationen sind schneller, weil ein Reißverschlussverfahren verwendet werden kann.\\ + \\ + 3. Implementierung mit dynamischem Array mit Sortierung der Elemente\\ + Bei der Implementierung einer Menge mit Hilfe einer einfach verketteten Liste profitierten allein die Mengenoperationen von der Sortierung.\\ + Vom Suchaufwand her ist eine Implementierung mithilfe eines Arrays eine günstigere Lösung, da direkt auf die einzelnen Elemente zugegriffen werden kann.\\ + Die Speicherung als Array spart Platz, weil keine Verzeigerung nötig ist. Sie kostet Laufzeit, weil beim Änderung der Größe Kopierarbeit anfällt.\\ + \\ + 4. Implementierung als Bitvektor ohne Sortierung der Elemente + \subsection{Streutabellen} + Idee: aus dem Wert eines zu speichernden Mengenelements wird seine Adresse im Speicher berechnet.\\ + Speicher zur Aufnahme der Mengenelemente ist Streutabelle der Größe m, wobei m i.d.R. sehr viel kleiner als der Wertebereich ist. Die zu speichernden Mengenelemente haben einen eindeutigen Schlüssel.\\ + Erforderlich ist die Hash-Funktion $h: G\rightarrow S$. h ist nicht injektiv, möglichst surjektiv, sollte Schlüssel gleichmäßig auf den Bereich der Tabellenindizes verteilen und sollte effizient zu berechnen sein. h bildet ein Datenelement auf einen Hashwert ab, benötigt dazu einen Schlüssel, der ein Element eindeutig identifiziert. Der errechnete Hashwert wird als Index in die Tabelle verwendet, das Element wird in entsprechendem Bucket der Tabelle gespeichert.\\ + Belegungsfaktor BF = tatsächlich eingetragene Schlüssel / Indexmenge\\ + BF $<$ 50 \% $\rightarrow$ Verschwendung von Speicherplatz, BF $>$ 80\% $\rightarrow$ höhere Kollisionswahrscheinlichkeit.\\ + + Offenes Hashing: Behälter kann beliebig viele kollidierende Schlüssel aufnehmen (verkettete Liste).\\ + Geschlossenes Hashing: Behälter kann nur kleine konstante Anzahl von kollidierenden Schlüsseln aufnehmen. + \section{Bäume} + \subsection{Allgemeine Bäume} + Bäume erlauben es, hierarchische Beziehungen zwischen Objekten darzustellen.\\ + \\ + Liste: jedes Element hat einen Nachfolger\\ + Baum: jedes Element hat mehrere Nachfolger\\ + \includegraphics[width=0.5\linewidth]{baum}\\ + \\ + Höhe eines Knotens: Anzahl der Kanten des längsten Pfades zu einem Blatt.\\ + Höhe des Baumes: Höhe der Wurzel\\ + Bäume enthalten keine Zyklen.\\ + Verzweigungsgrad: Zahl der mögl. Nachfolger pro Element\\ + X ist Vater/Mutter von Y, fall es eine Kante von X nach Y gibt.\\ + X ist Kind von Y, falls es eine Kante von Y nach X gibt.\\ + X ist sibling eines Knoten Y, falls X und Y denselben direkten Vorgänger haben.\\ + X ist indirekter Vorgänger/Nachfolger von Y, falls es einen Pfad von X nach Y/Y nach X gibt.\\ + X ist ein Blatt, falls X keine Kinder hat.\\ + X ist ein innerer Knoten, falls X mind. ein Kind hat. + \newpage + \begin{defi}[Binärbäume] + Jeder Knoten hat nur 2 Nachfolger\\ + Der leere Baum ist ein Binärbaum.\\ + Wenn x ein Knoten ist und A und B Binärbäume sind, dann ist auch (AxB) ein Binärbaum.\\ + x heißt Wurzel, A linker Teilbaum, B rechter Teilbaum.\\ + Die Wurzeln von A und B heißen Kinder/Söhne von x, x ist ihr Elternknoten/Vaterknoten.\\ + Ein Knoten, dessen beide Kinder leere Bäume sind, heißt Blatt. + \end{defi} + \begin{tabular}{ll} + adt& BinTree\\ + sorts& BinTree, E, Boolean\\ + ops&\\ + & \begin{tabular}{@{}lll} + create: && $\rightarrow$ Bintree\\ + maketree:& BinTree $\times$ E $\times$ BinTree& $\rightarrow$ BinTree\\ + value:& BinTree & $\rightarrow$ E\\ + left, right: & BinTree & $\rightarrow$ BinTree\\ + leaf:& E & $\rightarrow$ BinTree\\ + isEmpty:& BinTree& $\rightarrow$ Boolean + \end{tabular}\\ + axs&\\ + & left(maketree(l,e,r))=l\\ + & right(maketree(l,e,r))=r\\ + & value(maketree(l,e,r))=e\\ + & leaf(v)= maketree(create,v,create)\\ + & isEmpty(create)=true\\ + & isEmpty(maketree(l,e,r)=false\\ + & ...\\ + end& BinTree + \end{tabular} + \begin{defi}[Allgemeine Bäume] + Anzahl der Kinder eines Knotens ist beliebig.\\ + Keine festen Positionen für Kinder.\\ + Rekursive Definition:\\ + 1. Der leere Baum ist ein (allgemeiner) Baum.\\ + 2. Wenn x ein Knoten ist und A1, … , An Bäume sind, dann ist auch das + n+1-Tupel (x, A1, … , An) ein Baum.\\ + \\ + Jeder Knoten speichert eine ArrayList von Referenzen auf seine Kinder + \end{defi} + andere Interpretation der Referenzen:\\ + \\ + \includegraphics[width=1\linewidth]{baumref} + \subsection{Binäre Suchbäume} + Bäume kann man sich als strukturelles Abbild des Teile-und- + Herrsche-Prinzips vorstellen, wenn man beim Zugriff jeweils + pro Knoten nur ein Kind weiterverfolgt.\\ + Eine solche Datenstruktur in Form eines Baums wird Suchbaum + genannt. Ein binärer Suchbaum hat max. zwei Kinder je Knoten.\\ + \newpage + \textbf{Traversierung, Besuchsreihenfolgen:}\\ + Breitensuche: Geschwister vor Kindern besuchen, noch zu besuchende Kntoen in Queue\\ + Tiefensuche: Kinder vor Geschwistern besuchen, noch zu besuchende Knoten auf Stack\\ + \begin{tabular}{@{}lllll} + Inorder: &\ \ \ \ \ & Preorder:&\ \ \ \ \ & Postorder:\\ + 1. linker Unterbaum&& 1. aktueller Knoten && 1. linker Unterbaum\\ + 2. aktueller Knoten&& 2. linker Unterbaum&& 2. rechter Unterbaum\\ + 3. rechter Unterbaum&& 3. rechter Unterbaum&& 3. aktueller Knoten\\ + \end{tabular} + \\ + \\ + \textbf{Klassifikation von Suchbäumen:}\\ + 1. Anzahl m der Kinder pro Knoten (Knotengrad):\\ + - Binärbaum: Anzahl $0\leq m\leq 2$\\ + - allgemeiner Baum: Anzahl beliebig\\ + 2. Speicherort der Nutzdaten:\\ + - natürlicher Baum: Nutzdaten werden bei jedem Knoten gespeichert.\\ + - Blattbaum/hohler Baum: Nutzdaten werden nur in Blattknoten gespeichert\\ + 3. Ausgewogenheit/Ausgeglichenheit/Balanciertheit:\\ + - balancierter Baum: die Höhen aller Unterbäume unterscheiden sich höchstens um $\Delta n$\\ + - vollständig ausgewogen: ANzahl der Knoten in Unterbäumen unterscheidet sich höchstens um 1\\ + - unbalancierter Baum: keine derartigen Einschränkungen\\ + \\ + \textbf{Suche nach einem Element:}\\ + - fange bei der Wurzel an\\ + - solange gesuchter Wert nicht gefunden:\\ + - falls gesuchter Wert kleiner als aktueller Wert: steige nach links ab\\ + - falls gesuchter Wert größer als aktueller Wert: steige nach rechts ab\\ + - falls kein entsprechendes Kind vorhanden: Wert nicht enthalten\\ + \textbf{Einfügen eines Wertes:}\\ + - Suche nach Einfügeposition mittels Suchalgorithmus\\ + - Einhängen des neuen Knotens an dieser Position\\ + \textbf{Löschen eines Wertes:}\\ + - Suche nach Knoten mittels Such-Algorithmus\\ + - falls zu löschender Knoten Blatt-Knoten ist: aus Baum entfernen\\ + - falls zu löschender Knoten kein Blatt-Knoten ist: Knoten ersetzen durch...\\ + - größten Knoten im linken Teilbaum oder kleinsten Knoten im rechten Teilbaum\\ + - dadurch ggf. weitere Ersetzungen in den Teilbäumen notwendig! + \subsection{AVL-Bäume} + Ein AVL-Baum ist ein binärer Suchbaum, in dem sich die Höhen seiner zwei Teilbäume höchstens um 1 unterscheiden. Wird dies durch Aktualisierungen verletzt, so muss sie durch eine Rebalancieroperation wieder hergestellt werden.\\ + \\ + Einfügen und Löschen wie bei Binärbaum, danach Rebalancierphase:\\ + In jedem Knoten wird dessen Balance geprüft: falls Knoten aus der Balance geraten, wird das durch eine Rebalancieroperation korrigiert.\\ + Balancefaktor eines Knoten: Differenz zw. Höhe des linken und Höhe des rechten Teilbaums\\ + Nach einer Einfügung genügt eine einzige Rotation oder Doppelrotation, um die Strukturinvariante des AVL-Baums (AVL-Bedingung) wiederherzustellen (und dabei die Suchbaumeigenschaft zu erhalten).\\ + \\ + Nachteile von AVL-Bäumen: zusätzlicher Platzbedarf in den Knoten zur Speicherung der Höhe oder der Balancefaktoren, komplizierte Implementierung\\ + \includegraphics[width=1\linewidth]{rotationavl} + \subsection{Halden} + Heap ist ein partiell geordneter Binärbaum.\\ + Max-Heap: Wurzel jedes Teilbaums ist größer als andere Knoten des Teilbaums\\ + Min-Heap: Wurzel jedes Teilbaums ist kleiner als andere Knoten des Teilbaums\\ + Verwendung als Prioritätswarteschlange\\ + effiziente Speicherung in Array möglich\\ + \\ + \textbf{Einfügen eines neuen Elements}\\ + - Element an die nächste freie Position in der untersten Ebene einfügen\\ + - falls Ebene voll: Element wird erster Knoten einer neuen Ebene\\ + - solange Halden-Eigenschaft in einem Teilbaum verletzt ist: \\ + \hspace*{3mm}Element entsprechend der Halden-Eigenschaft nach oben wandern lassen\\ + \textbf{Löschen eines Elements}\\ + - zu löschendes Element mit dem letzten Element ersetzen\\ + - solange Halden-Eigenschaft in einem Teilbaum verletzt ist:\\ + - Element entsprechend der Halden-Eigenschaft nach unten wandern lassen\\ + - Min-Heap: Tauschen mit kleinerem Kind\\ + - Max-Heap: Tauschen mit größerem Kind + \section{Sortieralgorithmen} + \textbf{internes Sortierverfahren:} + Alle Datensätze können gleichzeitig im Hauptspeicher gehalten werden. Direkter Zugriff auf alle Elemente ist möglich und erforderlich.\\ + \textbf{externes Sortierverfahren:} Sortieren von Massendaten, die auf externen Speichermedien gehalten werden. Zugriff ist auf einen Ausschnitt der Datenelemente beschränkt\\ + Sortieren durch Auswählen, Einfügen, Fachverteilen oder mittels Divide-and-Conquer-Verfahren\\ + Laufzeit: Vergleichsbasierte Sortierverfahren haben mindestens Laufzeit von O(n log n).\\ + in-situ: Sortierverfahren verwendet nur ein Array\\ + stabiles Sortierverfahren: gleiche Werte ändern ihre relative Reihenfolge nicht.\\ + \includegraphics[width=0.9\linewidth]{sortier}\\ + \textbf{SelectionSort}\\ + Lösche nacheinander die Maxima aus einer Liste und füge sie vorne an eine anfangs leere Ergebnisliste an.\\ + \textbf{InsertionSort}\\ + Starte mit erstem Element. Nimm jeweils das nächste Element und füge dieses an der richtigen Stelle in die Reihe ein.\\ + \textbf{BubbleSort}\\ + Das Array wird immer wieder durchlaufen und dabei werden benachbarte Elemente in die richtige Reihenfolge gebracht. Größere Elemente überholen so die kleineren und drängen an das Ende der Folge.\\ + \textbf{HeapSort}\\ + Die n zu sortierenden Elemente werden in eine max-Halde (min-Halde) eingefügt: Aufwand O(n log n). Dann wird n-mal das Maximum (Minimum) aus der Halde entnommen: Aufwand O(n log n).\\ + \textbf{MergeSort}\\ + falls zu sortierende Liste mehr als ein Element beinhaltet:\\ + - teile Liste in zwei kleinere Listen auf\\ + - verfahre rekursiv mit den beiden Teillisten\\ + - Listen mit maximal einem Element sind trivialerweise sortiert\\ + - verschmelze jeweils zwei sortierte Listen (Sortierung in verschmolzener Liste beibehalten)\\ + \textbf{QuickSort}\\ + falls zu sortierende Liste mehr als ein Element beinhaltet:\\ + - wähle ein Pivot-Element (zufälliges oder Median aus x Elementen wählen)\\ + - schiebe alle kleineren Elemente vor das Pivot-Element\\ + - schiebe alle größeren Elemente hinter das Pivot-Element\\ + - verfahre rekursiv mit den beiden Teillisten\\ + \textbf{BucketSort}\\ + Zu sortierende Werte sind ganze Zahlen zw. 0 und m-1. Man nutzt eine Menge von Behältern $B_0$ bis $B_{m-1}$, in die man alle Elemente passend einfügt. Am Ende schreibt man die Elemente in die Ergebnisfolge.\\ + \textbf{RadixSort}\\ + Voraussetzung: Elemente sind Zeichenfolgen über endlichem Alphabet, auf dessen Zeichen eine Totalordnung definiert ist. Für jedes Zeichen steht ein Fach zur Verfügung. \\ + Einzelne Stellen aller Elemente von hinten nach vorne abarbeiten.\\ + Partitionieren: Elemente abhängig von der aktuellen Stelle in das passende Fach legen.\\ + Einsammeln: Elemente wieder aus den Fächern nehmen vom 'kleinsten' Fach zum 'größten' Fach. + \section{Graphen und Graphalgorithmen} + \subsection{Grundbegriffe} + Ein Graph ist ein Paar $G=(V, E)$, wobei gilt:\\ + 1. V ist eine endliche nichtleere Menge (Knoten)\\ + 2. E ist eine zweistellige Relation\\ + Ein Pfad von v nach w ist eine endliche Folge von Knoten. Die Länge eines Pfades ist die Anzahl der Kanten im Pfad. Ein Pfad heißt einfach, wenn alle Knoten im Pfad paarweise verschieden sind.\\ + \\ + In einem gerichteten Graphen heißt ein Pfad der Länge von v nach v ein Zyklus. Ein Zyklus von v nach v heißt minimaler Zyklus, wenn außer v kein anderer Knoten mehr als einmal vorkommt. Bei ungerichteten Graphen muss ein Zyklus mindestens drei Knoten enthalten.\\ + \\ + Die Anzahl der direkten Vorgänger eines Knotens heißt Eingangsgrad des Knotens. Die Anzahl der direkten Nachfolger eines Knotens heißt Ausgangsgrad des Knotens. Bei ungerichteten Graphen spricht man vom Grad des Knotens.\\ + Ein Knoten mit Grad 0/* heißt Quelle. Ein Knoten mit Grad */0 heißt Senke.\\ + \\ + Ein gerichteter Graph heißt stark zusammenhängend (oder: stark verbunden), wenn es für alle v, w einen Pfad von v nach w gibt. Bei ungerichteten Graphen lässt man stark weg. Anschaulich: Jeder Knoten ist von jedem anderen Knoten aus erreichbar. Ein gerichteter Graph heißt schwach zusammenhängend, + wenn der zugehörige ungerichtete Graph (der durch Hinzunahme aller Rückwärtskanten entsteht) zusammenhängend ist.\\ + \\ + Sind G ein Graph und S ein zyklenfreier Teilgraph von G, der alle Knoten von G enthält, und sind G und S beide zusammenhängend, dann heißt S Spannbaum (auch: aufspannender Baum) von G. S ist ein minimaler Spannbaum, falls S ein Spannbaum von G ist, und die Summe der Kantengewichte in S kleiner oder gleich der aller anderen Spannbäume S' von G ist.\\ + Ein Knoten v eines gerichteten azyklischen Graphen (DAG) heißt Wurzel, falls es keine auf ihn gerichteten Kanten gibt.\\ + \\ + Ein Graph wird zu einem bewerteten/gewichteten Graphen, indem man eine Gewichtsfunktion ergänzt, die jeder Kante ein positives Gewicht zuordnet. Die Kosten eines Pfades definiert man in gewichteten Graphen als die Summe der Gewichte seiner Kanten. + \subsection{Darstellungen von Graphen} + Adjazenzmatrix: boolesche $n\times n$ Matrix mit $A_{ij} = true$, falls $(v_i,v_j)\in E$ und $A_{ij} = false$ sonst.\\ + Bei bewerteten/gewichteten Graphen speichert man statt true/false die Kantengewichte in der Matrix. Fehlende Kanten stellt man mit einem Spezialwert für Unendlich dar.\\ + Diese Repräsentation wählen, wenn Tests auf das Vorhandensein von Kanten häufig vorkommen.\\ + Adjazenzliste: Man verwaltet für jeden der n Knoten eine Liste seiner Nachfolger/Nachbarknoten + \subsection{Graphdurchlauf} + Tiefensuche: entspricht einem Preorder-Durchlauf von G, der jeweils in einem bereits besuchten Knoten von G abgebrochen wird.\\ + Die Tiefensuche durchläuft die Knoten eines Graphen in einer bestimmten Reihenfolge. Wenn jedem Knoten die Position in dieser Reihenfolge zugeordnet wird, ist das eine DFS-Nummerierung.\\ + Wenn v.dfsNr $<$ w.dfsNr, dann ist v Vorfahre von w in T.\\ + Breitensuche: besucht die Knoten von G ebenenweise, also zuerst die Knoten, die über einen Pfad der Länge 1 von der Wurzel aus erreichbar sind, dann über einen Pfad der Länge 2 usw. In einem bereits besuchten Knoten von G wird abgebrochen. + \subsection{Kürzeste Wege in Graphen} + Dijkstra:\\ + für alle Knoten gespeichert: Distanz, Vorgänger und 'besucht'-Markierung\\ + - Initialisierung:\\ + - Startknoten mit Distanz 0, alle anderen Knoten mit Distanz $\infty $\\ + - markiere alle Knoten als unbesucht\\ + - solange es noch unbesuchte Knoten im Graphen gibt:\\ + \hspace*{5mm}- wähle den Knoten v mit der geringsten Distanz aus (Min-Heap)\\ + \hspace*{5mm}- markiere diesen Knoten v als besucht\\ + \hspace*{5mm}- für alle unbesuchten Nachbarn u des aktuellen Knoten v\\ + \hspace*{10mm}- berechne die Länge des Pfades über v nach u:\\ + \hspace*{13mm}Gewicht = Pfad vom Startknoten nach v + Gewicht der Kante (v, u)\\ + \hspace*{10mm}- falls das berechnete Gewicht kleiner als die gespeicherte Distanz ist, dann\\ + \hspace*{13mm}a) aktualisiere die Distanz auf den neuen Wert\\ + \hspace*{13mm}b) speichere Knoten v als Vorgänger von u\\ + \\ + Floyd:\\ + es existiert eine direkte Kante $v\rightarrow w$ genau dann, wenn ein Pfad von v nach w existiert, der nur Knoten aus der Ursprungsmenge als Zwischenknoten verwendet. Der kürzeste derartie Pfad hat Kosten $\alpha$. + \subsection{Minimaler Spannbaum} + Algorithmus von Prim:\\ + Wähle einen beliebigen Knoten als Startgraph T.\\ + Solange T noch nicht alle Knoten enthält, führe aus:\\ + \hspace*{5mm} Wähle eine Kante e mit minimalen Kosten aus, die einen noch nicht in T enthaltenen Knoten v \\ + \hspace*{5mm} mit T verbindet. Füge e und v dem Graphen T hinzu.\\ + \\ + Algorithmus von Kruskal:\\ + gieriger Algorithmus, in jedem Schritt die kleinste Kante wählen, die 'nichts kaputt macht'\\ + - Eingabe: Graph G, dessen minimaler Spannbaum S bestimmt werden soll\\ + - S enthält zu Anfang nur die Knoten von G und keine Kanten\\ + - sortiere die Kanten von G aufsteigend nach ihrem Gewicht\\ + - solange S noch nicht zusammenhängend ist, betrachte die nächste Kante\\ + - wenn diese zwei getrennte Komponenten von S verbindet, dann zu S hinzufügen, andernfalls nicht + \section{Geometrische Algorithmen} + \begin{tabular}{@{}ll} + Punkt:& n-Tupel im n-dimensionalen Raum (n Koordinaten)\\ + Gerade:& Linie durch zwei beliebige Punkte\\ + Strecke:& Linie durch ihre beiden Endpunkte\\ + Streckenzug:& Folge von Strecken (Kanten), bei der der Endpunkt (Knoten) einer Strecke der \\ + & Anfangspunkt der nächsten Strecke ist\\ + Polygon:& der Anfangspunkt der ersten Strecke ist Endpunkt der letzten\\ + innen:& Polygon, dessen Kanten sich nicht kreuzen, umrahmt eine Region, deren Punkte innen sind\\ + konvex:& Polygon ist konvex, wenn jede Strecke zwischen zwei seiner Punkte komplett im Inneren \\ + & des Polygons liegt + \end{tabular}\\ + \\ + + \textbf{Punkt-in-Polygon-Problem}\\ + Wähle Punkt 'unendlich' weit außen und verbinde ihn mit dem Punkt P. Gehe auf der Verbindung in Richtung P.\\ + Wenn eine Polygonkante gekreuzt wird und wir außen (a) sind, gehen wir nach innen (i). Wenn eine Polygonkante gekreuzt wird und wir innen (i) sind, gehen wir + nach außen (a).\\ + \textbf{Konstruktion von Polygonen}\\ + Punkte zu Polygon verbinden\\ + Radius einer Kreisscheibe über alle Punkte rotieren. Verbinde den letzten zum Polygon hinzugefügten Punkt mir dem nächsten vom Radius überstrichenen Punkt. Als Mittelpunkt einen Randpunkt der Punktmenge wählen.\\ + \\ + \textbf{Konvexe Hülle}\\ + Die konvexe Hülle einer Punktmenge ist das kleinste konv. Polygon, das alle Punkte in seinem Innern hat.\\ + Beginne mit einem Polygon aus beliebigen drei Punkten der gegebenen Punktmenge. Wähle beliebigen noch nicht bearbeiteten Punkt P (möglichst weit außen). Liegt er außerhalb des bisher erzeugten Polygons, dann modifiziere die bisher gefundene Hülle, sodass sie den neuen Punkt enthält.\\ + Einpackalgorithmus: Beginne mit einem äußersten Punkt (z.B. äußerst links) und Schnur nach unten. Verbinde immer zu dem Punkt, den bei Drehung gegen den + Uhrzeigersinn die Einwickelschnur als nächstes erreicht (stützende Kante).\\ + \\ + \textbf{Ballung und nächstes Paar}\\ + Problem: Punktmenge soll in homogene Klassen eingeteilt werden.\\ + Zweck: ohne a-priori-Wissen bestimmen, was zusammengehört bzw. was sich ähnlich ist.\\ + Scharfe Klassifikation: Punkte werden zu Klassen zusammengefasst, sodass jeder Punkt genau zu einer Klasse gehört.\\ + Der Heterogenitätsindex einer Klasse gibt an, wie gut die Punkte einer Klasse zusammenpassen. \\ + Der Güteindex einer Klassifikation bewertet eine Klassifikation.\\ + Hierarchische Verfahren:\\ + - zusammenfassend/agglomerativ: zu Beginn enthält jede Klasse genau einen Punkt. \\ + \hspace*{3mm}Füge in jedem Schritt die zwei am wenigsten entfernten Klassen zusammen.\\ + - zerteilend/divisiv: beginne mit einer Klasse, die alle Punkte enthält. \\ + \hspace*{3mm}Teile in jedem Schritt eine Klasse in zwei Unterklassen.\\ + Für jede pro Schritt entstehende Klassifikation wird ein Güteindex berechnet. Wähle die Klassifikation, bei der sich im nächsten Schritt der Güteindex nur noch gering verbessern würde.\\ + \\ + Finde nächstes Paar: Diejenigen zwei Punkte einer Punktmenge mit geringstem gegenseitigen Abstand. \\ + Einfacher Algorithmus: Berechne alle Distanzen, wähle das Paar mit der kleinsten Distanz. +\end{document} \ No newline at end of file diff --git a/horschtey_zusammenfassungen/aud/dowhile.png b/horschtey_zusammenfassungen/aud/dowhile.png new file mode 100644 index 0000000000000000000000000000000000000000..18a7fe39f3995f476ca7c5ecc4aec81137dea2ba GIT binary patch literal 9640 zcmd^_XH*kW*Y823C(=vkozNj5AVmlzbO^mllR!ieQJR2s14!rzNR!@s6G0G>CZUNG zDGJh*DqT9>;PdUSyWV@({dC`TKg`O^oXnY-ea`It+yAp;;f7j33Kj|i0s^3pwuT7- z0ii2?DB_OCzq`Yt<#vhYE*0z31Kya)3 zdL!)jDsv$q;8)hsfSLu^Z|8Kz%}u}m@N4vK&C|g}S?*zrnv%D+Lt>9I>BUj1FIV^9 zQaQg8@1?Y5CgEt^b^IV{ZS_`??1ml}Nb*4mNK&GgF>;ZCTk$P9HvlE7E_t5dlTgIF zuASyFpw)jC{L)2q@XUAQxNzr`J7|ZBAnZ#XfS`;3M-WEwf4hSWTMa(LDRKe>gA_=+ 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