[DMDB] Completed normal forms, start query optimization

2026-07-27 21:29:09 +02:00 · 2026-06-30 16:21:35 +02:00
parent 4b4b74fcb6
commit 59e4bcd5e9
19 changed files with 168 additions and 13 deletions
@@ -1,4 +1,5 @@
 \subsection{Functional Dependency}
+\label{sec:functional-dependency}
 As with relational algebra, we again define a schema to be a relation, e.g. $\cR(A:D_A, B:D_B, C:D_C, D: D_D)$,
 with an instance of it being $R \subseteq D_A \times D_B \times D_C \times D_D$.

@@ -12,9 +13,10 @@ and $R \vdash \alpha \rightarrow \beta$ if the same applies syntactically (i.e.
 \inlineintuition This means that there is a function that maps the values of columns $\alpha$ to the values of columns $\beta$.


-\inlinedefinition $\alpha \subseteq \R$ is a \bi{superkey} if and only if $\alpha \rightarrow \cR$.
+\inlinedefinition $\alpha \subseteq \R$ is a \gls{superkey} if and only if $\alpha \rightarrow \cR$.

-\inlineintuition This means that if we know the values of columns $\alpha$, we know the value of the rest of the columns in $\cR$
+\inlineintuition This means that if we know the values of columns $\alpha$, we know the value of the rest of the columns in $\cR$.
+It is a superset of some candidate keys (see \ref{sec:relations-keys})


 \inlinedefinition $\alpha \rightarrow \beta$ is minimal if and only if $\forall A \in \alpha : (\alpha - \{ A \}) \centernot{\rightarrow} \beta$.
@@ -26,7 +26,7 @@ We can use that to check the following:
    \item $F \vdash \alpha \rightarrow \gamma$: Calculate $\alpha^+$ and check $\gamma \in \alpha^+$
    \item $F \vdash G$: For each $\alpha \rightarrow \gamma \in G$, check $F \vdash \alpha \rightarrow \gamma$
    \item $F$ equivalent to $G$: Check $F \vdash G$ and $G \vdash F$
-    \item $K \subseteq \cR$ superkey for $F$: Check $F \vdash K \rightarrow \cR$:
+    \item $K \subseteq \cR$ \gls{superkey} for $F$: Check $F \vdash K \rightarrow \cR$:
 \end{itemize}


@@ -1,5 +1,5 @@
 \subsubsection{First Normal Form}
-In the first normal form, only atomic domains are allowed.
+In the first normal form, only atomic domains are allowed, i.e. keys are not allowed to be arrays.

 \inlineintuition This makes the concept of a key easier to define, or well defined

@@ -1,10 +1,16 @@
 \subsection{Second Normal Form}
-The two dependencies $\{ \text{Legi}, \text{Nr} \} \rightrightarrow \{ \text{Name}, \text{Semester} \}$,
-$\{ \text{Legi} \} \rightrightarrow \{ \text{Name}, \text{Semester} \}$ state the following:
+% The two dependencies $\{ \text{Legi}, \text{Nr} \} \rightrightarrow \{ \text{Name}, \text{Semester} \}$,
+% $\{ \text{Legi} \} \rightrightarrow \{ \text{Name}, \text{Semester} \}$ state the following:
+%
+% \begin{itemize}
+%     \item Legi determines the name and semester of a student.
+%     \item The first map has to happen, even though mapping $\{ \text{Legi}, \text{Nr} \}$ to $\{ \text{Legi} \}$ would be enough. We thus will have redundancy.
+%           However, this is the exact relationship that the table is meant to capture
+%           Thus, $\{ \text{Name}, \text{Semester} \}$ should not be in the table at all.
+% \end{itemize}

-\begin{itemize}
-    \item Legi determines the name and semester of a student.
-    \item The first map has to happen, even though mapping $\{ \text{Legi}, \text{Nr} \}$ to $\{ \text{Legi} \}$ would be enough. We thus will have redundancy.
-          However, this is the exact relationship that the table is meant to capture
-          Thus, $\{ \text{Name}, \text{Semester} \}$ should not be in the table at all.
-\end{itemize}
+For a relation to be in 2NF, every non-key attribute needs to minimally dependent on \bi{every} key, i.e. no attribute depends on part of a key.
+If relations are not in 2NF, they may experience insert, update and delete anomalies,
+which can lead to incorrect, redundant or inconsistent updates of the relations.
+
+However, some relations can still suffer from update and / or delete anomalies
@@ -0,0 +1,11 @@
+\subsubsection{Third Normal Form}
+A relation $R$ is in 3NF if and only if for all $\alpha \rightarrow B$, at least one of the following conditions holds:
+\begin{itemize}
+    \item $B \in \alpha$ (then $\alpha \rightarrow B$ is a trivial FD)
+    \item $B$ is an attribute of at least one key
+    \item $\alpha$ is a \gls{superkey} of $R$
+\end{itemize}
+\inlineintuition if $\alpha \rightarrow B$ does not satisfy any of these conditions, $\alpha$ is a concept in its own right,
+thus, the 3NF tries to get rid of ``transitive dependencies'' (e.g. $A \rightarrow B, B \rightarrow C$)
+
+The 3NF can still experience update and delete anomalies.
@@ -0,0 +1,13 @@
+\subsubsection{Boyce-Codd Normal Form (BCNF)}
+$R$ is in BCNF if and only if for all $\alpha \rightarrow B$ at least one of the following conditions holds:
+\begin{itemize}
+    \item $B \in \alpha$ (thus, $\alpha \rightarrow B$ is a trivial FD)
+    \item $\alpha$ is a \gls{superkey} of $R$
+\end{itemize}
+\inlineintuition In each relation, you only store the same information once.
+
+The following FD are okay in both 3NF and BCNF because both left hand sides are candidate keys for the table.
+\begin{itemize}
+    \item $\{ \text{PersNr} \} \rightarrow $ Name, Level, Room, City, Street, Canton
+    \item $\{ \text{Room} \} \rightarrow $ PersNr
+\end{itemize}
@@ -0,0 +1,25 @@
+\subsubsection{BCNF Decomposition Algorithm}
+\begin{algorithm}
+    \caption{Decomposition algorithm}
+    \begin{algorithmic}[1]
+        \Procedure{Decomposition}{$\cR$}
+            \State \texttt{result} $\gets \{ \cR \}$
+            \While{$\exists \cR_i$ in \texttt{result} such that $\cR_i$ is not in BCFN}
+                \State let $\alpha \rightarrow \beta$ be the evil FD
+                \State $\cR_i^1 \gets \alpha \cup \beta$
+                \State $\cR_i^2 \gets \cR_i - \beta$
+                \State $\texttt{result} \gets (\texttt{result} - \cR_i) \cup \{ \cR_i^1, \cR_i^2 \}$
+            \EndWhile
+            \State \Return \texttt{result}
+        \EndProcedure
+    \end{algorithmic}
+\end{algorithm}
+In words, this algorithm takes as input a schema $\cR$ and outputs a list of schemas $\cL = \cR_1, \ldots, \cR_n$,
+such that $\cL$ is a lossless decomposition of $\cR$ (i.e. joining them together will perfectly recover the information in $\cR$)
+and each of the $\cR_i$ are in BCNF.
+
+Of note is that this does \bi{not} preserve the functional dependencies and some data redundancies will still remain,
+only the ones caused by functional dependencies.
+
+In addition, the order of picking the evil FD matters, so we use heuristics, which pick the one with the largest right hand side.
+
@@ -0,0 +1,19 @@
+\subsubsection{3NF Synthesis Algorithm}
+\begin{algorithm}
+    \caption{Synthesis algorithm}
+    \begin{algorithmic}[1]
+        \Procedure{Synthesis}{$\cR$}
+            % TODO: what is the minimal basis?
+            \State Compute the minimal basis $F_c$ of $F$
+            \For{\textbf{all} $\alpha \rightarrow \beta \in F_c$}
+                \State create $R_{\alpha \cup \beta}(\alpha \cup \beta)$
+            \EndFor
+            \State If none of the above relations contains a superkey, add a relation with a key
+            \State Eliminate $R_\alpha$ if there exists $R_\alpha'$ such that $\alpha \subseteq \alpha'$
+        \EndProcedure
+    \end{algorithmic}
+\end{algorithm}
+In words, this algorithm takes as input a schema $\cR$ and outputs a list of schemas $\cL = \cR_1, \ldots, \cR_n$,
+such that $\cL$ is a lossless decomposition of $\cR$ and each of the $\cR_i$ are in 3NF.
+
+This algorithm preserves all functional dependencies, however does not remove redundancies because it is not BCNF.
@@ -14,6 +14,8 @@
 \input{parts/02_theory-background/01_normal-forms/02_second-nf.tex}
 \input{parts/02_theory-background/01_normal-forms/03_third-nf.tex}
 \input{parts/02_theory-background/01_normal-forms/04_bncf.tex}
+\input{parts/02_theory-background/01_normal-forms/05_decomposition-algorithm.tex}
+\input{parts/02_theory-background/01_normal-forms/06_synthesis-algorightm.tex}
 % \input{parts/02_theory-background/01_normal-forms/}

 % \input{parts/02_theory-background/}