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\chapter{Algorithm}
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\section{Idea}
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Since our approach to identifying candidates is large based on set theory, the algorithms for finding are also making heavy use of set operations, using essentially the same construction method for creating the set of optimization candidates as the definitions presented in \autoref{chap:candidates}.

\subsection{Graph Optimization}
Optimization of the theory graph happens one theory at a time, beginning at the outer most edges of the theory graphs, i.e. those theories that are not included by any other theory.

It may seem counter intuitive, since we are changing the theories before modifying the theories they depend upon, however our use of future lite code (see \autoref{sec:futurelite}) means that any change that might affect the outer theories should be discarded anyway, as it may be a desired part of the theory, even if it is not referenced in the theory itself, resulting in no effective gain for resolving inner theories first.

This is not true for the other direction, as removing unnecessary inclusions may significantly reduce the amount of dependencies that might limit our pruning of the inner theories.

As a result of these observations, it is a strictly better option to always prioritize outer theories and since the runtime will already will already be dominating the time needed for a sort, we can do so with little additional cost (see \autoref{sec:runtime}).

\subsection{Theory Optimization}
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Optimization of a theory itself happens in two separate passes. The first pass detects and eliminates superfluous inclusions. The second pass cleans up any remaining simply redundant inclusions.
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The reason for separating these passes is to avoid the removal of simply redundant inclusions, that will later have to be reinserted when eliminating a partially superfluous inclusion. An example for such a hazard is the example of an overlap between the two types of candidates in \autoref{fig:redundantoverlap}

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\subsubsection{First pass - Eliminating superfluous inclusions}

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In the first pass we first take a selection of optimization candidates, namely those theories that are directly included, but not used in the current theory. These are the theories that need to be replaced.

The algorithm then computes all the necessary replacements by looking at all indirect inclusions that result   from the inclusion to be replaced. The necessary replacements are those that are used by the theory or its future and are not already included by the theory or by one of its necessary dependencies. 
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For this algorithm it is not necessary to make a distinction between purely and partially superfluous inclusions, as the pure case is entirely subsumed by the partial case with an empty set of necessary inclusions.

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As we discussed in \autoref{sec:puresivia} this pass cannot be expected to remove all the inclusions that are unneeded for retaining a valid graph, but what we do expect it to leave a valid graph valid.

Proof sketch:
\begin{itemize}
\item Any theory used by the current theory or its future is either directly included by the current theory or indirectly.
\item Any directly included theory that is necessary isn't a candidate and will therefore remain.
\item Any indirectly included theory is either included by a theory that is retained or by one that is being replaced. Since the replacement of a theory inclusion is the necessary subset of its included theories, all necessary dependencies will remain included.
\end{itemize}

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\subsubsection{Second pass - Eliminating simply redundant inclusions}

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The idea behind the second pass is to collect all those theories that are included indirectly by the current theory and throwing away all of those direct inclusions that are part of this set. This will leave us exactly with those inclusions that are not simply redundant, without changing the flattened theory graph.
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Proof sketch:
\begin{itemize}
\item Simply redundant inclusions are those, that are both directly and indirectly included.
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We get rid of those direct inclusions that are also indirect, so clearly all simply redundant inclusions must be gone.
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\item For the invariance of the flattened theory graph, we must first assume that theories are partially ordered by inclusion and inclusion paths are of finite depth, ie. that cyclic inclusions are not present.\\
If we do so, all indirect inclusions must ultimately be the result of a direct inclusion, so all relevant direct inclusions are still in the set.\\
If these assumptions are wrong however, not only could we run into problems, but we will inevitably do so if the current theory is part of the cycle. It is therefore best to only perform this operation on a graph that satisfies our assumption.
\end{itemize}
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\subsection{Future Lite Code}
\label{sec:futurelite}
In \autoref{sec:viability} we already mentioned the problem of future lite code. The obvious method for aquiring the future lite code is traversing the entire theory graph and whenever we find a theory inclusion, we make a note for the included theory that the theory we are currently traversing is part of its future code. Unfortunately this requires traversing the graph in its entirety.

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For the actual algorithm we skip this part and instead put create a map of used theories in the future code, since this is the part we actually require. Since this means potentially going over every theory for every theory (or at least reasonably close to it), our worst case runtime for this part of the algorithm is quadratic in the number of theories in the graph.

This runtime is especially problematic, since we need to update the the information after every step of the optimization to make proper use of any improvements of the graph. We can demonstrate this by considering the following example.

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\begin{figure}[!htb]
\centering
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\begin{tikzpicture}[node distance=2cm]\footnotesize
\node[thy] (bottom) {\begin{tabular}{l}
                             \textsf{bottom}\\\hline
                             X\\\hline
                             ...
                           \end{tabular}};
\node[thy,  above of = bottom] (middle) {\begin{tabular}{l}
                             \textsf{middle}\\\hline
                             Y\\\hline
                             not X
                           \end{tabular}};
\node[thy, above of = middle] (top) {\begin{tabular}{l}
                             \textsf{top}\\\hline
                             ...\\\hline
                             X, but not Y
                           \end{tabular}};
\draw[include] (bottom) -- (middle);                 
\draw[include] (middle)  -- (top);
\end{tikzpicture}
\caption{Example of a graph where optimizing changes the future.}
\label{fig:changeablefuture}
\end{figure}

As we can immediately see in \autoref{fig:changeablefuture}, it is possible to replace the partially superfluous inclusion of middle in top with bottom. The result is the changed graph in \autoref{fig:changedfuture}.

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\begin{figure}[!htb]
\centering
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\begin{tikzpicture}[node distance=2cm]\footnotesize
\node[thy] (bottom) {\begin{tabular}{l}
                             \textsf{bottom}\\\hline
                             X\\\hline
                             ...
                           \end{tabular}};
\node[thy,  above of = bottom] (middle) {\begin{tabular}{l}
                             \textsf{middle}\\\hline
                             Y\\\hline
                             not X
                           \end{tabular}};
\node[thy, above of = middle] (top) {\begin{tabular}{l}
                             \textsf{top}\\\hline
                             ...\\\hline
                             X, but not Y
                           \end{tabular}};
\draw[include] (bottom) -- (middle); 
\draw[include, bend left] (bottom) edge (top);
\end{tikzpicture}
\caption{Example of a graph where optimizing changes the future.}
\label{fig:changedfuture}
\end{figure}

In the changed graph we can now optimize the purely superfluous inclusion of bottom in the theory middle. We could not have done this in the earlier graph, since X and therefore bottom was used in middle's future. Thus we need an updated version of the future to make full use of earlier optimizations.

To somewhat mitigate the effect this has on the performance, the future is created step by step, as we work our way from the outer edges of the graph towards the core.
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\section{Pseudo Code}
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\subsection{Optimizing graph}

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The following code applies the optimizations to the entire theory graph.
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\begin{algorithm}[H]
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\label{alg:graph}
 \KwData{theoryGraph = theory graph to be optimized}
 \KwResult{replacements = map from theories to maps of theory inclusions to their replacement}
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 \Begin{
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 	futureUses := empty map\;
 	replacements := empty map\;
 	theoryGraph := sort(theoryGraph)\;
 	theoryGraphRev := reverse(theoryGraph)\;
 	\For{theory $\leftarrow$ theoryGraph}{
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 		add theory $\rightarrow$ usedTheories(theory) $\cup$ futureUses(theory) to futureUses\;
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 		add theory $\rightarrow$ optimize(theory, futureUses(theory)) to replacements\;
 		\For{include $\leftarrow$ includes(theory)}{
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 			add include $\rightarrow$ futureUses(include)$\cup$usedTheories(include)$\cup$futureUses(theory) to futureUses\;
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 		}
 		\KwRet replacements \;
 	}
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 }
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 \caption{optimizeGraph(theorygraph)}
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\end{algorithm}

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\subsection{Optimizing theory}
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The following pseudo code applies optimizations to a given theory.
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\begin{algorithm}[H]
 \KwData{theory = theory from our theory graph\\
 futureUse = set of theories used by theories including $theory$ in future lite code\\
 pass changes that were already applied to the graph}
 \KwResult{replacements = map from theory inclusions to their replacement}
 \Begin{
 	replacements = superfluousIncludes(theory, futureUse)\;
 	\For{removal $\leftarrow$ redundantIncludes(theory, futureUse)}{
 		add removal $\rightarrow$ $\emptyset$ to replacements\;
 	}
 	\KwRet replacements\;
 } 
 \caption{optimize(theory, futureUse)}
\end{algorithm}
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\subsection{Finding superfluous inclusions}
\label{sec:alg_si}
The following pseudo code is for finding superfluous inclusions (see: \autoref{sec:superinc}).

\begin{algorithm}[H]
 \KwData{theory = theory from our theory graph\\
 futureUse = set of theories used by theories including $theory$ in future lite code\\
 pass changes that were already applied to the graph}
 \KwResult{replacements = map from theory inclusions to their replacement}
 \Begin{
 replacements := empty map\;
 futureUsedTheories := usedTheories(theory) $\cup$ futureUse\;
 candidates := DirectInludes(theory) $\setminus$ futureUsedTheories\;
 \For{candidate $\leftarrow$ candidates}{
 		neededCandidateIncludes := Includes(candidate) $\cap$ futureUsedTheories \;
		remainingIncludes := Includes((directIncludes(theory) $\cap$ futureUsedTheories))$\cup$ (directIncludes(theory) $\cap$ futureUsedTheories) \;
		neededCandidateIncludes := neededCandidateIncludes $\setminus$ remainingIncludes\;
		add candidate $\rightarrow$ neededCandidateIncludes to replacements\;
 }
 \KwRet superfluousIncludes\;
 \caption{superfluousIncludes(theory, futureUse)}
 }
\end{algorithm}

Note that in \autoref{sec:superinc} we made a destinction between purely and partially superfluous inclusions. However we do not need to make this distinction while searching for them, as we can search for them by using the criteria for generally superfluous inclusion. Since they only differ in the set of inclusions that needs to be retained and we write that set in our result anyway, both can be accomplished by the same routine.

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\subsection{Finding simply redundant inclusions}
\label{sec:alg_sri}
The following pseudo code is for finding simply redundant inclusions (see: \autoref{sec:redinc}).
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\begin{algorithm}[H]
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 \KwData{$theory$ = theory from our theory graph\\
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 pass changes that were already applied to the graph}
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 \KwResult{$redundantIncludes$ = set of simply redundant inclusions}
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 \Begin{
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 $subIncludes := \emptyset$\;
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 $redundantIncludes := \emptyset$\;
 \For{$i \leftarrow directIncludes(theory)$}{
	$subIncludes := subIncludes \cup includes(i)$\; 
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 }
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 \For{$i \leftarrow directIncludes(theory)$}{
 	\If{$i \in subIncludes$} {
		$redundantIncludes := redundantIncludes \cup \{i\}$\;
	}
 }
 \KwRet $redundantIncludes$\;
 }
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 \caption{simplyRedundantIncludes(theory)}
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\end{algorithm}

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\section{Performance analysis}
Since the performance will be heavily dependent on the size of the graph to be optimized, we will measure runtime and memory requirements depending on the variable t, which denotes the number of theories in the graph.

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\subsection{Runtime}
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\label{sec:runtime}
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Exact runtime is specific to the library implementations of used functions, but after looking at the relevant Scala documentation we can make a few helpful assumptions \cite{scaladoccolperf}.

With effective constant time to lookup, add and remove in a hashset, we can deduce that likely runtimes for the set-operations $\cup$, $\cap$ and $\setminus$ are O(n), where n is the number of elements in the sets involved. 

\subsubsection{First pass}
\begin{algorithm}[H]
 \Begin{
 replacements := empty map \tcp*{1}
 futureUsedTheories := usedTheories(theory) $\cup$ futureUse \tcp*{t}
 candidates := DirectInludes(theory) $\setminus$ futureUsedTheories \tcp*{t}
 \For{candidate $\leftarrow$ candidates\tcp*{$\times t$}}{
 		neededCandidateIncludes := Includes(candidate) $\cap$ futureUsedTheories \tcp*{t}
		remainingIncludes := Includes((directIncludes(theory) $\cap$ futureUsedTheories))$\cup$ (directIncludes(theory) $\cap$ futureUsedTheories) \tcp*{5$\cdot{}$t}
		neededCandidateIncludes := neededCandidateIncludes $\setminus$ remainingIncludes\tcp*{t}
		add candidate $\rightarrow$ neededCandidateIncludes to replacements\tcp{1}
 }
 \KwRet superfluousIncludes\tcp{1}
 \caption{superfluousIncludes(theory, futureUse) - runtime}
 }
\end{algorithm}

The loop runs up to $t$ times over up to $7 \cdot t+1$.\\
This results in an overall worst case performance of $7\cdot t^2+t+3 = O(t^2)$.

\subsubsection{Second pass}

\begin{algorithm}[H]
 \Begin{
 $subIncludes := \emptyset$\tcp*{1}
 $redundantIncludes := \emptyset$\tcp*{1}
 \For{$i \leftarrow directIncludes(theory)$\tcp*{$\times t +t$}}{
	$subIncludes := subIncludes \cup includes(i)$\tcp*{t}
 }
 \For{$i \leftarrow directIncludes(theory)$\tcp*{$\times t +t$}}{
 	\If{$i \in subIncludes$\tcp*{1}} {
		$redundantIncludes := redundantIncludes \cup \{i\}$\tcp*{t}
	}
 }
 \KwRet $redundantIncludes$\tcp*{1}
 \caption{simplyredundantIncludes(theory) - runtime}
 }
\end{algorithm}

The first loop runs up to $t$ times over up to $t$.\\
The second loop runs up to $t$ times over up to $t+1$.\\
This results in an overall worst case performance of $2\cdot t^2+t+3 = O(t^2)$.

\subsubsection{Theory optimization}

\begin{algorithm}[H]
 \Begin{
 	replacements = superfluousIncludes(theory, futureUse)\tcp*{$t^2$}
 	\For{removal $\leftarrow$ redundantIncludes(theory, futureUse)\tcp*{$\times t + t^2$}}{
 		add removal $\rightarrow$ $\emptyset$ to replacements\tcp*{$1$}
 	}
 	\KwRet replacements\tcp*{$1$}
 } 
 \caption{optimize(theory, futureUse)}
\end{algorithm}

The loop runs up to $t$ times over 1.\\
This results in an overall worst case performance of $2\cdot O(t^2)+t+1 = O(t^2)$.

\subsubsection{Graph optimization}

\begin{algorithm}[H]
 \Begin{
 	futureUses := empty map\tcp*{1}
 	replacements := empty map\tcp*{1}
 	theoryGraph := sort(theoryGraph)\tcp*{$t^2$}
 	theoryGraphRev := reverse(theoryGraph)\tcp*{$t$}
 	\For{theory $\leftarrow$ theoryGraph\tcp*{$\times t$}}{
 		add theory $\rightarrow$ usedTheories(theory) $\cup$ futureUses(theory) to futureUses\tcp*{$t$}
 		add theory $\rightarrow$ optimize(theory, futureUses(theory)) to replacements\tcp*{$O(t^2)$}
 		\For{include $\leftarrow$ includes(theory)\tcp*{$\times t$}}{
 			add include $\rightarrow$ futureUses(include)$\cup$usedTheories(include)$\cup$futureUses(theory) to futureUses\tcp*{$t$}
 		}
 		\KwRet replacements \tcp*{1}
 	}
 }
 \caption{optimizeGraph(theorygraph) - runtime}
\end{algorithm}

The inner loop runs up to $t$ times over up to $t$.\\
The outer loop runs up to $t$ times over up to $t+O(t^2+)+t^2$.\\
This results in an overall worst case performance of $t^2+t+t\cdot (t+O(t^2)+t^2) +3= O(t^3)$.
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Since the total number of theories can be quite large a cubic runtime is hardly ideal. However it should be noted that the worst case requires the average theory to include most of the other theories.

In graphs where no theory includes more than the square root of the number of theories, the runtime should remain within quadratic bounds.
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\subsection{Memory}
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Making precise assumptions about the memory usage is even harder, as the documentation is insufficiently specific. We will therefore assume that the memory required by a HashSet or HashMap is in the same order of magnitude as it's elements.

\subsubsection{First pass}
\begin{algorithm}[H]
 \KwData{\\
 theory = O(1)\\
 futureUse = O(t)\\
 replacements = O($t^2$)\\
 futureUsedTheories = O(t)\\
 candidates = O(t)\\
 neededCandidateIncludes = O(t)\\
 remainingIncludes = O(t)\\
 neededCandidateIncludes = O(t)
 }
 \caption{superfluousIncludes(theory, futureUse) - memory}
\end{algorithm}

Assuming none of the library functions need larger memory, the overall memory usage lies in O($t^2$).

\subsubsection{Second pass}
\begin{algorithm}[H]
 \KwData{\\
 theory = O(1)\\
 subIncludes = O(t)\\
 redundantIncludes = O(t)\\
 $i_{first\ loop}$ = O(1)\\
 $i_{second\ loop}$ = O(1)
 }
 \caption{simplyredundantIncludes(theory) - memory}
\end{algorithm}

Assuming none of the library functions need larger memory, the overall memory usage lies in O(t).

\subsubsection{Theory optimization}
\begin{algorithm}[H]
 \KwData{theory = theory from our theory graph\\
 futureUse = set of theories used by theories including $theory$ in future lite code\\
 pass changes that were already applied to the graph}
 \KwResult{\\
 	replacements = O($t^2$)\\
 	superfluousIncludes(theory, futureUse) = O($t^2$)\\
 	removal = O(t)\\
 	redundantIncludes(theory, futureUse) = O(t)
 } 
 \caption{optimize(theory, futureUse) - memory}
\end{algorithm}

Assuming none of the library functions need larger memory, the overall memory usage lies in O($t^2$).

\subsubsection{Graph optimization}
\begin{algorithm}[H]
\label{alg:graph}
 \KwData{\\
 	theoryGraph = O(t)\\
 	futureUses = O($t^2$)\\
 	replacements = O($t^2$)\\
 	theoryGraphRev = O(t)\\
 	theory = O(1)\\
 	optimize(theory, futureUse) = O($t^2$)
 }
 \caption{optimizeGraph(theorygraph) - memory}
\end{algorithm}

Assuming none of the library functions need larger memory, the overall memory usage lies in O($t^2$).