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Diffstat (limited to 'tex/thesis')
| -rw-r--r-- | tex/thesis/Makefile | 3 | ||||
| -rw-r--r-- | tex/thesis/contribution/contribution.tex | 423 | ||||
| -rw-r--r-- | tex/thesis/references.bib | 6 | ||||
| -rw-r--r-- | tex/thesis/thesis.tex | 6 | ||||
| -rw-r--r-- | tex/thesis/usydthesis.cls | 3 |
5 files changed, 437 insertions, 4 deletions
diff --git a/tex/thesis/Makefile b/tex/thesis/Makefile index 1ccb5c4..8609c3a 100644 --- a/tex/thesis/Makefile +++ b/tex/thesis/Makefile @@ -3,7 +3,7 @@ PROJ = thesis .PHONY: all pdf wc clean -all: pdf +all: pdf clean pdf: pdflatex $(PROJ) @@ -26,4 +26,3 @@ clean: rm -f `find . -name '*.lot'` rm -f `find . -name '*.out'` rm -f `find . -name '*~'` - diff --git a/tex/thesis/contribution/contribution.tex b/tex/thesis/contribution/contribution.tex new file mode 100644 index 0000000..6d4aeca --- /dev/null +++ b/tex/thesis/contribution/contribution.tex @@ -0,0 +1,423 @@ +\floatname{algorithm}{Listing} + +\newcommand\stable{\mathsf{stable}} +\newcommand\eval{\mathsf{\textsc{eval}}} +\newcommand\solve{\mathsf{\textsc{solve}}} +\newcommand\system{\mathsf{system}} +\algblockx[Globals]{Globals}{EndGlobals}{\textbf{Globals:\\}}{} +\algblockx[Assume]{Assumptions}{EndAssumptions}{\textbf{Assume:\\}}{} + + +\chapter{Contribution} \label{chap:contribution} + +The main theoretical contribution of this paper is an improvement on a +$\max$-strategy improvement algorithm for solving fixpoint equations +over the integers with monotonic +operators\cite{Gawlitza:2007:PFC:1762174.1762203}. The original +algorithm is presented in Section \ref{section:basic-algorithm}. We +employ the ideas of Seidl, et al. to design an algorithm which runs in +considerably less time than the existing solver. + +In this chapter we will begin by presenting the Work-list Depth First +Search (W-DFS) fixpoint algorithm developed by Seidl, et +al.\cite{DBLP:tr/trier/MI96-11}. We will then present a modification +to the algorithm to allow it to perform $\max$-strategy iteration +rather than fixpoint iteration. The chapter will then conclude with +our Local Demand-driven Strategy Improvement (LDSI) algorithm. + +The existing algorithm as presented in Section +\ref{section:basic-algorithm} consists of two iterative operations: +fixpoint iteration and max-strategy iteration. Each of these +operations consists of naively ``evaluating'' the system repeatedly +until a further evaluation yields no change. It is shown by Gawlitza, +et al. that these iterations must converge in a finite number of +steps\cite{Gawlitza:2007:PFC:1762174.1762203}, but in practice this +naive approach performs many more operations than are necessary, in +many cases merely re-calculating results which are already known. + +By making use of some data-dependencies within the equation systems it +is possible to reduce the amount of work that is to be done quite +considerably. + +In order to aid our explanation of these algorithms we will now define +a few terms and notations. All variables are taken from the set $X$ +and all values from the set $\D$. + +\begin{definition} + \textbf{Variable Assignments:} $X \to \D$. A function from a + variable to a value in our domain. An underlined value + (eg. $\underline{\infty}$) indicates a variable assignment mapping + everything to that value. Variable assignments may be combined with + $\oplus$ in the following way: + \begin{align*} + \rho \oplus \varrho = \left\{\begin{array}{lc} + \varrho(x) & x \in \mathsf{domain}(\varrho) \\ + \rho(x) & \mbox{otherwise} + \end{array}\right. + \end{align*} +\end{definition} + +\begin{definition} + \textbf{Expressions:} For the purposes of this discussion we will + consider expressions, $e \in E$, as $e : (X \to \D) \to \D$, a + mapping from a variable assignment to the expression's value in that + assignment. +\end{definition} + +\begin{definition} + \textbf{Equation System:} $\{ x = e_x \mid x \in X, e_x \in E \}$. An + equation system can also be considered as a function $\varepsilon : + (X \to D) \to (X \to D)$; $\varepsilon[\rho](x) = e_x(\rho)$. +\end{definition} + +\begin{definition} + \textbf{Dependencies:} A variable $x$ is said to \emph{depend on} + $y$ if a change to the value of $y$ induces a change in the value of + $x$. +\end{definition} + +\section{Fixpoint Iteration} +\subsection{Kleene Iteration} + +A simple approach to fixpoint iteration over monotonic equations is to +simply iterate over the system repeatedly until a reevaluation results +in no change to the values of any variables. This approach will always +reach the least/greatest solution if there is one to be found, but it +will often perform many more evaluations than are necessary. This +algorithm is presented in Listing \ref{algo:kleene}. + +\begin{algorithm} + \begin{algorithmic} + \Assumptions + \begin{tabularx}{0.9\textwidth}{rX} + $\rho $:&$ X \to \D$, a variable assignment \\ + $\varepsilon $:&$ (X \to \D) \to (X \to \D)$, an equation system + \end{tabularx} + \EndAssumptions + + \State $n = 0$ + \State $\rho_0 = \underline{\infty}$ + \Repeat + \State $\rho_{n+1} = \varepsilon[ \rho_{n} ]$ + \State $n = n + 1$ + \Until {$\rho_{n-1} = \rho_n$} + \State \Return $\rho_n$ + \end{algorithmic} + \caption{The Kleene iteration algorithm for solving fixpoint + equations for their greatest solutions.} + \label{algo:kleene} +\end{algorithm} + +For each iteration the entire system is evaluated, irrespective of +whether it could possibly have changed value. This results in a +considerable inefficiency in practice and thus an approach which can +evaluate smaller portions of the system in each iteration would be a +sensible improvement. + +\subsection{W-DFS algorithm} + +The W-DFS algorithm presented by Seidl, et al. takes into account some +form of data-dependencies as it solves the system. This gives it the +ability to leave portions of the system unevaluated when it is certain +that those values have not changed. + +\begin{algorithm} + \begin{algorithmic} + \Globals + \begin{tabularx}{0.9\textwidth}{rX} + $D : X \to \D$ & a mapping from variables to their current values, starting + at $\{ x \mapsto \infty | \forall x \in X \}$ \\ + I & A mapping from a variable to the variables which depend on + it in their evaluation \\ + stable & The set of all variables whose values have stabilised \\ + system & The equation system, a mapping from a variable to its + associated function \\ + \end{tabularx} + \EndGlobals + something something + \end{algorithmic} + + \begin{algorithmic} + \Function {eval} {$x$, $y$} + \Comment{Evaluate $y$ for its value and note that when $y$ + changes, $x$ must be re-evaluated} + \State $\solve(y)$ + \State $I[y] = I[y] \cup \{x\}$ + \State \Return $D[y]$ + \EndFunction + \end{algorithmic} + + \begin{algorithmic} + \Function {solve} {$x$} + \Comment{Solve a specific variable and place its value in $D$} + \If {$x \not \in \stable$} + \State $f = \system[x]$ + \State $\stable = \stable \cup \{x\}$ + \State $v = f( \lambda y . \eval(x, y) )$ + \If {$v \ne D[x]$} + \State $D = \{ x \mapsto v, \alpha \mapsto D[\alpha] \} + \forall \alpha \ne x$ + \State $W = I[x]$ + \State $I(x) = \emptyset$ + \State $\stable = \stable \backslash W$ + \For {$v \in W$} + \State $\solve(v)$ + \EndFor + \EndIf + \EndIf + \EndFunction + \end{algorithmic} + + \caption{The W-DFS alluded to in \cite{DBLP:tr/trier/MI96-11} and + presented in \cite{fixpoint-slides}, modified to find + greatest-fixpoints of monotonic fixpoint equations} + \label{algo:w-dfs} +\end{algorithm} + + + +\section{$\max$-strategy Iteration} +\subsection{Naive approach} + +TODO: Explanation of the naive approach + +\subsection{Adapted W-DFS algorithm} + +The $\max$-strategy iteration can be viewed as a fixpoint problem. We +are attempting to find a strategy, $\sigma: E_{\max} \to E$ that will +result in the greatest value for each $e \in E_{\max}$. Therefore if +we consider our ``variables'' to be $\max$-expressions, our ``values'' +to be their subexpressions and our ``comparison'' to be carried out +using the result of evaluating the system with that strategy. + +This, then, allows us to use the W-DFS algorithm to re-evaluate only +those parts of the strategy which have changed. Listing +\ref{algo:w-dfs-max} presents this variation on W-DFS. + +\begin{algorithm} + \begin{algorithmic} + \Globals + \begin{tabularx}{0.9\textwidth}{rX} + $\sigma$ & A mapping from $\max$-expressions to their current + sub-expressions, starting by mapping to the first + sub-expression \\ + I & A mapping from a $\max$-expression to the sub-expressions + which depend on it in their evaluation \\ + stable & The set of all $\max$-expressions whose strategies have + stabilised \\ + system & The equation system, a mapping from a variable to its + associated function \\ + bestStrategy & A function $(E_{\max}, (X \to D)) \to E$ mapping + from an expression and a variable \\& assignment to the greatest + subexpression in that context + \end{tabularx} + \EndGlobals + + \Function {eval} {$x$, $y$} + \Comment{Evaluate $y$ for its value and note that when $y$ + changes, $x$ must be re-evaluated} + \State $\solve(y)$ + \State $I[y] = I[y] \cup \{x\}$ + \State \Return $\sigma[y]$ + \EndFunction + + \Function {solve} {$x$} + \Comment{Solve a specific expression and place its value in $\sigma$} + \If {$x \not \in \stable$} + \State $f = \system[x]$ + \State $\stable = \stable \cup \{x\}$ + \State $v = bestStrategy(x, \lambda y . \eval(x, y))$ + \If {$v \ne \sigma[x]$} + \State $\sigma = \{ x \mapsto v, \alpha \mapsto \sigma[\alpha] + \} \forall \alpha \ne x $ + \State $W = I[x]$ + \State $I(x) = \emptyset$ + \State $\stable = \stable \backslash W$ + \For {$v \in W$} + \State $\solve(v)$ + \EndFor + \EndIf + \EndIf + \EndFunction + \end{algorithmic} + + \caption{W-DFS, this time modified to find the best $\max$-strategy.} + \label{algo:w-dfs-max} +\end{algorithm} + + +\section{Combined W-DFS} + +W-DFS can be used to speed up both the $\max$-strategy iteration and +the fixpoint iteration as two independent processes, but each +fixpoint-iteration step still requires at least one complete +evaluation of the equation system per $\max$-strategy improvement +step. Ideally we would be able to adapt the W-DFS algorithm so that +the fixpoint-iteration and $\max$-strategy iteration steps could +provide some information to each other about what values have changed +so that at each step only a subset of the entire system would have to +be evaluated. + +The new algorithm, \emph{Combined W-DFS} seeks to do this. By adding +an ``invalidate'' method to both W-DFS instances we provide an +interface for the two sides of the algorithm to indicate which values +have changed. This gives the other side enough information to avoid +evaluating irrelevant portions of the equation system. + +This algorithm is presented in two parts. Listing +\ref{algo:combined-fixpoint} presents the fixpoint-iteration portion +of the algorithm, while Listing \ref{algo:combined-max} presents the +$\max$-strategy portion. The correctness of this new algorithm is +argued in \ref{sec:combined-correctness}. + + +\begin{algorithm} + \begin{algorithmic} + \Globals + \begin{tabularx}{0.9\textwidth}{rX} + D & A mapping from variables to their current values, starting + at $\{ x \mapsto \infty \}$ \\ + I & A mapping from a variable to the variables which depend on + it in their evaluation \\ + stable & The set of all variables whose values have stabilised \\ + system & The equation system, a mapping from a variable to its + associated function \\ + \end{tabularx} + \EndGlobals + + \Function {eval} {$x$, $y$} + \Comment{Evaluate $y$ for its value and note that when $y$ + changes, $x$ must be re-evaluated} + \State $\solve(y)$ + \State $I[y] = I[y] \cup \{x\}$ + \State \Return $D[y]$ + \EndFunction + + \Function {invalidate} {$x$} + \If {$x \in \stable$} + \State $\stable = \stable \backslash \{x\}$ + \State $D[x] = \infty$ + \State $W = I[x]$ + \State $I[x] = \emptyset$ + \For {$v \in W$} + invalidate(v) + \EndFor + \EndIf + \EndFunction + + \Function {solve} {$x$} + \Comment{Solve a specific expression and place its value in $D$} + \If {$x \not \in \stable$} + \State $f = \system[x]$ + \State $\stable = \stable \cup \{x\}$ + \State $v = f( \lambda y . \eval(x, y) )$ + \If {$v \ne D[x]$} + \State $D = \{ x \mapsto v, \alpha \mapsto D[\alpha] \} + \forall \alpha \ne x$ + \State $W = I[x]$ + \State $I(x) = \emptyset$ + \State strategy::invalidate($x$) + \State $\stable = \stable \backslash W$ + \For {$v \in W$} + \State $\solve(v)$ + \EndFor + \EndIf + \EndIf + \EndFunction + \end{algorithmic} + + \caption{The fixpoint portion of the Combined W-DFS.} + \label{algo:combined-fixpoint} +\end{algorithm} + + +\begin{algorithm} + \begin{algorithmic} + \Globals + \begin{tabularx}{0.9\textwidth}{rX} + $\sigma$ & A mapping from $\max$-expressions to their current + sub-expressions, starting by \\& mapping to the first + sub-expression \\ + D & A \emph{dynamic} variable assignment which will stay updated + as $\sigma$ changes \\ + $I$ & A mapping from a $\max$-expression to the sub-expressions + which depend on it \\& in their evaluation \\ + $I_V$ & A mapping from a variable to the $\max-$ expressions which + depend on it in their \\& evaluation \\ + stable & The set of all $\max$-expressions whose strategies have + stabilised \\ + system & The equation system, a mapping from a variable to its + associated function \\ + bestStrategy & A function $(E_{\max}, (X \to D)) \to E$ mapping + from an expression and a variable \\& assignment to the greatest + subexpression in that context + \end{tabularx} + \EndGlobals + + \Function {eval} {$x$, $y$} + \Comment{Evaluate $y$ for its value and note that when $y$ + changes, $x$ must be re-evaluated} + \State $\solve(y)$ + \State $I[y] = I[y] \cup \{x\}$ + \State \Return $\sigma[y]$ + \EndFunction + + \Function {invalidate} {$x \in X$} \Comment{X is vars} + \State $\stable = \stable \backslash I[x]$ + \State $W = I_v[x]$ + \State $I_V = \emptyset$ + \For {$v \in W$} + \State solve(v) + \EndFor + \EndFunction + + \Function {solve} {$x$} + \Comment{Solve a specific variable and place its value in $\sigma$} + \If {$x \not \in \stable$} + \State $f = \system[x]$ + \State $\stable = \stable \cup \{x\}$ + \State $v = bestStrategy(x, + \lambda y . \eval(x, y))$ + \If {$v \ne \sigma[x]$} + \State $\sigma = \{ x \mapsto v, \alpha \mapsto \sigma[\alpha] + \} \forall \alpha \ne x $ + \State $W = I[x]$ + \State $I(x) = \emptyset$ + \State fixpoint::invalidate$(\mathsf{lookupVarFor}(x))$ + \State $\stable = \stable \backslash W$ + \For {$v \in W$} + \State $\solve(v)$ + \EndFor + \EndIf + \EndIf + \EndFunction + \end{algorithmic} + + \caption{The $\max$-strategy portion of the Combined W-DFS.} + \label{algo:combined-max} +\end{algorithm} + + +\subsection{Correctness} \label{sec:combined-correctness} + +This algorithm relies on the correctness of the underlying W-DFS +algorithm. This algorithm was presented in +\cite{DBLP:tr/trier/MI96-11}. + +If we assume that W-DFS is correct then we only have to prove that the +combination algorithm is correct. For this it is sufficient to show +that the invalidate calls in both directions preserve the correctness +of the original algorithm. + +// TODO finish this. +General idea: +\begin{itemize} + \item + Invalidate calls from fixpoint $\to$ max strategy are correct if + the calls the other way are, because it completely re-solves the + equations + \item + Invalidate calls from max strategy $\to$ fixpoint are correct + because they effectively ``reset'' that part of the system, + creating it to be entirely re-calculated. +\end{itemize} diff --git a/tex/thesis/references.bib b/tex/thesis/references.bib index d03c67a..e743703 100644 --- a/tex/thesis/references.bib +++ b/tex/thesis/references.bib @@ -134,4 +134,10 @@ publisher = {OCG}, OPTnote = {}, OPTannote = {} +} +@misc{fixpoint-slides, + author = {Helmut Seidl}, + title = {Lecture Slides in Program Optimisation}, + url = {http://www2.in.tum.de/~seidl/Courses/WS2011/Optimierung/all2011.pdf}, + year = {2011} }
\ No newline at end of file diff --git a/tex/thesis/thesis.tex b/tex/thesis/thesis.tex index 6983d79..8f64c07 100644 --- a/tex/thesis/thesis.tex +++ b/tex/thesis/thesis.tex @@ -37,6 +37,7 @@ \usepackage{xcolor} \usepackage{xspace} \usepackage{stmaryrd,tikz} +\usepackage{tabularx} \usepackage{float} \floatstyle{boxed} @@ -44,6 +45,8 @@ \lstset{language={Python}, literate={<=}{$\le$}{1}}%, frame=tlrb} \usetikzlibrary{arrows} + +\newcommand\D{\mathbb{D}} \newcommand\R{\mathbb{R}} \newcommand\Z{\mathbb{Z}} \newcommand\CZ{\overline{\Z}} @@ -136,6 +139,7 @@ \input{introduction/introduction.tex} \input{litreview/litreview.tex} +\input{contribution/contribution.tex} \input{evaluation/evaluation.tex} \input{experiments/experiments.tex} \input{results/results.tex} @@ -146,7 +150,7 @@ % End % Bibliography -\bibliographystyle{style/mybibstyle} +\bibliographystyle{abbrvnat} %style/mybibstyle} { \setstretch{1.25} \cleardoublepage diff --git a/tex/thesis/usydthesis.cls b/tex/thesis/usydthesis.cls index e18eb53..5b1a4d4 100644 --- a/tex/thesis/usydthesis.cls +++ b/tex/thesis/usydthesis.cls @@ -32,7 +32,8 @@ \usepackage{amssymb,amsmath} \newcommand{\theHalgorithm}{\arabic{algorithm}} \usepackage[ruled]{algorithm} -\usepackage{algorithmic} +\usepackage{algorithmicx} +\usepackage{algpseudocode} % Fancy footnotes \usepackage[stable]{style/footmisc} |