Merge branch 'master' of bitbucket.org:winfried/hpc
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Armin Friedl
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reduce/report/nodes
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reduce/report/nodes
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reduce/report/report.tex
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\documentclass[a4paper, DIV=12]{scrartcl}
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\newrefformat{tbl}{\hyperref[#1]{Table~\ref*{#1}}}
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\newrefformat{lst}{\hyperref[#1]{Listing~\ref*{#1}}}
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\lstset{ %
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backgroundcolor=\color{white}, % choose the background color; you must add \usepackage{color} or
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basicstyle=\ttfamily, % the size of the fonts that are used for the code
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escapeinside={(*}{*)}, % if you want to add LaTeX within your code
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rulecolor=\color{black}, % if not set, the frame-color may be changed on line-breaks within not-black text (e.g. comments (green here))
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title=\lstname, % show the filename of files included with \lstinputlisting; also try caption instead of title
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emph=[3]{int:,array,set,of,int,if,then,else,constraint,var,union,endif,function,where,in,div,predicate,let,opt,full,format,def,for,True,False,return,or},
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emphstyle=[3]\color{ForestGreen},
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emph=[2]{length,max,forall,startEmptyBuffer,fix,startEmptyBufferShow,exactly,cumulative,occurs,deopt,sum,,all},
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stringstyle =\color{red},
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}
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\begin{document}
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\subject{High Performance Computing}
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\title{Reduction trees for MPI Reductions}
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\subtitle{Project 2}
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\author{Johannes Winklehner\\1226104 \and Armin Friedl\\1053597}
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\date{\today}
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\maketitle
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\tableofcontents
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\newpage
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\section{Problem Description}
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\label{sec:description}
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The purpose of this project is to compare different implementations of the collective communication call MPI\_Reduce.
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The compared implementations should all use different forms of Tree Reduction algorithms.
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As a baseline for the comparison serves a given implementation of the MPI standard, which is in our case NEC MPI.
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\begin{description}
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\item[Binomial Tree]
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A binomial tree has a non-fixed degree where each tree $B_i$ has exactly $i$ subtrees of size $B_0$ to $B_{i-1}$.
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The number of nodes in such a tree is equal to $2^i$ and the depth is $i$.
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\item[Fibonacci Tree]
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The Fibonacci tree uses a fixed degree of $2$ where a tree of size $F_i$ has one subtree of size $T_{i-1}$ and one of $T_{i-2}$.
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Therefore the number of nodes in this kind of tree is $fib(i+3)-1$ using the Fibonacci function $fib(x) = fib(x-1)+fib(x-2)$ and its depth is as well $i$.
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\item[Binary Tree]
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The binary tree used for reduction is a common complete binary tree where a tree $T_i$ has two subtrees $T_{i-1}$.
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Such a tree has $2^{i+1}-1$ nodes and its depth is as for the other types $i$.
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\end{description}
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\begin{center}
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\begin{minipage}{.4\textwidth}
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\begin{tikzpicture}
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\node [circle,draw]{$B_i$}
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child { node [circle,draw]{$B_{i-1}$}}
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child {node [circle,draw] {$B_{i-2}$}}
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child {node {\dots} edge from parent[draw=none]}
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child {node [circle,draw] {$B_0$}};
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\end{tikzpicture}
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%\caption{Binomial Tree of size $i$}
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\end{minipage}
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\begin{minipage}{.2\textwidth}
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\begin{tikzpicture}
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\node [circle,draw]{$F_i$}
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child { node [circle,draw]{$F_{i-1}$}}
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child {node [circle,draw] {$F_{i-2}$}};
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\end{tikzpicture}
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%\caption{Fibonacci Tree of size $i$}
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\end{minipage}
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\begin{minipage}{.2\textwidth}
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\begin{tikzpicture}
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\node [circle,draw]{$T_i$}
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child { node [circle,draw]{$T_{i-1}$}}
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child {node [circle,draw] {$T_{i-2}$}};
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\end{tikzpicture}
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%\caption{Complete Binary Tree of size $i$}
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\end{minipage}
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\end{center}
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All three implementations of the reduce function must use exactly the same interface as the MPI standard defines it.
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This interface is shown in \prettyref{lst:reduce}.
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This requires that all implementations support any arbitrary MPI datatype as well as operations.
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The standard also provides some constraints regarding the associativity and commutativity of executable operations.
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Every MPI operation must be associative, but does not necessarily have to be commutative.
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This means that all results of the operation must be computed in the MPI rank order of all processes.
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\begin{lstlisting}[language=C, caption=MPI Reduce interface, label=lst:reduce]
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int MPI_Reduce(const void *sendbuf, void *recvbuf, int count, MPI_Datatype datatype, MPI_Op op, int root, MPI_Comm comm)
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\end{lstlisting}
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The standard also defines additional features of the reduce function, for example an in place operator for the root process.
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However since those details where not mentioned in the assignment description, we did not consider them as part of the project.
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The basic algorithm for a tree reduction, which will be shown in the next section, is very similar for all kinds of trees and uses Point-to-Point communication between tree nodes.
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The assumption for our implementations to be efficient is that the underlying communication network is fully connected and allows for bidirectional communication.
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\FloatBarrier
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\section{Implemented Algorithms}
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\label{sec:algorithms}
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The basic algorithm for a tree reduction is very simple and is shown in \prettyref{alg:reduce}.
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At first the parent and all child nodes have to be determined to know the communication partners of each process.
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Then each process receives the partial results from all of its children and calculates its own result from the received data.
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To ensure the correctness of the result for non commutative operations the iteration of child nodes has to be done in rank order.
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Processes which are leaf nodes in the tree have no children and therefore skip the receiving part of the algorithm.
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If a process has a parent and is therefore not the root process, it sends its result to the determined parent node.
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However if the process is the root process the reduction is finished and can be returned.
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\begin{algorithm}
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\caption{Tree Reduce}
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\label{alg:reduce}
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\KwIn{An array $\vec{a}$ of a given $datatype$ with size $count$ for each process}
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\KwOut{The result of the reduction on the $root$ process}
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determine $parent$ and $children$\;
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$result = \vec{a}$\;
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\ForAll{child in children}{
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receive $result$ from $child$\;
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$result =$ local reduce of received array and $result$\;
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}
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\eIf{parent exists}{
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send $result$ to $parent$\;
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}{
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$output = result$\;
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}
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\end{algorithm}
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The calculation of the parent and child nodes is the only aspect which has to be changed for all possible kinds of trees.
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However there are of course certain optimizations possible to use some knowledge of a concrete tree.
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\FloatBarrier
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\section{Implementation Details}
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\label{sec:kernels}
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\FloatBarrier
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\section{Results}
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\label{sec:results}
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\FloatBarrier
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\section{Analysis}
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\label{sec:analysis}
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\section{Appendix}
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\lstinputlisting[language=C]{../binom_reduce.c}
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\lstinputlisting[language=C]{../fib_reduce.c}
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\lstinputlisting[language=C]{../bin_reduce.c}
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\end{document}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: t
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%%% End:
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