From 6dd6de8e052da23ebd254828437f47d2d610a9c5 Mon Sep 17 00:00:00 2001
From: Rob van Nieuwpoort <nieuwpoort@astron.nl>
Date: Wed, 24 Jun 2009 13:27:27 +0000
Subject: [PATCH] Bug 1198: performance results
---
.gitattributes | 1 -
doc/papers/2010/SPM/Makefile | 2 +-
.../2010/SPM/figures/correlation-triangle.fig | 88 --
.../2010/SPM/figures/correlation-triangle.pdf | Bin 6558 -> 9631 bytes
.../2010/SPM/figures/correlation-triangle.svg | 1043 +++++++++--------
doc/papers/2010/SPM/spm.tex | 156 +--
6 files changed, 604 insertions(+), 686 deletions(-)
delete mode 100644 doc/papers/2010/SPM/figures/correlation-triangle.fig
diff --git a/.gitattributes b/.gitattributes
index 35b3d0bdb39..00f10aea77e 100644
--- a/.gitattributes
+++ b/.gitattributes
@@ -1526,7 +1526,6 @@ doc/papers/2009/SC-09/transpose.fig -text
doc/papers/2010/SPM/IEEEbib.bst -text
doc/papers/2010/SPM/Makefile -text
doc/papers/2010/SPM/figures/LBA-small.jpg -text
-doc/papers/2010/SPM/figures/correlation-triangle.fig -text
doc/papers/2010/SPM/figures/correlation-triangle.pdf -text
doc/papers/2010/SPM/figures/correlation-triangle.svg -text
doc/papers/2010/SPM/figures/pastedpic_11252008_163347.png -text
diff --git a/doc/papers/2010/SPM/Makefile b/doc/papers/2010/SPM/Makefile
index 8368724ca36..eba742c283b 100644
--- a/doc/papers/2010/SPM/Makefile
+++ b/doc/papers/2010/SPM/Makefile
@@ -2,7 +2,7 @@ TEX_SOURCES = spm.tex
BIB_SOURCES = spm.bib
-FIG_SOURCES = figures/correlation-triangle.fig
+FIG_SOURCES =
JGR_SOURCES =
diff --git a/doc/papers/2010/SPM/figures/correlation-triangle.fig b/doc/papers/2010/SPM/figures/correlation-triangle.fig
deleted file mode 100644
index 2f05d430cde..00000000000
--- a/doc/papers/2010/SPM/figures/correlation-triangle.fig
+++ /dev/null
@@ -1,88 +0,0 @@
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diff --git a/doc/papers/2010/SPM/spm.tex b/doc/papers/2010/SPM/spm.tex
index 6f59ccb2820..e38bf8b8dfd 100644
--- a/doc/papers/2010/SPM/spm.tex
+++ b/doc/papers/2010/SPM/spm.tex
@@ -586,126 +586,82 @@ that share a cache should be tailored accordingly. On GPUs, for
example, this can be done by \emph{coalescing} memory accesses. This
means that different concurrent threads read subsequent memory
locations. This can be counter-intuitive, since traditionally, it was
-better to have linear memory access patterns within a thread.
-In the next section, we explain the techniques described above by applying them to the correlator application.
+better to have linear memory access patterns within a thread. In the
+next section, we explain the techniques described above by applying
+them to the correlator application.
\section{Optimizing the correlator}
+% TODO add text about mapping from alg to arch
+
+\begin{figure}[t]
+\begin{center}
+\includegraphics[width=4.2cm]{figures/correlation-triangle.pdf}
+\end{center}
+\vspace{-0.5cm}
+\caption{An example correlation triangle.}
+\label{fig-correlation}
+\end{figure}
+Although in reality the receivers are dual-polarized, and the samples are complex numbers,
+we use single-polarized real samples in the following example for simplicity.
An unoptimized correlator would read the samples from two receivers and
-multiply them, requiring two memory reads for one complex multiplication.
+multiply them, requiring two memory loads for one multiplication.
The correlator can be optimized by reusing a sample that is read from memory
as often as possible, by using it for multiple correlations (see
-Figure~\ref{fig:correlator-triangle}).
+Figure~\ref{fig-correlation}).
+The figure is triangular, because we compute
+the correlation of each pair of receivers only once. The squares labeled \emph{A} are
+autocorrelations.
+
For example, the samples from receivers 8, 9, 10, and 11 can be correlated
with the samples from receivers 4, 5, 6, and 7 (the red square in the figure),
-using each fetched sample for four complex multiplications.
-This way, eight memory accesses are required for sixteen complex
+reusing each fetched sample four times.
+This way, eight samples are read from memory for 16
multiplications, reducing the amount of memory operations already by a factor
of four.
-Correlating even higher numbers of receivers at one go would reduce the
+Correlating even higher numbers of receivers simultaneously would reduce the
memory bandwidth usage further, but the maximum number of receivers that can
-be correlated this way is quite limited by the size of the register file.
-The accumulated correlations are best kept in registers, and the number of
+be correlated this way is limited by the number of registers that an architecture has.
+The accumulated correlations are also best kept in registers, and the number of
required registers grows rapidly with the number of receiver inputs.
-The example above already requires sixteen complex accumulators.
-Additional registers are needed to load samples from memory.
+The example above already requires 16 accumulators.
+Additional registers are needed to load samples from memory (8 in this case).
To obtain good performance, it is important to tune the tile size to the
architecture.
-Caches and memory prefetch units can also improve the performance.
-However, a cache-size dependent tradeoff must be made.
+%Caches and memory prefetch units can also improve the performance.
+%However, a cache-size dependent tradeoff must be made.
%On the one hand, correlating and integrating over long periods of time
%is good for pipelined FPU operation, on the other hand, the
-
-%- optimaliseren van het algorithme: tiles, etc
-
-% TODO add text about mapping from alg to arch
-
-\begin{figure}[t]
-\begin{center}
-\includegraphics[width=4.2cm]{figures/correlation-triangle.pdf}
-\end{center}
-\vspace{-0.5cm}
-\caption{An example correlation triangle.}
-\label{fig-correlation}
-\end{figure}
-
-An important optimization that we implemented is the reduction
-of memory loads by the correlator.
-A sample can be used multiple times by correlating it
-with the samples from multiple other receivers in the same loop iteration.
-For example, a sample from receiver A in the X polarization
-that is loaded into a register pair can be correlated with the X and
-Y polarizations of receivers B, C and D, using it six times.
-Figure~\ref{fig-correlation} shows how we correlate multiple
-receivers at the same time. Each square represents the XX, XY,
-YX, and YY correlations of the receivers as indicated by row and
-column number. The figure is triangular, because we compute
-the correlation of each pair of receivers only once. The squares labeled \emph{A} are
-autocorrelations, which can be treated specially since they require only half
-the amount of computations.
-The triangle is divided into larger tiles, in this case
-2x3 tiles (the dark gray boxes), but arbitrary sizes are possible.
-A tile is correlated as a unit. For example, the lower
-right-hand-side rectangle correlates receivers 9 and 10 with receivers
-0, 1, and 2.
-
-It is important to tune the tile size to the architecture. We want to
-make the tile size as large as possible, while still fitting in the
-register file. This offers the highest level of data reuse.
-If we have a $w \times h$ tile size, the number of operations is given by $flops = 32wh$.
-The number of bytes that has to loaded from memory is $16(w+h)$.
-The minimum number of registers that is required is $4 (1 + min(w,h)) + 8 w h$.
-This is the total number of registers, including accumulators, while reusing
-registers if a value is no longer needed (hence the $min$ operation). However,
-this formula does not count additional registers that could be needed for data prefetching,
-address calculations and loop counters.
-The number of registers is expressed in single-precision float registers. If an architecture has vector
-registers, the result can be divided by the vector length.
-Table~\ref{tile-size-table} shows the properties of different tile sizes.
-
-Despite the division of the correlation triangle in tiles, there
-still is opportunity for additional data reuse \emph{between} tiles.
-The tiles
+Even when dividing the correlation triangle in tiles there still is
+opportunity for additional data reuse \emph{between} tiles. The tiles
within a row or column in the triangle still need the same samples.
-In
-addition to registers, caches can thus also be used to increase data
-reuse. Since we know exactly what data can be reused at what moment, we
-found it is important to have direct influence on the caches and the thread scheduler. This
-way, we can make sure that tiles in same row or column are calculated
-at the same time by different threads.
-Because the algorithm is
-extremely data intensive, the resulting optimized implementation on
-many-cores is typically limited by the architecture's memory
-bandwidth. The memory aspects of the algorithm are twofold.
-There is an algorithmic part, the tile size, which is limited
-by the number of registers. The second aspect is architectural in nature: the cache
-sizes, cache hierarchy and hit ratio. Together, these two aspects dictate the
-memory bandwidth that is needed to keep the FPUs busy.
-
-\begin{table}
-\begin{center}
-{\small
-\begin{tabular}{l|r|r|r|r}
-tile & floating point & memory loads & arithmetic & minimum nr. \\
-size & operations & (bytes) & intensity & registers (floats) \\
-\hline
-1x1 & 32 & 32 & 1.00 & 16 \\
-1x2 & 64 & 48 & 1.33 & 24 \\
-2x2 & 128 & 64 & 2.00 & 44 \\
-3x2 & 192 & 80 & 2.40 & 60 \\
-3x3 & 288 & 96 & 3.00 & 88 \\
-4x3 & 384 & 112 & 3.43 & 112 \\
-4x4 & 512 & 128 & 4.00 & 148 \\
-\end{tabular}
-} %\small
-\end{center}
-\vspace{-0.5cm}
-\caption{Properties of different tile sizes.}
-\label{tile-size-table}
-\end{table}
+In addition to registers, caches can thus also be used to increase
+data reuse.
+
+%% \begin{table}
+%% \begin{center}
+%% {\small
+%% \begin{tabular}{l|r|r|r|r}
+%% tile & floating point & memory loads & arithmetic & minimum nr. \\
+%% size & operations & (bytes) & intensity & registers (floats) \\
+%% \hline
+%% 1x1 & 32 & 32 & 1.00 & 16 \\
+%% 1x2 & 64 & 48 & 1.33 & 24 \\
+%% 2x2 & 128 & 64 & 2.00 & 44 \\
+%% 3x2 & 192 & 80 & 2.40 & 60 \\
+%% 3x3 & 288 & 96 & 3.00 & 88 \\
+%% 4x3 & 384 & 112 & 3.43 & 112 \\
+%% 4x4 & 512 & 128 & 4.00 & 148 \\
+%% \end{tabular}
+%% } %\small
+%% \end{center}
+%% \vspace{-0.5cm}
+%% \caption{Properties of different tile sizes.}
+%% \label{tile-size-table}
+%% \end{table}
It is important to realize that the
correlator itself is \emph{trivially parallel}, since tens of thousands of
--
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