THREADING(1) User Contributed Perl Documentation THREADING(1)NAMEPDL::Threading - Tutorial for PDL's Threading feature
INTRODUCTION
One of the most powerful features of PDL is threading, which can
produce very compact and very fast PDL code by avoiding multiple nested
for loops that C and BASIC users may be familiar with. The trouble is
that it can take some getting used to, and new users may not appreciate
the benefits of threading.
Other vector based languages, such as MATLAB, use a subset of threading
techniques, but PDL shines by completely generalizing them for all
sorts of vector-based applications.
TERMINOLOGY: PIDDLE
MATLAB typically refers to vectors, matrices, and arrays. Perl already
has arrays, and the terms "vector" and "matrix" typically refer to one-
and two-dimensional collections of data. Having no good term to
describe their object, PDL developers coined the term "piddle" to give
a name to their data type.
A piddle consists of a series of numbers organized as an N-dimensional
data set. Piddles provide efficient storage and fast computation of
large N-dimensional matrices. They are highly optimized for numerical
work.
THINKING IN TERMS OF THREADING
If you have used PDL for a little while already, you may have been
using threading without realising it. Start the PDL shell (type
"perldl" or "pdl2" on a terminal). Most examples in this tutorial use
the PDL shell. Make sure that PDL::NiceSlice and PDL::AutoLoader are
enabled. For example:
% pdl2
perlDL shell v1.352
...
ReadLines, NiceSlice, MultiLines enabled
...
Note: AutoLoader not enabled ('use PDL::AutoLoader' recommended)
pdl>
In this example, NiceSlice was automatically enabled, but AutoLoader
was not. To enable it, type "use PDL::AutoLoader".
Let's start with a two-dimensional piddle:
pdl> $a = sequence(11,9)
pdl> p $a
[
[ 0 1 2 3 4 5 6 7 8 9 10]
[11 12 13 14 15 16 17 18 19 20 21]
[22 23 24 25 26 27 28 29 30 31 32]
[33 34 35 36 37 38 39 40 41 42 43]
[44 45 46 47 48 49 50 51 52 53 54]
[55 56 57 58 59 60 61 62 63 64 65]
[66 67 68 69 70 71 72 73 74 75 76]
[77 78 79 80 81 82 83 84 85 86 87]
[88 89 90 91 92 93 94 95 96 97 98]
]
The "info" method gives you basic information about a piddle:
pdl> p $a->info
PDL: Double D [11,9]
This tells us that $a is an 11 x 9 piddle composed of double precision
numbers. If we wanted to add 3 to all elements in an "n x m" piddle, a
traditional language would use two nested for-loops:
# Pseudo-code. Traditional way to add 3 to an array.
for (x=0; x < n; x++) {
for (y=0; y < m; y++) {
a(x,y) = a(x,y) + 3
}
}
Note: Notice that indices start at 0, as in Perl, C and Java (and
unlike MATLAB and IDL).
But with PDL, we can just write:
pdl> $b = $a + 3
pdl> p $b
[
[ 3 4 5 6 7 8 9 10 11 12 13]
[ 14 15 16 17 18 19 20 21 22 23 24]
[ 25 26 27 28 29 30 31 32 33 34 35]
[ 36 37 38 39 40 41 42 43 44 45 46]
[ 47 48 49 50 51 52 53 54 55 56 57]
[ 58 59 60 61 62 63 64 65 66 67 68]
[ 69 70 71 72 73 74 75 76 77 78 79]
[ 80 81 82 83 84 85 86 87 88 89 90]
[ 91 92 93 94 95 96 97 98 99 100 101]
]
This is the simplest example of threading, and it is something that all
numerical software tools do. The "+ 3" operation was automatically
applied along two dimensions. Now suppose you want to to subtract a
line from every row in $a:
pdl> $line = sequence(11)
pdl> p $line
[0 1 2 3 4 5 6 7 8 9 10]
pdl> $c = $a - $line
pdl> p $c
[
[ 0 0 0 0 0 0 0 0 0 0 0]
[11 11 11 11 11 11 11 11 11 11 11]
[22 22 22 22 22 22 22 22 22 22 22]
[33 33 33 33 33 33 33 33 33 33 33]
[44 44 44 44 44 44 44 44 44 44 44]
[55 55 55 55 55 55 55 55 55 55 55]
[66 66 66 66 66 66 66 66 66 66 66]
[77 77 77 77 77 77 77 77 77 77 77]
[88 88 88 88 88 88 88 88 88 88 88]
]
Two things to note here: First, the value of $a is still the same. Try
"p $a" to check. Second, PDL automatically subtracted $line from each
row in $a. Why did it do that? Let's look at the dimensions of $a,
$line and $c:
pdl> p $line->info => PDL: Double D [11]
pdl> p $a->info => PDL: Double D [11,9]
pdl> p $c->info => PDL: Double D [11,9]
So, both $a and $line have the same number of elements in the 0th
dimension! What PDL then did was thread over the higher dimensions in
$a and repeated the same operation 9 times to all the rows on $a. This
is PDL threading in action.
What if you want to subtract $line from the first line in $a only? You
can do that by specifying the line explicitly:
pdl> $a(:,0) -= $line
pdl> p $a
[
[ 0 0 0 0 0 0 0 0 0 0 0]
[11 12 13 14 15 16 17 18 19 20 21]
[22 23 24 25 26 27 28 29 30 31 32]
[33 34 35 36 37 38 39 40 41 42 43]
[44 45 46 47 48 49 50 51 52 53 54]
[55 56 57 58 59 60 61 62 63 64 65]
[66 67 68 69 70 71 72 73 74 75 76]
[77 78 79 80 81 82 83 84 85 86 87]
[88 89 90 91 92 93 94 95 96 97 98]
]
See PDL::Indexing and PDL::NiceSlice to learn more about specifying
subsets from piddles.
The true power of threading comes when you realise that the piddle can
have any number of dimensions! Let's make a 4 dimensional piddle:
pdl> $piddle_4D = sequence(11,3,7,2)
pdl> $c = $piddle_4D - $line
Now $c is a piddle of the same dimension as $piddle_4D.
pdl> p $piddle_4D->info => PDL: Double D [11,3,7,2]
pdl> p $c->info => PDL: Double D [11,3,7,2]
This time PDL has threaded over three higher dimensions automatically,
subtracting $line all the way.
But, maybe you don't want to subtract from the rows (dimension 0), but
from the columns (dimension 1). How do I subtract a column of numbers
from each column in $a?
pdl> $cols = sequence(9)
pdl> p $a->info => PDL: Double D [11,9]
pdl> p $cols->info => PDL: Double D [9]
Naturally, we can't just type "$a - $cols". The dimensions don't match:
pdl> p $a - $cols
PDL: PDL::Ops::minus(a,b,c): Parameter 'b'
PDL: Mismatched implicit thread dimension 0: should be 11, is 9
How do we tell PDL that we want to subtract from dimension 1 instead?
MANIPULATING DIMENSIONS
There are many PDL functions that let you rearrange the dimensions of
PDL arrays. They are mostly covered in PDL::Slices. The three most
common ones are:
xchg
mv
reorder
Method: "xchg"
The "xchg" method "exchanges" two dimensions in a piddle:
pdl> $a = sequence(6,7,8,9)
pdl> $a_xchg = $a->xchg(0,3)
pdl> p $a->info => PDL: Double D [6,7,8,9]
pdl> p $a_xchg->info => PDL: Double D [9,7,8,6]
| |
V V
(dim 0) (dim 3)
Notice that dimensions 0 and 3 were exchanged without affecting the
other dimensions. Notice also that "xchg" does not alter $a. The
original variable $a remains untouched.
Method: "mv"
The "mv" method "moves" one dimension, in a piddle, shifting other
dimensions as necessary.
pdl> $a = sequence(6,7,8,9) (dim 0)
pdl> $a_mv = $a->mv(0,3) |
pdl> V _____
pdl> p $a->info => PDL: Double D [6,7,8,9]
pdl> p $a_mv->info => PDL: Double D [7,8,9,6]
----- |
V
(dim 3)
Notice that when dimension 0 was moved to position 3, all the other
dimensions had to be shifted as well. Notice also that "mv" does not
alter $a. The original variable $a remains untouched.
Method: "reorder"
The "reorder" method is a generalization of the "xchg" and "mv"
methods. It "reorders" the dimensions in any way you specify:
pdl> $a = sequence(6,7,8,9)
pdl> $a_reorder = $a->reorder(3,0,2,1)
pdl>
pdl> p $a->info => PDL: Double D [6,7,8,9]
pdl> p $a_reorder->info => PDL: Double D [9,6,8,7]
| | | |
V V v V
dimensions: 0 1 2 3
Notice what happened. When we wrote "reorder(3,0,2,1)" we instructed
PDL to:
* Put dimension 3 first.
* Put dimension 0 next.
* Put dimension 2 next.
* Put dimension 1 next.
When you use the "reorder" method, all the dimensions are shuffled.
Notice that "reorder" does not alter $a. The original variable $a
remains untouched.
GOTCHA: LINKING VS ASSIGNMENT
Linking
By default, piddles are linked together so that changes on one will go
back and affect the original as well.
pdl> $a = sequence(4,5)
pdl> $a_xchg = $a->xchg(1,0)
Here, $a_xchg is not a separate object. It is merely a different way of
looking at $a. Any change in $a_xchg will appear in $a as well.
pdl> p $a
[
[ 0 1 2 3]
[ 4 5 6 7]
[ 8 9 10 11]
[12 13 14 15]
[16 17 18 19]
]
pdl> $a_xchg += 3
pdl> p $a
[
[ 3 4 5 6]
[ 7 8 9 10]
[11 12 13 14]
[15 16 17 18]
[19 20 21 22]
]
Assignment
Some times, linking is not the behaviour you want. If you want to make
the piddles independent, use the "copy" method:
pdl> $a = sequence(4,5)
pdl> $a_xchg = $a->copy->xchg(1,0)
Now $a and $a_xchg are completely separate objects:
pdl> p $a
[
[ 0 1 2 3]
[ 4 5 6 7]
[ 8 9 10 11]
[12 13 14 15]
[16 17 18 19]
]
pdl> $a_xchg += 3
pdl> p $a
[
[ 0 1 2 3]
[ 4 5 6 7]
[ 8 9 10 11]
[12 13 14 15]
[16 17 18 19]
]
pdl> $a_xchg
[
[ 3 7 11 15 19]
[ 4 8 12 16 20]
[ 5 9 13 17 21]
[ 6 10 14 18 22]
]
PUTTING IT ALL TOGETHER
Now we are ready to solve the problem that motivated this whole
discussion:
pdl> $a = sequence(11,9)
pdl> $cols = sequence(9)
pdl>
pdl> p $a->info => PDL: Double D [11,9]
pdl> p $cols->info => PDL: Double D [9]
How do we tell PDL to subtract $cols along dimension 1 instead of
dimension 0? The simplest way is to use the "xchg" method and rely on
PDL linking:
pdl> p $a
[
[ 0 1 2 3 4 5 6 7 8 9 10]
[11 12 13 14 15 16 17 18 19 20 21]
[22 23 24 25 26 27 28 29 30 31 32]
[33 34 35 36 37 38 39 40 41 42 43]
[44 45 46 47 48 49 50 51 52 53 54]
[55 56 57 58 59 60 61 62 63 64 65]
[66 67 68 69 70 71 72 73 74 75 76]
[77 78 79 80 81 82 83 84 85 86 87]
[88 89 90 91 92 93 94 95 96 97 98]
]
pdl> $a->xchg(1,0) -= $cols
pdl> p $a
[
[ 0 1 2 3 4 5 6 7 8 9 10]
[10 11 12 13 14 15 16 17 18 19 20]
[20 21 22 23 24 25 26 27 28 29 30]
[30 31 32 33 34 35 36 37 38 39 40]
[40 41 42 43 44 45 46 47 48 49 50]
[50 51 52 53 54 55 56 57 58 59 60]
[60 61 62 63 64 65 66 67 68 69 70]
[70 71 72 73 74 75 76 77 78 79 80]
[80 81 82 83 84 85 86 87 88 89 90]
]
General Strategy:
Move the dimensions you want to operate on to the start of your
piddle's dimension list. Then let PDL thread over the higher
dimensions.
EXAMPLE: CONWAY'S GAME OF LIFE
Okay, enough theory. Let's do something a bit more interesting: We'll
write Conway's Game of Life in PDL and see how powerful PDL can be!
The Game of Life is a simulation run on a big two dimensional grid.
Each cell in the grid can either be alive or dead (represented by 1 or
0). The next generation of cells in the grid is calculated with simple
rules according to the number of living cells in it's immediate
neighbourhood:
1) If an empty cell has exactly three neighbours, a living cell is
generated.
2) If a living cell has less than two neighbours, it dies of
overfeeding.
3) If a living cell has 4 or more neighbours, it dies from starvation.
Only the first generation of cells is determined by the programmer.
After that, the simulation runs completely according to these rules. To
calculate the next generation, you need to look at each cell in the 2D
field (requiring two loops), calculate the number of live cells
adjacent to this cell (requiring another two loops) and then fill the
next generation grid.
Classical implementation
Here's a classic way of writing this program in Perl. We only use PDL
for addressing individual cells.
#!/usr/bin/perl -w
use PDL;
use PDL::NiceSlice;
# Make a board for the game of life.
my $nx = 20;
my $ny = 20;
# Current generation.
my $a = zeroes($nx, $ny);
# Next generation.
my $n = zeroes($nx, $ny);
# Put in a simple glider.
$a(1:3,1:3) .= pdl ( [1,1,1],
[0,0,1],
[0,1,0] );
for (my $i = 0; $i < 100; $i++) {
$n = zeroes($nx, $ny);
$new_a = $a->copy;
for ($x = 0; $x < $nx; $x++) {
for ($y = 0; $y < $ny; $y++) {
# For each cell, look at the surrounding neighbours.
for ($dx = -1; $dx <= 1; $dx++) {
for ($dy = -1; $dy <= 1; $dy++) {
$px = $x + $dx;
$py = $y + $dy;
# Wrap around at the edges.
if ($px < 0) {$px = $nx-1};
if ($py < 0) {$py = $ny-1};
if ($px >= $nx) {$px = 0};
if ($py >= $ny) {$py = 0};
$n($x,$y) .= $n($x,$y) + $a($px,$py);
}
}
# Do not count the central cell itself.
$n($x,$y) -= $a($x,$y);
# Work out if cell lives or dies:
# Dead cell lives if n = 3
# Live cell dies if n is not 2 or 3
if ($a($x,$y) == 1) {
if ($n($x,$y) < 2) {$new_a($x,$y) .= 0};
if ($n($x,$y) > 3) {$new_a($x,$y) .= 0};
} else {
if ($n($x,$y) == 3) {$new_a($x,$y) .= 1}
}
}
}
print $a;
$a = $new_a;
}
If you run this, you will see a small glider crawl diagonally across
the grid of zeroes. On my machine, it prints out a couple of
generations per second.
Threaded PDL implementation
And here's the threaded version in PDL. Just four lines of PDL code,
and one of those is printing out the latest generation!
#!/usr/bin/perl -w
use PDL;
use PDL::NiceSlice;
my $a = zeroes(20,20);
# Put in a simple glider.
$a(1:3,1:3) .= pdl ( [1,1,1],
[0,0,1],
[0,1,0] );
my $n;
for (my $i = 0; $i < 100; $i++) {
# Calculate the number of neighbours per cell.
$n = $a->range(ndcoords($a)-1,3,"periodic")->reorder(2,3,0,1);
$n = $n->sumover->sumover - $a;
# Calculate the next generation.
$a = ((($n == 2) + ($n == 3))* $a) + (($n==3) * !$a);
print $a;
}
The threaded PDL version is much faster:
Classical => 32.79 seconds.
Threaded => 0.41 seconds.
Explanation
How does the threaded version work?
There are many PDL functions designed to help you carry out PDL
threading. In this example, the key functions are:
Method: "range"
At the simplest level, the "range" method is a different way to select
a portion of a piddle. Instead of using the "$a(2,3)" notation, we use
another piddle.
pdl> $a = sequence(6,7)
pdl> p $a
[
[ 0 1 2 3 4 5]
[ 6 7 8 9 10 11]
[12 13 14 15 16 17]
[18 19 20 21 22 23]
[24 25 26 27 28 29]
[30 31 32 33 34 35]
[36 37 38 39 40 41]
]
pdl> p $a->range( pdl [1,2] )
13
pdl> p $a(1,2)
[
[13]
]
At this point, the "range" method looks very similar to a regular PDL
slice. But the "range" method is more general. For example, you can
select several components at once:
pdl> $index = pdl [ [1,2],[2,3],[3,4],[4,5] ]
pdl> p $a->range( $index )
[13 20 27 34]
Additionally, "range" takes a second parameter which determines the
size of the chunk to return:
pdl> $size = 3
pdl> p $a->range( pdl([1,2]) , $size )
[
[13 14 15]
[19 20 21]
[25 26 27]
]
We can use this to select one or more 3x3 boxes.
Finally, "range" can take a third parameter called the "boundary"
condition. It tells PDL what to do if the size box you request goes
beyond the edge of the piddle. We won't go over all the options. We'll
just say that the option "periodic" means that the piddle "wraps
around". For example:
pdl> p $a
[
[ 0 1 2 3 4 5]
[ 6 7 8 9 10 11]
[12 13 14 15 16 17]
[18 19 20 21 22 23]
[24 25 26 27 28 29]
[30 31 32 33 34 35]
[36 37 38 39 40 41]
]
pdl> $size = 3
pdl> p $a->range( pdl([4,2]) , $size , "periodic" )
[
[16 17 12]
[22 23 18]
[28 29 24]
]
pdl> p $a->range( pdl([5,2]) , $size , "periodic" )
[
[17 12 13]
[23 18 19]
[29 24 25]
]
Notice how the box wraps around the boundary of the piddle.
Method: "ndcoords"
The "ndcoords" method is a convenience method that returns an
enumerated list of coordinates suitable for use with the "range"
method.
pdl> p $piddle = sequence(3,3)
[
[0 1 2]
[3 4 5]
[6 7 8]
]
pdl> p ndcoords($piddle)
[
[
[0 0]
[1 0]
[2 0]
]
[
[0 1]
[1 1]
[2 1]
]
[
[0 2]
[1 2]
[2 2]
]
]
This can be a little hard to read. Basically it's saying that the
coordinates for every element in $piddle is given by:
(0,0) (1,0) (2,0)
(1,0) (1,1) (2,1)
(2,0) (2,1) (2,2)
Combining "range" and "ndcoords"
What really matters is that "ndcoords" is designed to work together
with "range", with no $size parameter, you get the same piddle back.
pdl> p $piddle
[
[0 1 2]
[3 4 5]
[6 7 8]
]
pdl> p $piddle->range( ndcoords($piddle) )
[
[0 1 2]
[3 4 5]
[6 7 8]
]
Why would this be useful? Because now we can ask for a series of
"boxes" for the entire piddle. For example, 2x2 boxes:
pdl> p $piddle->range( ndcoords($piddle) , 2 , "periodic" )
The output of this function is difficult to read because the "boxes"
along the last two dimension. We can make the result more readable by
rearranging the dimensions:
pdl> p $piddle->range( ndcoords($piddle) , 2 , "periodic" )->reorder(2,3,0,1)
[
[
[
[0 1]
[3 4]
]
[
[1 2]
[4 5]
]
...
]
Here you can see more clearly that
[0 1]
[3 4]
Is the 2x2 box starting with the (0,0) element of $piddle.
We are not done yet. For the game of life, we want 3x3 boxes from $a:
pdl> p $a
[
[ 0 1 2 3 4 5]
[ 6 7 8 9 10 11]
[12 13 14 15 16 17]
[18 19 20 21 22 23]
[24 25 26 27 28 29]
[30 31 32 33 34 35]
[36 37 38 39 40 41]
]
pdl> p $a->range( ndcoords($a) , 3 , "periodic" )->reorder(2,3,0,1)
[
[
[
[ 0 1 2]
[ 6 7 8]
[12 13 14]
]
...
]
We can confirm that this is the 3x3 box starting with the (0,0) element
of $a. But there is one problem. We actually want the 3x3 box to be
centered on (0,0). That's not a problem. Just subtract 1 from all the
coordinates in "ndcoords($a)". Remember that the "periodic" option
takes care of making everything wrap around.
pdl> p $a->range( ndcoords($a) - 1 , 3 , "periodic" )->reorder(2,3,0,1)
[
[
[
[41 36 37]
[ 5 0 1]
[11 6 7]
]
[
[36 37 38]
[ 0 1 2]
[ 6 7 8]
]
...
Now we see a 3x3 box with the (0,0) element in the centre of the box.
Method: "sumover"
The "sumover" method adds along only the first dimension. If we apply
it twice, we will be adding all the elements of each 3x3 box.
pdl> $n = $a->range(ndcoords($a)-1,3,"periodic")->reorder(2,3,0,1)
pdl> p $n
[
[
[
[41 36 37]
[ 5 0 1]
[11 6 7]
]
[
[36 37 38]
[ 0 1 2]
[ 6 7 8]
]
...
pdl> p $n->sumover->sumover
[
[144 135 144 153 162 153]
[ 72 63 72 81 90 81]
[126 117 126 135 144 135]
[180 171 180 189 198 189]
[234 225 234 243 252 243]
[288 279 288 297 306 297]
[216 207 216 225 234 225]
]
Use a calculator to confirm that 144 is the sum of all the elements in
the first 3x3 box and 135 is the sum of all the elements in the second
3x3 box.
Counting neighbours
We are almost there!
Adding up all the elements in a 3x3 box is not quite what we want. We
don't want to count the center box. Fortunately, this is an easy fix:
pdl> p $n->sumover->sumover - $a
[
[144 134 142 150 158 148]
[ 66 56 64 72 80 70]
[114 104 112 120 128 118]
[162 152 160 168 176 166]
[210 200 208 216 224 214]
[258 248 256 264 272 262]
[180 170 178 186 194 184]
]
When applied to Conway's Game of Life, this will tell us how many
living neighbours each cell has:
pdl> $a = zeroes(10,10)
pdl> $a(1:3,1:3) .= pdl ( [1,1,1],
..( > [0,0,1],
..( > [0,1,0] )
pdl> p $a
[
[0 0 0 0 0 0 0 0 0 0]
[0 1 1 1 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
]
pdl> $n = $a->range(ndcoords($a)-1,3,"periodic")->reorder(2,3,0,1)
pdl> $n = $n->sumover->sumover - $a
pdl> p $n
[
[1 2 3 2 1 0 0 0 0 0]
[1 1 3 2 2 0 0 0 0 0]
[1 3 5 3 2 0 0 0 0 0]
[0 1 1 2 1 0 0 0 0 0]
[0 1 1 1 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
]
For example, this tells us that cell (0,0) has 1 living neighbour,
while cell (2,2) has 5 living neighbours.
Calculating the next generation
At this point, the variable $n has the number of living neighbours for
every cell. Now we apply the rules of the game of life to calculate the
next generation.
If an empty cell has exactly three neighbours, a living cell is
generated.
Get a list of cells that have exactly three neighbours:
pdl> p ($n == 3)
[
[0 0 1 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0]
[0 1 0 1 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
]
Get a list of empty cells that have exactly three neighbours:
pdl> p ($n == 3) * !$a
If a living cell has less than 2 or more than 3 neighbours, it dies.
Get a list of cells that have exactly 2 or 3 neighbours:
pdl> p (($n == 2) + ($n == 3))
[
[0 1 1 1 0 0 0 0 0 0]
[0 0 1 1 1 0 0 0 0 0]
[0 1 0 1 1 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
]
Get a list of living cells that have exactly 2 or 3 neighbours:
pdl> p (($n == 2) + ($n == 3)) * $a
Putting it all together, the next generation is:
pdl> $a = ((($n == 2) + ($n == 3)) * $a) + (($n == 3) * !$a)
pdl> p $a
[
[0 0 1 0 0 0 0 0 0 0]
[0 0 1 1 0 0 0 0 0 0]
[0 1 0 1 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
]
Bonus feature: Graphics!
If you have PDL::Graphics::TriD installed, you can make a graphical
version of the program by just changing three lines:
#!/usr/bin/perl
use PDL;
use PDL::NiceSlice;
use PDL::Graphics::TriD;
my $a = zeroes(20,20);
# Put in a simple glider.
$a(1:3,1:3) .= pdl ( [1,1,1],
[0,0,1],
[0,1,0] );
my $n;
for (my $i = 0; $i < 100; $i++) {
# Calculate the number of neighbours per cell.
$n = $a->range(ndcoords($a)-1,3,"periodic")->reorder(2,3,0,1);
$n = $n->sumover->sumover - $a;
# Calculate the next generation.
$a = ((($n == 2) + ($n == 3))* $a) + (($n==3) * !$a);
# Display.
nokeeptwiddling3d();
imagrgb [$a];
}
But if we really want to see something interesting, we should make a
few more changes:
1) Start with a random collection of 1's and 0's.
2) Make the grid larger.
3) Add a small timeout so we can see the game evolve better.
4) Use a while loop so that the program can run as long as it needs to.
#!/usr/bin/perl
use PDL;
use PDL::NiceSlice;
use PDL::Graphics::TriD;
use Time::HiRes qw(usleep);
my $a = random(100,100);
$a = ($a < 0.5);
my $n;
while (1) {
# Calculate the number of neighbours per cell.
$n = $a->range(ndcoords($a)-1,3,"periodic")->reorder(2,3,0,1);
$n = $n->sumover->sumover - $a;
# Calculate the next generation.
$a = ((($n == 2) + ($n == 3))* $a) + (($n==3) * !$a);
# Display.
nokeeptwiddling3d();
imagrgb [$a];
# Sleep for 0.1 seconds.
usleep(100000);
}
CONCLUSION: GENERAL STRATEGY
The general strategy is: Move the dimensions you want to operate on to
the start of your piddle's dimension list. Then let PDL thread over the
higher dimensions.
Threading is a powerful tool that helps eliminate for-loops and can
make your code more concise. Hopefully this tutorial has shown why it
is worth getting to grips with threading in PDL.
COPYRIGHT
Copyright 2010 Matthew Kenworthy (kenworthy@strw.leidenuniv.nl) and
Daniel Carrera (dcarrera@gmail.com). You can distribute and/or modify
this document under the same terms as the current Perl license.
See: http://dev.perl.org/licenses/
perl v5.18.1 2013-05-12 THREADING(1)
