in reply to Data visualisation.

BrowserUk:

I think I have my code in fairly decent shape, functionality-wise. (It's still ugly right now, though.)

However, to get most of the points in your dataset located, I had to open up the tolerance to accept up to a 15% error. (It misses only 3 points at that setting.)

Do you have a way to check whether the coordinates look reasonable against your original data? I'm curious at how good the program works. I've still got some testing to do, but so far, it's been giving decent results with my test sets.

I've uglified the code a bit more to draw the coordinates in a bitmap so I could do a visual check with my original coordinate sets, and it comes out good on them (three different point sets).

Code in readmore tags...

#!/usr/bin/perl
#
#   trv_slsmn_triangulate.pl <FName>
#
#   Given the distance matrix, compute a set of 2D locations for the p
+oints that yields
#   the specified distance map.
#
#   Since any coordinates will do, we'll first set point A at (0,0), a
+nd point B at (aDb,0).
#   Finally, we'll make sure the angle of point C is counterclockwise 
+from the line AB.
#   Obviously, we have to be sure that aDc+bDc > aDb, or it won't make
+ a triangle, so we
#   have to use a bit of care to select our first points.
#
use strict;
use warnings;
use Image::Magick;

my $tolerance = 0.15;       # 5% error
my @Quadrants = ( [1, 1], [1, -1], [-1, 1], [-1, -1] );

# Prepare data source
my $FName = shift;
my $IFH;
if (defined $FName) {
    open $IFH, '<', $FName;
}
else {
    $IFH = \*DATA;
}

# Read data
my %D;
my ($R, $C) = (0);
my $max_dist = [ 0 ];
my @LOCs;
while (<$IFH>) {
    my @t = split /\s+/, $_;
    $C=0;
    $LOCs[$R] = [ ];
    while (@t) {
        my ($dist, $a, $b);
        $dist = shift @t;
        ($a,$b) = ($R>$C) ? ($C, $R) : ($R, $C);
        if ($a != $b) {
            if (exists $D{$a}{$b}) {
                die "Mismatch ($R,$C)=$dist, but ($C,$R)=$D{$a}{$b}\n"
+ if $dist != $D{$a}{$b};
            }
            $D{$a}{$b} = $dist;
            if ($dist > $$max_dist[0]) {
                $max_dist = [ $dist, $a, $b ];
            }
        }
        ++$C;
    }
    ++$R;
}

# OK, points A and B are the points in max_dist, set their initial loc
+ations:
my ($ptA, $ptB) = ($max_dist->[1], $max_dist->[2]);
$LOCs[$ptA] = [0, 0];
$LOCs[$ptB] = [$max_dist->[0], 0];
print "Point A is $ptA (0, 0); point B is $ptB ($max_dist->[0], 0)\n";
my $dAB = dist($ptA, $ptB);

# For point C, we'll find the point giving the triangle with the large
+st area.
# Presumably, that'll get us a point sufficiently far away from the AB
+ baseline
# to give us the best ability to resolve the other points.
$max_dist = [ 0 ];
for my $i (0 .. $#LOCs) {
    next if defined $LOCs[$i][0]; # skip A and B
    my ($dCA, $dCB) = (dist($i,$ptA), dist($i,$ptB) );
    my $s = ($dCA+$dCB+$dAB)/2.0;
    my $dsq = sqrt( $s * ($s-$dCA) * ($s-$dCB) * ($s-$dAB) );
    #print "$i: dCA=$dCA, dCB=$dCB, Area=$dsq";
    if ($dsq > $max_dist->[0]) {
        $max_dist = [$dsq, $i];
        #print "  new best!";
    }
    #print "\n";
}
my $ptC = $max_dist->[1];

# Find position of c via law of cosines (CAB) and sin(arccos x)=sqrt(1
+-x**2)
my ($dCA, $dCB) = (dist($ptC,$ptA), dist($ptC,$ptB));
print "dCA=$dCA, dAB=$dAB, dCB=$dCB\n";
my $cosCAB = ($dCA*$dCA + $dAB*$dAB - $dCB*$dCB) / (2*$dCA*$dAB);
my $sinCAB = sqrt(1 - $cosCAB**2);
$LOCs[$ptC] = [ $dCA * $cosCAB, $dCA * $sinCAB ];
print "Point C is $ptC ($LOCs[$ptC][0], $LOCs[$ptC][1])\n";

#dump_LOCs();
#dump_distance_map();

# remaining locations can be found by triangulation from the three kno
+wn points
for my $i (0 .. $#LOCs) {
    next if $i==$ptA or $i==$ptB or $i==$ptC;
    my ($dIA, $dIB, $dIC) = (dist($i,$ptA), dist($i,$ptB), dist($i,$pt
+C));

    if ($dIA+$dIB < $dAB) {
        print "$i: ($dIA, $dIB, $dIC) *** Fails triangle inequality **
+*\n";
        next;
    }

    print "$i: ($dIA, $dIB, $dIC)\n";
    my $cosIAB = ($dIA*$dIA + $dAB*$dAB - $dIB*$dIB) / (2*$dIA*$dAB);
    my $sinIAB = sqrt(1 - $cosIAB**2);

    my @best = (1000000);
    for my $q (@Quadrants) {
        my ($x_sign, $y_sign) = @$q;
        my ($x, $y) = ($dIA*$cosIAB*$x_sign, $dIA*$sinIAB*$y_sign);
        my $check = sqrt( ($LOCs[$ptC][0]-$x)**2 + ($LOCs[$ptC][1]-$y)
+**2 );
        my $err = abs($check-$dIC);
        my $err_pct = 100*$err / $dIC;
        if ($err < $dIC*$tolerance) {
            if ($err < $best[0]) {
                @best = ($err, $err_pct, $x, $y);
            }
            print "\t($x, $y) <case 1: $err, $err_pct>\n";
            $LOCs[$i] = [$x, $y];
            #last;
        }
        else {
            print "\t($x, $y) miss ($dIC, $err)\n";
        }
    }
    $LOCs[$i] = [ $best[2], $best[3] ] if @best>1;
}
dump_LOCs();
my ($x_min, $x_max, $y_min, $y_max);
($x_min,$y_min) = ($x_max,$y_max) = @{$LOCs[0]};
for my $i (1 .. $#LOCs) {
    my ($x, $y) = @{$LOCs[$i]};
    next if ! defined $y;
    $x_min = $x if $x_min>$x;
    $y_min = $y if $y_min>$y;
    $x_max = $x if $x_max<$x;
    $y_max = $y if $y_max<$y;
}
print "($x_min,$y_min) - ($x_max,$y_max)\n";
my ($x_offs, $y_offs, $x_scale, $y_scale);
$x_scale = 1400 / ($x_max - $x_min);
$y_scale = 900 / ($y_max - $y_min);
$x_offs = 50 - $x_min;
$y_offs = 50 - $y_min;
if ($x_scale < $y_scale) {
    ($x_scale, $y_scale) = ($x_scale, $x_scale);
}
else {
    ($x_scale, $y_scale) = ($y_scale, $y_scale);
}

print "$x_scale . $y_scale;  $x_offs, $y_offs\n";
for my $i (0 .. $#LOCs) {
    if (defined $LOCs[$i][0]) {
        my ($x,$y) = @{$LOCs[$i]};
        
        my ($x1,$y1,$x2,$y2) = (
            $x_scale*($x+$x_offs),
            1000-$y_scale*($y+$y_offs)
        );
        $LOCs[$i][2]=$x1;
        $LOCs[$i][3]=$y1;
    }
}
my $q = Image::Magick->new;
$q->Set(size=>'1500x1000');
$q->Read('gradient:#000185-#0053f8');
my $pts="$LOCs[$ptA][2],$LOCs[$ptA][3] $LOCs[$ptB][2],$LOCs[$ptB][3]";
$q->Draw(stroke=>'white', fill=>'white', primitive=>'line', points=>$p
+ts);
$pts="$LOCs[$ptA][2],$LOCs[$ptA][3] $LOCs[$ptC][2],$LOCs[$ptC][3]";
$q->Draw(stroke=>'red', fill=>'red', primitive=>'line', points=>$pts);
$pts="$LOCs[$ptB][2],$LOCs[$ptB][3] $LOCs[$ptC][2],$LOCs[$ptC][3]";
$q->Draw(stroke=>'green', fill=>'green', primitive=>'line', points=>$p
+ts);
for my $i (0 .. $#LOCs) {
    if (defined $LOCs[$i][0]) {
        my (undef, undef, $x,$y) = @{$LOCs[$i]};
        
        my ($x1,$y1,$x2,$y2) = ($x-5, $y-5, $x+5, $y+5);

        my $fill = 'black';
        $fill = 'red' if $i eq $ptA;
        $fill = 'green' if $i eq $ptB;
        $fill = 'yellow' if $i eq $ptC;

        my $pts = "$x1,$y1 $x2,$y2";
        print "$i: $pts\n";
        $q->Draw(stroke=>'white', fill=>$fill, primitive=>'rectangle',
+ points=>$pts);
    }
}
$q->Write('map.gif');

sub dist {
    my ($r, $c) = @_;
    return 0 if $r eq $c;

    if (defined($LOCs[$r][0]) and defined($LOCs[$c][0]) ) {
        return sqrt( ($LOCs[$r][0]-$LOCs[$c][0])**2 + ($LOCs[$r][1]-$L
+OCs[$c][1])**2 );
    }

    ($r, $c) = ($c, $r) if $c < $r;
    die "$r,$c entry DNE?\n" unless exists $D{$r}{$c};
    $D{$r}{$c};
}

sub dump_LOCs {
    for my $i (0 .. $#LOCs) {
        print dump_helper($i), ": ";
        if (defined($LOCs[$i][0])) {
            print "($LOCs[$i][0], $LOCs[$i][1])\n";
        }
        else {
            print "undef\n";
        }
    }
}

sub dump_helper {
    my $i = shift;
    return "<A>" if $i eq $ptA;
    return "<B>" if $i eq $ptB;
    return "<C>" if $i eq $ptC;
    return "<$i>";
}

sub dump_distance_map {

    print "    ";
    printf "%6s ", dump_helper($_) for 0 .. $#LOCs;
    print "\n";

    for my $i (0 .. $#LOCs) {
        printf "%-4.4s", dump_helper($i);
        for my $j (0 .. $#LOCs) {
            my $d = dist($i,$j);
            printf "% 6u ", $d;
        }
        print "\n";
    }
}

# Matrix shows distance from pt on left to dest column
__DATA__
0  3812 13902  8619 15811  5015  5230  9615 10624 13346  7575  6170  6
+812  9487 18135  8030  5409 17959 12822 17136  3267 12882 11223 11078
3812     0 11527 12431 15446  8057  4519  8761 14398 12569  3764  6668
+ 10603 11117 14805  5154  8276 18175  9367 14840  7056  9698 13603  7
+266
13902 11527     0 13638 10220 18405  8675  4611 12993 11970  8798  822
+6 15087 16591  6859  6381 11223  7602  3236  3457 14535  8830 14748  
+6655
8619 12431 13638     0  9965  5555 11256 11549  2157 10917 16194  9609
+  1926  7111 12565 14906  5737  9378 16868 11899  5402 15921  5765 19
+675
15811 15446 10220  9965     0 11122 18683 14599  7873  2940 13119 1779
+3 11057  6374  3517 14973 15565  3627 10307  7077 14478  6475  5155 1
+0235
5015  8057 18405  5555 11122     0 10109 13856  6744  9617 11348 10270
+  3811  4858 14594 12975  7186 13212 17301 17106  3881 12658  6210 14
+138
5230  4519  8675 11256 18683 10109     0  4584 13284 16923  6018  2299
+ 10267 14709 15205  3741  5549 15863  8377 11981  6865 12543 16177  8
+600
9615  8761  4611 11549 14599 13856  4584     0 12503 16548  8240  3619
+ 11888 18524 11444  4627  6948 11282  6102  7523  9992 12321 16782  8
+550
10624 14398 12993  2157  7873  6744 13284 12503     0  9231 18070 1140
+5  3813  6447 10427 16699  7735  7340 15982 10421  7492 14151  4374 1
+8092
13346 12569 11970 10917  2940  9617 16923 16548  9231     0 11052 1922
+0 11199  4874  5269 14093 16358  6543 10709  9458 13488  5008  5170  
+9146
7575  3764  8798 16194 13119 11348  6018  8240 18070 11052     0  8220
+ 14354 12490 11186  3614 11306 14452  5954 11533 10801  6649 14987  3
+508
6170  6668  8226  9609 17793 10270  2299  3619 11405 19220  8220     0
+  9124 15115 15059  5330  3975 14165  9100 10994  6469 14453 15268 10
+326
6812 10603 15087  1926 11057  3811 10267 11888  3813 11199 14354  9124
+     0  6645 14126 13999  5168 11153 17968 13808  3728 15782  6168 17
+749
9487 11117 16591  7111  6374  4858 14709 18524  6447  4874 12490 15115
+  6645     0  9754 15942 11588  9288 15400 13331  8652  9157  2621 12
+782
18135 14805  6859 12565  3517 14594 15205 11444 10427  5269 11186 1505
+9 14126  9754     0 11725 16484  3379  6883  4250 17835  5311  8643  
+7729
8030  5154  6381 14906 14973 12975  3741  4627 16699 14093  3614  5330
+ 13999 15942 11725     0  9173 13841  4874  9780 10451  9122 18555  5
+003
5409  8276 11223  5737 15565  7186  5549  6948  7735 16358 11306  3975
+  5168 11588 16484  9173     0 13413 12955 12642  3450 17931 11293 14
+142
17959 18175  7602  9378  3627 13212 15863 11282  7340  6543 14452 1416
+5 11153  9288  3379 13841 13413     0  9181  4145 14775  8581  7147 1
+0945
12822  9367  3236 16868 10307 17301  8377  6102 15982 10709  5954  910
+0 17968 15400  6883  4874 12955  9181     0  5593 15242  6278 15447  
+3419
17136 14840  3457 11899  7077 17106 11981  7523 10421  9458 11533 1099
+4 13808 13331  4250  9780 12642  4145  5593     0 15914  8478 11291  
+8453
3267  7056 14535  5402 14478  3881  6865  9992  7492 13488 10801  6469
+  3728  8652 17835 10451  3450 14775 15242 15914     0 15624  9326 14
+308
12882  9698  8830 15921  6475 12658 12543 12321 14151  5008  6649 1445
+3 15782  9157  5311  9122 17931  8581  6278  8478 15624     0 10157  
+4139
11223 13603 14748  5765  5155  6210 16177 16782  4374  5170 14987 1526
+8  6168  2621  8643 18555 11293  7147 15447 11291  9326 10157     0 1
+4265
11078  7266  6655 19675 10235 14138  8600  8550 18092  9146  3508 1032
+6 17749 12782  7729  5003 14142 10945  3419  8453 14308  4139 14265  
+   0

[download]

...roboticus

When your only tool is a hammer, all problems look like your thumb.

Replies are listed 'Best First'.
Re^2: Data visualisation.
by BrowserUk (Patriarch) on Jan 04, 2014 at 08:44 UTC
    Do you have a way to check whether the coordinates look reasonable against your original data? I'm curious at how good the program works.

    Sorry for the delay in getting back to you Roboticus, but I was caught up with my own efforts. I tried C&ping my dataset into your code but something broke and I couldn't immediately see what.

    I finally just got back to it and it was simply that you use split /\s+/ rather than split ' ' and my data had leading whitespace which screwed everything up. Fixed that, ran it and got:

    <0>: (600.804026845638, 199.30760478734)
<A>: (0, 0)
<2>: undef
<3>: (628.038255033557, 206.128479872154)
<4>: undef
<5>: (443.404697986577, 203.804498977408)
<6>: (518.110738255034, 242.374220792608)
<7>: (479.277852348993, 226.258127473328)
<8>: (551.842953020134, 59.1130713295962)
<C>: (113.638255033557, 265.720430138385)
<10>: (228.714765100671, 164.965318248851)
<11>: (629.140939597315, 105.9513007122)
<12>: (532.210067114094, 195.554198273552)
<13>: undef
<14>: (334.319463087248, 254.225287099954)
<B>: (745, 0)
<16>: (476.814093959732, 202.416204394214)

    [download]

    Those undefs are the points you had trouble with I assume. Everyone has found problems with the dataset -- it comes from TSPLIB and there was never any guarantee that it was a plottable dataset; that's part of the reason for whating to try and visualise it.

    However, everyone seems to have problems with different points.

    I eventually gave up with the Law of Cosines method. Not only are there four quadrants that each point could be in, there are two points (+-y) that need to be considered. Instead I went the Circle-Circle intersection route, which proved to be far simpler.

    This animated gif shows the problem with the dataset quite nicely. The green arcs are the distances from B & P. The cyan arc is from the 3rd point (in this case the A point), which is used to decide (nearest wins) which of the two intersect points of the green arcs is chosen. It also highlights the accuracy or lack thereof of the correspondence between them.

    It shows that -- with A as the third point -- D is a particularly bad fit; but chose a different 3rd reference point and D might be spot on but some other previously good fit point becomes bad.

    Anyway, many thanks for your code -- you're first attempt was the cluebat I needed to get started. I've added my code below, but it is also very obfuscated with the image generation code. I need to separate the two, but it was very useful when coding.

    My code:

    #! perl -slw
use strict;
use Data::Dump qw[ pp ]; $Data::Dump::WIDTH = 200;
use List::Util qw[ min max sum ];
use GD;
use enum qw[ X Y ];
use constant PI => 3.1415926535897932384626433832795;

my @N2A; @N2A[ 0 .. 25 ] = 'A'..'Z';

sub acos { atan2( sqrt( 1 - $_[0] * $_[0] ), $_[0] ) }
sub asin { atan2( $_[0], sqrt( 1 - $_[0] * $_[0] ) ) }

sub rgb2n{ local $^W; unpack 'N', pack 'CCCC', 0, @_ }
my $BLACK   = rgb2n( 0,0,0 );
my $RED     = rgb2n( 255, 0, 0 );
my $GREEN   = rgb2n( 0, 255, 0 );
my $BLUE    = rgb2n( 0, 0, 255 );
my $YELLOW  = rgb2n( 255, 255, 0 );
my $MAGENTA = rgb2n( 255, 0, 255 );
my $CYAN    = rgb2n( 0, 255, 255 );
my $WHITE   = rgb2n( 255,255,255 );

my( $xOrg, $yOrg, @pts, @dists );
sub plotPt{
    my( $im, $pt, $label ) = @_;
    $im->filledArc(           $pt->[X]  +$xOrg, $pt->[Y]  +$yOrg, 14, 
+14, 0, 360, $RED );
    $im->string( gdSmallFont, $pt->[X]-1+$xOrg, $pt->[Y]-7+$yOrg, $lab
+el, $BLACK )
}
sub plotRoute{
    my( $im, $pt1, $pt2 ) = @_;
    $im->line( $pt1->[X]+$xOrg, $pt1->[Y]+$yOrg, $pt2->[X]+$xOrg, $pt2
+->[Y]+$yOrg, $BLUE );
}
sub plotArc {
    my( $im, $p1, $p2, $color ) = @_;
    $im->arc( $pts[ $p1 ][X]+$xOrg, $pts[ $p1 ][Y]+$yOrg,
        ( $dists[ $p1 ][ $p2 ]*2 )x2,
        0,360, $color
    );
}

@dists = map[ split ' ' ], <DATA>; shift @dists; shift @$_ for @dists;
sub d{ $dists[ $_[0] ][ $_[1] ] }
my $dMax = max( map max( @$_ ), @dists );

my $xMax = 200+$dMax;
my $yMax = 40+sqrt( $dMax**2 - ( $dMax / 2 )**2 ) * 2;
( $xOrg, $yOrg ) = ( 100, $yMax / 2 );

my $im = GD::Image->new( $xMax, $yMax, 1 ); $im->fill( 0,0, $WHITE );
$im->line( $xOrg, 0, $xOrg, $yMax, $BLACK );
$im->line( 0, $yOrg, $xMax, $yOrg, $BLACK );

my( $p1, $p2 ) = map{
    my $y = $_;
    map{
        $dists[$y][$_] == $dMax ? ( $_, $y ) : ()
    } 0 .. $#dists;
} 0 .. $#dists;

print "$p1, $p2";
$pts[ $p2 ] = [ 0, 0];
$pts[ $p1 ] = [ $dMax, 0 ];
$pts[ 0 ] = do {
    my( $d, $r, $R ) = ( $dMax, d( $p1, 0 ), d( $p2, 0 ) );
    my $x = ( $d**2 - $r**2 + $R**2 ) / ( 2 * $d );
    my $y = ( 1/$d * sqrt( (-$d+$r-$R)*(-$d-$r+$R)*(-$d+$r+$R)*($d+$r+
+$R) ) ) / 2;
    [ $x, $y ];
};
plotPt( $im, $pts[ 0 ], $N2A[ 0 ] );
plotRoute( $im, @pts[ $p1, $p2 ] );
plotPt( $im, $pts[ $p1 ], $N2A[ $p1 ] );
plotPt( $im, $pts[ $p2 ], $N2A[ $p2 ] );

my $ani = $im->gifanimbegin( 1, 10 );
for my $p ( 1 .. $#dists ) {
    next if $p == $p1 or $p == $p2 ;

    my( $d, $r, $R ) = ( $dMax, d( $p1, $p ), d( $p2, $p ) );
    my $x = ( $d**2 - $r**2 + $R**2 ) / ( 2 * $d );
    my $y = ( 1/$d * sqrt( (-$d+$r-$R)*(-$d-$r+$R)*(-$d+$r+$R)*($d+$r+
+$R) ) ) / 2;
    plotArc( $im, $p1, $p, $GREEN );
    plotArc( $im, $p2, $p, $GREEN );
    plotArc( $im, 0, $p, $CYAN );

    my $checkD1 = sqrt( ( $pts[0][X] - $x )**2 + ( $pts[0][Y] - $y )**
+2 );
    my $checkD2 = sqrt( ( $pts[0][X] - $x )**2 + ( $pts[0][Y] + $y )**
+2 );
    printf "$N2A[ $p ]: %u $checkD1 $checkD2\n", d( 0, $p );
    $pts[ $p ] = [ $x, ( abs( $checkD1 - d( 0, $p ) ) < abs( $checkD2 
+- d( 0, $p ) ) ) ? $y : -$y ];
    plotPt( $im, $pts[ $p ], $N2A[ $p ] );
    $ani .= $im->gifanimadd( 1, 0, 0, 300, );
    open PNG, '>:raw', "$0.png" or die $!;
    print PNG $im->png;
    close PNG;
    system "$0.png";
    plotArc( $im, $p1, $p, $WHITE );
    plotArc( $im, $p2, $p, $WHITE );
    plotArc( $im, 0, $p, $WHITE );
}
$ani .= $im->gifanimend();
open GIF, '>:raw', "$0.gif" or die $!;
print GIF $ani;
close GIF;
system "$0.gif";

my @route = ( 0, 15, 11, 8, 3, 12, 6, 7, 5, 2, 10, 4, 1, 9, 14, 13, 16
+ );
plotRoute( $im, @pts[ @route[ $_-1, $_ ] ] ) for 1..$#route;

plotPt( $im, $pts[ $_ ], $N2A[ $_ ] ) for 0 .. $#pts;

open PNG, '>:raw', "$0.png" or die $!;
print PNG $im->png;
close PNG;
system "$0.png";

print '      ', join ' ' x 26, 'B' .. 'Q';
for my $i ( 0 .. $#dists ) {
    printf "$N2A[ $i ]: ";
    for my $j ( $i+1 .. $#dists ) {
        my $checkD1 = sqrt( ( $pts[$i][X] - $pts[$j][X] )**2 + ( $pts[
+$i][Y] - $pts[$j][Y] )**2 );
        my $checkD2 = sqrt( ( $pts[$i][X] - $pts[$j][X] )**2 + ( $pts[
+$i][Y] + $pts[$j][Y] )**2 );
        my $checkD3 = sqrt( ( $pts[$i][X] + $pts[$j][X] )**2 + ( $pts[
+$i][Y] - $pts[$j][Y] )**2 );
        my $checkD4 = sqrt( ( $pts[$i][X] + $pts[$j][X] )**2 + ( $pts[
+$i][Y] + $pts[$j][Y] )**2 );
        printf "%4u (%4.f %4.f %4.f %4.f) ", d( $i, $j ), $checkD1, $c
+heckD2, $checkD3, $checkD4;
    }
    print '';
}

__DATA__
     A    B    C    D    E    F    G    H    I    J    K    L    M    
+N    O    P    Q
A:   0  633  257   91  412  150   80  134  259  505  353  324   70  21
+1  268  246  121
B: 633    0  390  661  227  488  572  530  555  289  282  638  567  46
+6  420  745  518
C: 257  390    0  228  169  112  196  154  372  262  110  437  191   7
+4   53  472  142
D:  91  661  228    0  383  120   77  105  175  476  324  240   27  18
+2  239  237   84
E: 412  227  169  383    0  267  351  309  338  196   61  421  346  24
+3  199  528  297
F: 150  488  112  120  267    0   63   34  264  360  208  329   83  10
+5  123  364   35
G:  80  572  196   77  351   63    0   29  232  444  292  297   47  15
+0  207  332   29
H: 134  530  154  105  309   34   29    0  249  402  250  314   68  10
+8  165  349   36
I: 259  555  372  175  338  264  232  249    0  495  352   95  189  32
+6  383  202  236
J: 505  289  262  476  196  360  444  402  495    0  154  578  439  33
+6  240  685  390
K: 353  282  110  324   61  208  292  250  352  154    0  435  287  18
+4  140  542  238
L: 324  638  437  240  421  329  297  314   95  578  435    0  254  39
+1  448  157  301
M:  70  567  191   27  346   83   47   68  189  439  287  254    0  14
+5  202  289   55
N: 211  466   74  182  243  105  150  108  326  336  184  391  145    
+0   57  426   96
O: 268  420   53  239  199  123  207  165  383  240  140  448  202   5
+7    0  483  153
P: 246  745  472  237  528  364  332  349  202  685  542  157  289  42
+6  483    0  336
Q: 121  518  142   84  297   35   29   36  236  390  238  301   55   9
+6  153  336    0

    [download]
    With the rise and rise of 'Social' network sites: 'Computers are making people easier to use everyday'
    Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.
    "Science is about questioning the status quo. Questioning authority".
    In the absence of evidence, opinion is indistinguishable from prejudice.
[reply]
[d/l]
[select]

In Section Seekers of Perl Wisdom