Re: ulam's spiral too slow

After seeing the difference in outputs between the OP code and liverpole's version, and then actually looking up the references at wikipedia, ~~I realized that there were other problems besides execution speed: way too many spots were being drawn in black, which meant way too many numbers were being categorized as primes~~ I felt compelled to figure out who was wrong; I have updated my earlier reply accordingly.

I also noticed that the OP code actually draws all the points (from 1 to $max), both the supposed primes and the supposed non-primes. But you really only need to draw one set, and just leave the other set as "background color".

Then, I thought it would be helpful to get a sense of what took longest -- computing or drawing -- so I decided to refactor the code as follows:

run through all the desired numeric range before doing any drawning, keeping track of the x/y coords as we go;
each time we find a prime number, store that as both a hash key and as an item in an ascending array; use the hash value to store the x/y coords where the dot should be drawn for that prime;
once all the primes have been found, set up the Tk::Canvas, then run through the sorted list of prime values (which are hash keys), and draw a dot for each one.

As it turns out, the computing takes about twice as long as the drawing (4 sec vs. 2 sec for a max number of 150,000, which pretty well fills the given grid); but bear in mind the compute step loops over all integers in the given range, while the drawing step only loops over the primes.

And luckily, the resulting canvas looks a lot more like the pictures I saw over at wikipedia (it still might not be entirely correct -- but you could try dumping a portion of the %primecoord hash structure to make sure the values are as intended).

#!/usr/local/bin/perl 

use strict; 
use Tk;

die "Usage:  $0  max_number\n"  # use the command-line, Luke!
    if ( @ARGV != 1 or $ARGV[0] !~ /^\d+/ );
my $lastnumber = shift;

my ( $mincoord, $midcoord, $maxcoord ) = ( 50, 250, 450 );

plot_grid( compute_primes( $lastnumber ));

MainLoop;

{  # closure for building prime number hash structure

    my @sorted_primes = ( 1 ); # primes in ascending order
    my %primecoord = ( 1 => [$midcoord, $midcoord] );
     # HoA: keys are primes, values are x/y coords in grid

    sub ck_prime {
        my ( $num, $x, $y ) = @_;
        my $end = int( sqrt( $num ));
        my $keep = 1;
        for my $prime ( @sorted_primes ) {
            if ( 0 == $num % $prime ) {
                $keep = 0;
                last;
            }
            elsif ( $prime > $end ) {
                last;
            }
        }
        if ( $keep ) {
            $primecoord{$num} = [ $x, $y ];
            push @sorted_primes, $num;
        }
    }

    sub compute_primes
    {
# list of directions for incrementing x, y coordinates:
        my %vector = ( 0 => [ 1, 0 ],  # rightward
                       1 => [ 0, 1 ],  # downward
                       2 => [ -1, 0 ], # leftward
                       3 => [ 0, -1 ], # upward
                     );

        my $x = my $y = 250;  # starting position
        my $direction = 0;  # changes when we've gone $pathlen steps
        my $traveled = 0;  # no. of points drawn in current direction
        my $pathlen = 1;  # increments on every second direction chang
+e
        my $changes = 0;  # no. of dir. changes since last $pathlen in
+crement
        my $number = 1;

        my $bgn = time;
        while ( ++$number <= $lastnumber ) 
        {
            $x += $vector{$direction}[0];
            $y += $vector{$direction}[1];
            last if ( $x < $mincoord or $x > $maxcoord );

            if ( ++$traveled == $pathlen ) {
                $direction = ++$direction % 4;
                $traveled = 0;
                if ( ++$changes % 2 == 0 ) {
                    $changes = 0;
                    $pathlen++;
                }
            }
            ck_prime( $number, $x, $y );
        }
        my $end = time;
        warn "Checked $lastnumber integers in ", $end - $bgn, " sec\n"
+;
        return( \@sorted_primes, \%primecoord );
    }
}

sub plot_grid
{
    my ( $primes, $coords ) = @_;

    my $mw = MainWindow->new; 
    my $c = $mw->Canvas(-width => $maxcoord+50, -height => $maxcoord+5
+0,
                        -background => 'white' )->pack;
    $c->createLine( $mincoord, $midcoord, $maxcoord, $midcoord ); 
    $c->createText( $mincoord-30, $midcoord, -fill => 'blue', -text =>
+ 'X'); 
    $c->createLine( $midcoord, $mincoord, $midcoord, $maxcoord ); 
    $c->createText( $midcoord, $mincoord-30, -fill => 'blue', -text =>
+ 'Y'); 

    my $bgn = time;

    for my $prime ( @$primes ) {
        my ( $x, $y ) = @{$$coords{$prime}};
        $c->createText( $x, $y, -fill => 'black', -text => "\xb7" );
    }

    my $end = time;
    warn "Plotted ", scalar @$primes, " primes in ", $end - $bgn, " se
+c\n";
}
[download]

Note that I used "\xb7" (centered dot) as the character -- seemed like "." (period) was off-center.

(updated to correct the comments about the "%vector" hash, to reflect the fact that coords in Tk::Canvas put 0,0 at the upper left corner)

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