Also, I like the heavy usage of map and grep. A little harder to read at first (as a programmer who's functional at play, not at work), but very concise.

Question - how does the regex engine not choose a free square in d4 when looking for free squares in d3? I'm guessing it has to do with [\x00-\xFF]*, but I'm not positive as to how that's working ...

how does the regex engine not choose a free square in d4 when looking for free squares in d3?

I do not know what you mean by this. Do you mean "why doesn't the regex engine never try to place a queen on both d4 and d3?" In that case, the answer is simple - it will never even attempt to place two queens on the same column. The first part of the string is:

,a1,a2,a3,a4,a5,a6,a7,a8,
,b1,b2,b3,b4,b5,b6,b7,b8,
,c1,c2,c3,c4,c5,c6,c7,c8,
,d1,d2,d3,d4,d5,d6,d7,d8,
,e1,e2,e3,e4,e5,e6,e7,e8,
,f1,f2,f3,f4,f5,f6,f7,f8,
,g1,g2,g3,g4,g5,g6,g7,g8,
,h1,h2,h3,h4,h5,h6,h7,h8,
[download]

is being matched against the first part of the regex:

.*,(\w+),.*
.*,(\w+),.*
.*,(\w+),.*
.*,(\w+),.*
.*,(\w+),.*
.*,(\w+),.*
.*,(\w+),.*
.*,(\w+),.*
[download]

The newlines in the string and regex are significant here (. won't match a newline). This part will make \1 one of 'a1' .. 'a8', \2 one of 'b1' .. 'b8', \3 one of 'c1' .. 'c8', etc. So, it will never try to place two queens on the 'd' column.

But you may ask, "why doesn't it put a queen on d3 and one on e4?". That's where the second part of the string and regex come in. \4 contains the position of the queen on the d column, so, in this case, \4 equals 'd3'. Then there's this line in the second part of the regex:

[\x00-\xFF]*\n\4:.*\1,.*\2,.*\3,.*\5,.*\6,.*\7,.*\8,.*
[download]

We know that \4 equals 'd3', so part of this lines read '\nd3:'. Looking back at the second part of the string, there is only one line that starts with 'd3:':

d3:a1,a2,a4,a5,a7,a8,b2,b4,b6,b7,b8,c1,c5,c6,c7,c8,e1,e5,e6,e7,e8,f2,f
+4,f6,f7,f8,g1,g2,g4,g5,g7,g8,h1,h2,h4,h5,h6,h8,
[download]

If you look carefully, after the colon are all the fields that aren't attacked by a queen on d3. Specifically, the field 'e4' is missing. But the line

[\x00-\xFF]*\n\4:.*\1,.*\2,.*\3,.*\5,.*\6,.*\7,.*\8,.*
[download]

is saying "match a line that starts with 'd3', and has after the colon a list of fields that include the positions of all other 7 queens". The [\x00-\xFF]* just skips enough lines to get to the next queen.

Abigail

The [\x00-\xFF]* just skips enough lines to get to the next queen.

Your explanation above is good. However, the above quote is the actual meat of my question ...

------
We are the carpenters and bricklayers of the Information Age.

The idea is a little like C++ templates, except not quite so brain-meltingly complicated. -- TheDamian, Exegesis 6

Please remember that I'm crufty and crochety. All opinions are purely mine and all code is untested, unless otherwise specified.

Since it's slow like hell (but I've some ideas to speed it up)

Speeding it up turned out to be easier than I thought it was. Below is reworked program that is dramatically faster that the original. But first a table comparing running times of three versions, the original, pure regex solution from the parent node, the faster (still pure regex) solution presented below, and the non-pure variant presented last year. The latter is still the faster solution though.

Timings (values in wall clock seconds):

  N    Original    Faster   Non-pure
 
  4       0.035     0.034      0.035
  5       0.045     0.036      0.036
  6       0.769     0.041      0.038
  7       4.833     0.042      0.038
  8                 0.082      0.049
  9                 0.072      0.044
 10                 0.113      0.056
 11                 3.504      0.051
 12                            0.096
 13                            0.071
 14                            0.577
 15                            0.467
 16                            3.864
 17                            2.289
 18                           19.630
 19                            1.324
 20                          117.227

Before giving the program, some sample output:

$ ./queens -p -n 4
';,a1,a2,a3,a4,
;,b1,b2,b3,b4,
b1:,a3,a4,
b2:,a4,
b3:,a1,
b4:,a1,a2,
;,c1,c2,c3,c4,
c1:,a2,a4,b3,b4,
c2:,a1,a3,b4,
c3:,a2,a4,b1,
c4:,a1,a3,b1,b2,
;,d1,d2,d3,d4,
d1:,a2,a3,b2,b4,c3,c4,
d2:,a1,a3,a4,b1,b3,c4,
d3:,a1,a2,a4,b2,b4,c1,
d4:,a2,a3,b1,b3,c1,c2,
' =~ 
/^;.*,(\w+),.*
;.*,(\w+),.*
[^;]*\2:.*,\1[^;]*
;.*,(\w+),.*
[^;]*\3:.*,\1.*,\2[^;]*
;.*,(\w+),.*
[^;]*\4:.*,\1.*,\2.*,\3[^;]*
/
[a3 b1 c4 d2]
$ ./queens -n 8
[a8 b4 c1 d3 e6 f2 g7 h5]
[download]

And here's the program:

#!/usr/bin/perl

use strict;
use warnings 'all';
    
use Getopt::Long;
              
Getopt::Long::Configure ("bundling");
GetOptions ('p|print'    =>  \my $print,
            'P|Print'    =>  \my $Print,
            'n|number=i' => \(my $nr_of_queens = 8)
);
    
my @rows =               1 .. $nr_of_queens;
my @cols = ('a' .. 'z') [0 .. $nr_of_queens - 1];
    
sub a2i {ord ($_ [0]) - ord ('a') + 1}
sub i2a {chr ($_ [0] + ord ('a') - 1)}
 
# Given a square, return all non-attacked squares on columns to
# the *left* of the given square. (a1 is the lower left corner).
sub free {
    my ($C, $R) = $_ [0] =~ /(\D)(\d+)/;
    $C = a2i $C;   
    map {join "" => i2a ($_ -> [0]), $_ -> [1]}
        grep {$_ -> [0] != $C &&
              $_ -> [1] != $R &&
              abs ($_ -> [0] - $C) != abs ($_ -> [1] - $R)}
        map {my $c = a2i $_; map {[$c, $_]} @rows} @cols [0 .. $C - 1]
}
            
my ($str, $re) = ("", "");
    
foreach my $c (@cols) {
    $str .= ";," . (join "," => map {"$c$_"} @rows) . ",\n";
    $re  .= ";.*,(\\w+),.*\n";

    next if $c eq 'a';
 
    map {$str .= "$_:," . join ("," => free ($_)) . ",\n"} map {"$c$_"
+} @rows;

    my $C = a2i $c;
    
    $re .= "[^;]*\\$C:" . join ("" => map {".*,\\$_"} 1 .. $C - 1) . "
+[^;]*\n";
}
        
if ($print || $Print) {
    print "'$str' =~ \n/^$re/\n";
    exit if $Print;
}
            
if (my @a = $str =~ /^$re/) {
    print "[@a]\n";
}
    
__END__
[download]

Abigail

N    Original    Faster   Non-pure
 
  4       0.035     0.034      0.035
  5       0.045     0.036      0.036
  6       0.769     0.041      0.038
  7       4.833     0.042      0.038
  8                 0.082      0.049
  9                 0.072      0.044
 10                 0.113      0.056
 11                 3.504      0.051
 12                            0.096
 13                            0.071
 14                            0.577
 15                            0.467
 16                            3.864
 17                            2.289
 18                           19.630
 19                            1.324
 20                          117.227
[download]

I'm curious - have you had a chance to look at why the speeds actually improve when going from 8 to 9 for both Faster and Non-pure and from 10-11 for Non-pure, but slows down 30x for Faster? And, what's with 17, 18, and 19 when it's 2.289 -> 19.630 -> 1.324??

------
We are the carpenters and bricklayers of the Information Age.

The idea is a little like C++ templates, except not quite so brain-meltingly complicated. -- TheDamian, Exegesis 6

Please remember that I'm crufty and crochety. All opinions are purely mine and all code is untested, unless otherwise specified.

It has to do with how many positions are rejected before a suitable one is found. The solutions found for n = 8 and n = 9 are:

[a8 b4 c1 d3 e6 f2 g7 h5]
[a9 b7 c4 d2 e8 f6 g1 h3 i5]
[download]

As you can see, for n = 8, it never has to backtrack for the first queen (a8 is choosen), but for the seconde queen, b8, b7, b6, and b5 need to be rejected. b8 and b7 will be rejected right away (as they are attacked by a8), but for b6 and b5 to be rejected, lots of other queens will be have to be placed. For n = 9, no backtracking for the first queen is needed, and for the second queen, the positions b9 and b8 are rejected immediately. It's only the third queen were there's some real backtracking going on - c9, c8, c7, and c6 are rejected immediately, and only for c5 more queens will be tried before rejecting it.

The timings for 'faster' with n >= 10 cannot be trusted, as the program contained a bug for n >= 10 (see elsewhere in this thread - the bug is now fixed). Here's a new table (done on a different computer, and recording user times, not wall clock time), with the fixed programs:

  N       Original         Faster       Non-Pure
  4           0.06           0.05           0.04
  5           0.07           0.04           0.05
  6           1.57           0.07           0.05
  7           9.29           0.06           0.05
  8                          0.23           0.06
  9                          0.16           0.06
 10                          0.50           0.07
 11                          0.41           0.07
 12                          2.64           0.14
 13                          1.58           0.10
 14                         37.23           0.82
 15                         35.45           0.70
 16                                         5.45
 17                                         3.18
 18                                        27.17
 19                                         1.89
 20
[download]

And, in case you are interested, the code that generated the table:

#!/usr/bin/perl
            
use strict;
use warnings;
no warnings qw /syntax/;

$| = 1;

my $width    =  15;
my $time_out = 120;

my @cmds = ("./queens2 -n ",
            "./queens3 -n ",
            "./queens1 -f -n ");
my $nr_of_commands = @cmds;
            
my $N = 4;
print "  N";
printf "%${width}s" => $_ for qw /Original Faster Non-Pure/;
print "\n";
while ($nr_of_commands) {
    printf "%3d" => $N;
    foreach my $cmd (@cmds) {
        unless (defined $cmd) {
            print " " x $width;
            next;
        }
        local $SIG {ALRM} = sub {die "Time out!"};
        alarm ($time_out);
        eval {
            my $time = (`/usr/bin/time -f "%U" $cmd $N 2>&1`) [-1];
            alarm (0);
            chomp $time;
            printf "%$width.2f" => $time;
        }; 
        if ($@ && $@ =~ /Time out/) {
            undef $cmd; 
            $nr_of_commands --;
            print " " x $width;
        }
    }
    print "\n";
    $N ++;
}
[download]

Home work question: the code above is lacking something vital. What is it not doing what it should do?

Abigail

For any odd N odd N not divisible by 3: $a_solution = [ map { chr(ord('a') + $_ - 1).(((2 * $_) - 1) % $N); } (1..$N) ]

(e.g., for $N = 11: [ a1, b3, c5, d7, e9, f11, g2, h4, i6, j8, k10 ])

This is equivalent to starting in the bottom-left corner of the board and moving right one square and up two squares (wrapping when you hit the edge) N-1 times.

If the regex picks the first space not-already-capturable in each column (From brief inspection, it appears to do so -- It finds an equivalent solution for odd-N), this is the first solution it will find. In the even-N case, this process will not leave any not-already-capturable squares in the last column on the first pass, so it must then backtrack.

Update: Whoa there, Ben. I spoke way too soon. The above solution only applies when N is odd AND not divisible by 3. (So, for N = (1,5) mod 6)

More update: I was wrong about most of the analysis, too. This won't be the first solution found.

You are quite correct. The faulty solution can only happen with n >= 10; it happens because 'e10' isn't attacked by 'f1', and "e10" =~ /e1/. Luckely, the fix is simple:

#!/usr/bin/perl

use strict;
use warnings 'all';

use Getopt::Long;

Getopt::Long::Configure ("bundling");
GetOptions ('p|print'    =>  \my $print,
            'P|Print'    =>  \my $Print,
            'n|number=i' => \(my $nr_of_queens = 8)
);
 
my @rows =               1 .. $nr_of_queens;
my @cols = ('a' .. 'z') [0 .. $nr_of_queens - 1];
    
sub a2i {ord ($_ [0]) - ord ('a') + 1}
sub i2a {chr ($_ [0] + ord ('a') - 1)}

# Given a square, return all non-attacked squares on columns to
# the *left* of the given square. (a1 is the lower left corner).
sub free {
    my ($C, $R) = $_ [0] =~ /(\D)(\d+)/;
    $C = a2i $C;
    map {join "" => i2a ($_ -> [0]), $_ -> [1]}
        grep {$_ -> [0] != $C &&
              $_ -> [1] != $R &&
              abs ($_ -> [0] - $C) != abs ($_ -> [1] - $R)}
        map {my $c = a2i $_; map {[$c, $_]} @rows} @cols [0 .. $C - 1]
}
 
my ($str, $re) = ("", "");
 
foreach my $c (@cols) {
    $str .= ";," . (join "," => map {"$c$_"} @rows) . ",\n";
    $re  .= ";.*,(\\w+),.*\n";

    next if $c eq 'a';

    map {$str .= "$_:," . join (",," => free ($_)) . ",\n"} map {"$c$_
+"} @rows;

    my $C = a2i $c;
            
    $re .= "[^;]*\\$C:" . join ("" => map {".*,\\$_,"} 1 .. $C - 1) . 
+"[^;]*\n";
}

if ($print || $Print) {
    print "'$str' =~ \n/^$re/\n";
    exit if $Print;
}

if (my @a = $str =~ /^$re/) {
    print "[@a]\n";
}
    
__END__
[download]

Abigail

#!/usr/bin/perl

use strict;
use warnings 'all';

use Getopt::Long;

Getopt::Long::Configure ("bundling");
GetOptions ('p|print'    =>  \my $print,
            'P|Print'    =>  \my $Print,
            'n|number=i' => \(my $nr_of_queens = 8)
);
 
my @rows =               1 .. $nr_of_queens;
my @cols = ('a' .. 'z') [0 .. $nr_of_queens - 1];
    
sub a2i {ord ($_ [0]) - ord ('a') + 1}
sub i2a {chr ($_ [0] + ord ('a') - 1)}

# Given a square, return all non-attacked squares on columns to
# the *left* of the given square. (a1 is the lower left corner).
sub free {
    my ($C, $R) = $_ [0] =~ /(\D)(\d+)/;
    $C = a2i $C;
    map {join "" => i2a ($_ -> [0]), $_ -> [1]}
        grep {$_ -> [0] != $C &&
              $_ -> [1] != $R &&
              abs ($_ -> [0] - $C) != abs ($_ -> [1] - $R)}
        map {my $c = a2i $_; map {[$c, $_]} @rows} @cols [0 .. $C - 1]
}
 
my ($str, $re) = ('', '');

my $solrow = 0;

foreach my $c (@cols) {
    $str .= ";," . (join "," => map {"$c$_"} @rows) . ",\n";

    $solrow += 2;
    $solrow = 1 if $solrow > $nr_of_queens;

    $re .= ".*\n" unless $c eq 'a';
    $re .= ";.*,($c$solrow),";

    next if $c eq 'a';

    $re .= sprintf "(?:.*\n){%d}\\%d:", $solrow, a2i($c);

    for my $i (1..a2i($c)-1) {
      $re .= ".*,\\$i,";
    }
    $re .= sprintf "(?:.*\n){%d}", $nr_of_queens - $solrow;

    map {$str .= "$_:," . join (",," => free ($_)) . ",\n"} map {"$c$_
+"} @rows;

}

if ($print || $Print) {
    print "'$str' =~ \n/^$re/\n";
    exit if $Print;
}

if (my @a = $str =~ /^$re/) {
    print "[@a]\n";
}
    
__END__
[download]