TL;DR

On with TASK #2 from the Perl Weekly Challenge #086. Enjoy!

# The challenge

You are given Sudoku puzzle (9x9). Write a script to complete the puzzle and must respect the following rules: …

Well… we all know the rules for Sudoku, right?!?

# The questions

Questions I would ask here are all related to the input and output formats. Apart from this… nothing more.

# The solution

There’s a ton of solvers for Sudoku and I definitely remember that there’s even a regular expression to do this. I think I know who will propose this solution.

As always, I’ve gone the good old boring way. Which is recognize that this is a constraint programming problem, remember that is something I already addressed in the past in the blog, and that I’ve written a generic solver (More constraint programming).

So… it’s a matter of filling in the voids! This is the main function:

``````sub sudoku_puzzle (\$puzzle) {
\$puzzle = dclone(\$puzzle); # don't mess with the original!
my %missing; # records how many alternatives are for undecided positions
for my \$row (0 .. 8) {
for my \$col (0 .. 8) {
next unless \$puzzle->[\$row][\$col] eq '_';
\$puzzle->[\$row][\$col] = [ 1 .. 9 ];
\$missing{"\$row-\$col"} = 9;
}
}
my \$state = solve_by_constraints(
is_done => sub (\$state) { # we're done when there's no more missing
return keys \$state->{missing}->%* == 0;
},
constraints => [
constraint_group_factory( # rows
[map { [\$_, 0] } 0 .. 8], # outer loop
[map { [0, \$_] } 0 .. 8], # inner loop
),
constraint_group_factory( # columns
[map { [0, \$_] } 0 .. 8], # outer loop
[map { [\$_, 0] } 0 .. 8], # inner loop
),
constraint_group_factory( # 3x3 blocks
[map { ([\$_, 0], [\$_, 3], [\$_, 6]) } (0, 3, 6)], # outer
[map { ([\$_, 0], [\$_, 1], [\$_, 2]) } (0, 1, 2)], # inner
),
],
search_factory => \&search_factory,
start => {
field => \$puzzle,
missing => \%missing,
},
);
return \$state->{field};
}
``````

It begins with a little preparation, getting all locations where there is the need for a digit and changing the puzzle to store all possible alternatives. Apart from this… it’s all handed over to `solve_by_constraints`.

## State tracking data structure

``````      start => {
field => \$puzzle,
missing => \%missing,
},
``````

The state is tracked with an anonymous hash that keeps the following:

• `field` is an Array of Arrays holding the current status of the field. Each undecided slots are further array references that keep the possible candidates for that specific location; this array is pruned by the application of conditions or, when no more pruning is possible, is fixed with an attempt (that might be backtracked later);
• `missing` is a hash whose keys are locations inside the field (in the form of a string with the row number, a dash, the column number) and whose values are the number of candidates in that location. It only keeps locations where a decision might be necessary.

## Ending condition

``````      is_done => sub (\$state) { # we're done when there's no more missing
return keys \$state->{missing}->%* == 0;
},
``````

The ending condition is simple: when there’s nothing left as `missing`, every location has been assigned a value and the whole thing is compliant to the rules. Hence, it suffices to check that the `missing` hash reference in the state is empty.

## Constraints

``````      constraints => [
constraint_group_factory( # rows
[map { [\$_, 0] } 0 .. 8], # outer loop
[map { [0, \$_] } 0 .. 8], # inner loop
),
constraint_group_factory( # columns
[map { [0, \$_] } 0 .. 8], # outer loop
[map { [\$_, 0] } 0 .. 8], # inner loop
),
constraint_group_factory( # 3x3 blocks
[map { ([\$_, 0], [\$_, 3], [\$_, 6]) } (0, 3, 6)], # outer
[map { ([\$_, 0], [\$_, 1], [\$_, 2]) } (0, 1, 2)], # inner
),
],
``````

This comes a little cryptic, but bear with me a moment.

Each of the three constraints according to the rules apply to a partition of the field, where each group contains exactly nine elements. Moreover, there is some structure: the elements in a set either lie on the same row, or in the same column, or in a tight 3x3 block.

The juice of the constraint is the same in all cases: make sure that each group only contains nine distinct values. Hence, apart from figuring out the items that belong to a group, the check to be done is the same in the three cases.

This is why we leverage a factory function `constraint_group_factory`, which takes as input a way to do the right iteration over the group and inside each group, and returns a subroutine that does exactly that and applies the constraint rules. Here is the function:

`````` 1 sub constraint_group_factory (\$bases, \$deltas) {
2    return sub (\$state) {
3       my \$field = \$state->{field};
4       my \$changes = 0;
5       for my \$group (0 .. 8) {
6          my (\$row, \$col) = \$bases->[\$group]->@*;
7          my (%present, @vague);
8          for my \$delta (\$deltas->@*) {
9             my (\$r, \$c) = (\$row + \$delta->, \$col + \$delta->);
10             my \$item = \$field->[\$r][\$c];
11             if (ref \$item) { push @vague, [\$r, \$c] }
12             elsif (\$present{\$item}) { die 'overlap!' }
13             else { \$present{\$item} = 1 }
14          }
15          for my \$pair (@vague) {
16             my (\$r, \$c, @kept) = \$pair->@*;
17             for my \$candidate (\$field->[\$r][\$c]->@*) {
18                if (\$present{\$candidate}) { \$changes++ }
19                else { push @kept, \$candidate }
20             }
21             if (@kept == 0) { die 'no way forward here' }
22             elsif (@kept == 1) {
23                \$field->[\$r][\$c] = \$kept;
24                \$present{\$kept} = 1;
25                delete \$state->{missing}{"\$r-\$c"};
26             }
27             else {
28                \$field->[\$r][\$c] = \@kept;
29                \$state->{missing}{"\$r-\$c"} = scalar @kept;
30             }
31          }
32       }
33       return \$changes;
34    };
35 }
``````

As we said, we use `\$bases` and `\$deltas` to do the group iteration (via `\$bases`) and the iteration inside a group (via `\$deltas`). This will eventually resolve to iterating by row, by column, or by block.

Lines 7 to 14 do a first pass to collect what the pre-existing constraints might be, i.e. to collect which items are already `%present` and which are still undecided (`@vague`). Line 12 is very important, because it makes sure to complain loudly if there is any overlapping, triggering a backtrack (when the exception is caught).

After this first pass, it’s time to do some pruning by eliminating the `%present` items from the undecided ones (lines 15 through 32). The list of remaining candidate for each of the is computed (lines 17 through 20), then analyzed:

• no more candidates? No solution! (line 21)
• one single candidate? Very well, we have decided something (lines 22 through 26). Note that we have one more `%present` at this point (line 24), as well as one less `missing` (line 25);
• still several candidates? No worries, let’s keep them (lines 27 through 30).

Note that line 18 increases the number of changes we did in this pass: this is because we are removing a candidate, which is a change!

## Guessing

Alas, constraints in constraint programming can only do this much. Sometimes we can get stuck in a situation where all constraints are satisfied, and yet we’re not on a final solution.

For these cases, the algorithm needs a way to iterate through different possible guesses. In practice, one of the not-yet-decided positions is selected, and one of the elements inside is tried. If this leads us to a solution… good for us. If this choice eventually leads us to break our constraints… we backtrack and try another guess.

In our case, the guessing function is the following:

`````` 1 sub search_factory (\$state) {
2    my \$field = \$state->{field};
3    my %missing = \$state->{missing}->%*;
4    my (\$target, \$tn);
5    for my \$candidate (keys %missing) {
6       (\$target, \$tn) = (\$candidate, \$missing{\$candidate})
7          if (! defined \$target) || (\$tn > \$missing{\$candidate});
8    }
9    delete \$missing{\$target};
10    my (\$row, \$col) = split m{-}mxs, \$target;
11    my @values = \$field->[\$row][\$col]->@*;
12    return sub (\$state) {
13       return unless @values;
14       \$state->{missing} = { %missing };
15       my \$f = \$state->{field} = dclone(\$field);
16       \$f->[\$row][\$col] = shift @values;
17       return 1;
18    },
19 }
``````

In ConstraintSolver.pod we are told that our guessing machine should be a function that produces other functions (i.e. a factory); the produced functions will be passed a `\$state` and are supposed to modify it according to the next item to be tried at a certain level. Hence, the factory function and the produced function have to make sure that the state is the correct one, without making assumption as to what is the previous state - unless, of course, they know they can.

As an example, we get a reference to the `\$field` (line 2) and later use it to restore the field by doing a deep copy (line 15) and changing it a bit to actually make the guess.

The hash of missing items is the real star here. We use it to select our best candidate, for whatever best means. Here, we are assuming that it’s better to take a candidate with as few remaining choices as possible, hoping this will reduce the branching factor. Are we too optimistic? I really don’t know.

The selected not-yet-decided cell contains a few items that are put in array `@values` (line 11); this array is then used inside the generated sub to iterate through all of them (line 13 and line 16).

## Everything together

I guess it’s time at this point to get all pieces together:

``````#!/usr/bin/env perl
use 5.024;
use warnings;
use experimental qw< postderef signatures >;
no warnings qw< experimental::postderef experimental::signatures >;
use Storable qw< dclone >;
use autodie;

sub sudoku_puzzle (\$puzzle) {
\$puzzle = dclone(\$puzzle); # don't mess with the original!
my %missing; # records how many alternatives are for undecided positions
for my \$row (0 .. 8) {
for my \$col (0 .. 8) {
next unless \$puzzle->[\$row][\$col] eq '_';
\$puzzle->[\$row][\$col] = [ 1 .. 9 ];
\$missing{"\$row-\$col"} = 9;
}
}
my \$state = solve_by_constraints(
is_done => sub (\$state) { # we're done when there's no more missing
return keys \$state->{missing}->%* == 0;
},
constraints => [
constraint_group_factory( # rows
[map { [\$_, 0] } 0 .. 8], # outer loop
[map { [0, \$_] } 0 .. 8], # inner loop
),
constraint_group_factory( # columns
[map { [0, \$_] } 0 .. 8], # outer loop
[map { [\$_, 0] } 0 .. 8], # inner loop
),
constraint_group_factory( # 3x3 blocks
[map { ([\$_, 0], [\$_, 3], [\$_, 6]) } (0, 3, 6)], # outer
[map { ([\$_, 0], [\$_, 1], [\$_, 2]) } (0, 1, 2)], # inner
),
],
search_factory => \&search_factory,
start => {
field => \$puzzle,
missing => \%missing,
},
);
return \$state->{field};
}

# this sub generates sub references that can be used to iterate over
# different "alternatives" in undecided locations.
sub search_factory (\$state) {
my \$field = \$state->{field};
my %missing = \$state->{missing}->%*;
my (\$target, \$tn);
for my \$candidate (keys %missing) {
(\$target, \$tn) = (\$candidate, \$missing{\$candidate})
if (! defined \$target) || (\$tn > \$missing{\$candidate});
}
delete \$missing{\$target};
my (\$row, \$col) = split m{-}mxs, \$target;
my @values = \$field->[\$row][\$col]->@*;
return sub (\$state) {
return unless @values;
\$state->{missing} = { %missing };
my \$f = \$state->{field} = dclone(\$field);
\$f->[\$row][\$col] = shift @values;
return 1;
},
}

sub constraint_group_factory (\$bases, \$deltas) {
return sub (\$state) {
my \$field = \$state->{field};
my \$changes = 0;
for my \$group (0 .. 8) {
my (\$row, \$col) = \$bases->[\$group]->@*;
my (%present, @vague);
for my \$delta (\$deltas->@*) {
my (\$r, \$c) = (\$row + \$delta->, \$col + \$delta->);
my \$item = \$field->[\$r][\$c];
if (ref \$item) { push @vague, [\$r, \$c] }
elsif (\$present{\$item}) { die 'overlap!' }
else { \$present{\$item} = 1 }
}
for my \$pair (@vague) {
my (\$r, \$c, @kept) = \$pair->@*;
for my \$candidate (\$field->[\$r][\$c]->@*) {
if (\$present{\$candidate}) { \$changes++ }
else { push @kept, \$candidate }
}
if (@kept == 0) { die 'no way forward here' }
elsif (@kept == 1) {
\$field->[\$r][\$c] = \$kept;
\$present{\$kept} = 1;
delete \$state->{missing}{"\$r-\$c"};
}
else {
\$field->[\$r][\$c] = \@kept;
\$state->{missing}{"\$r-\$c"} = scalar @kept;
}
}
}
return \$changes;
};
}

# https://github.com/polettix/cglib-perl/blob/master/ConstraintSolver.pm
# https://github.com/polettix/cglib-perl/blob/master/ConstraintSolver.pod
sub solve_by_constraints {
my %args = (@_ && ref(\$_)) ? %{\$_} : @_;
my @reqs = qw< constraints is_done search_factory start >;
exists(\$args{\$_}) || die "missing parameter '\$_'" for @reqs;
my (\$constraints, \$done, \$factory, \$state, @stack) = @args{@reqs};
my \$logger = \$args{logger} // undef;
while ('necessary') {
last if eval {    # eval - constraints might complain loudly...
\$logger->(validating => \$state) if \$logger;
my \$changed = -1;
while (\$changed != 0) {
\$changed = 0;
\$changed += \$_->(\$state) for @\$constraints;
\$logger->(pruned => \$state) if \$logger;
}
\$done->(\$state) || (push(@stack, \$factory->(\$state)) && undef);
};
\$logger->(backtrack => \$state, \$@) if \$logger;
while (@stack) {
last if \$stack[-1]->(\$state);
pop @stack;
}
return unless @stack;
}
return \$state;
}

sub debug_puzzle (\$puzzle) {
my \$i = 1;
my \$is_solving = 0;
CHECK_FINAL:
for my \$row (\$puzzle->@*) {
for my \$item (\$row->@*) {
next unless ref \$item;
\$is_solving = 1;
last CHECK_FINAL;
}
}
for my \$row (\$puzzle->@*) {
my @row = \$row->@*;
my @line = map { join ' ', '[', map ({
\$is_solving ? sprintf('%19s', ref \$_ ? "{@\$_}" : \$_) : \$_
} splice(@row, 0, 3)), ']' } 1 .. 3;
say {*STDERR} join ' ', @line;
print {*STDERR} "\n" if (\$i % 3 == 0) && (\$i < 9);
++\$i;
} ## end for my \$row (\$puzzle->@*)
return;
}

sub print_puzzle (\$puzzle) {
say {*STDOUT} join ' ', '[', \$_->@*, ']' for \$puzzle->@*;
return;
}

sub main (\$filename = undef) {
my \$fh =
!defined(\$filename) ? \*DATA
: (\$filename eq '-')  ? \*STDIN
:                       do { open my \$fh, '<', \$filename; \$fh };
my @puzzle;
while (<\$fh>) {
my @line = grep { m{[_1-9]} } split m{\s+}mxs;
die "wrong number of elements in line \$.\n" unless @line == 9;
push @puzzle, \@line;
last if \$. == 9;
} ## end while (<\$fh>)
die "not enough rows\n" unless @puzzle == 9;
debug_puzzle(\@puzzle);
print {*STDERR} "\n";
my \$solved_puzzle = sudoku_puzzle(\@puzzle);
print_puzzle(\$solved_puzzle);
print {*STDERR} "\n";
debug_puzzle(\$solved_puzzle);
return;
} ## end sub main (\$filename = undef)

main(@ARGV);

__DATA__
[ _ 4 9 7 3 _ _ _ _ ]
[ _ _ 8 _ _ _ 6 7 _ ]
[ _ 7 6 _ 5 _ _ _ _ ]
[ _ _ 7 9 _ _ _ _ _ ]
[ _ 6 _ _ _ _ _ 5 _ ]
[ _ _ _ _ _ 1 7 _ _ ]
[ _ _ _ _ 1 _ 8 2 _ ]
[ _ 9 1 _ _ _ 4 _ _ ]
[ _ _ _ _ 2 7 5 1 _ ]
``````

It’s a huge program compared to other Perl Weekly Challenge solutions… but it gave me the opportunity to look back at an interesting topic and testing the flexibility of ConstraintSolver.pod, a piece of code I wrote some time ago!

Cheers!

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