my $ColumnCount = 4; # or whatever we need it to be
my $CategoryCount = 6; # or whatever we need it to be

my %Weight = ();
# TODO: initialize %Weight hash with values from 0 to Q-1
my %WeightCache = (); # to be filled with weights
# TODO: initialize weight cache to handle references to 
# hashes with indexes 1..Q-1

# brute-force would iterate across combinations of 1...Q-1;
# for 6 categories and 4 columns, this is 5 choose 3 i.e. 10

sub WeightSpan
{
    my ($LowBound,$HighBound) = @_;
    my $WeightSum = 0;
    $WeightSum += $Weight{$LowBound++}
                                 while $LowBound<$HighBound;
    return $WeightSum;
}

sub IncrementCombination
{
    my ($LowBound,$HighBound,@Combination) = @_;
    my $Idx = 0;
    while($Combination[$Idx-1]=$HighBound-$Idx)
        {
        return () if 1==$Idx+scalar @Combination;
        $Idx--;
        };
    $Combination[$Idx-1]++;
    $Combination[++$Idx -1]++ while $Idx;
    return @Combination;
}

sub MaximumHeight
{
    my ($LowBound,$HighBound,@Combination) = @_;
    my @IndexSpan = ();
    # TODO: auto-init @IndexSpan: 0...scalar @Combination-1
    return max(map
       {$WeightCache{$Combination[$_]}{$Combination[$_+1]}
    ||= &WeightSpan($Combination[$_],$Combination[$_+1])}
               @IndexSpan);
}

sub CrunchBestCombination
{
    my ($LowBound,$HighBound,$N) = @_;
    my @Combination = ();
    # TODO: auto-init: fill @Combination with 1...$N-1
    my @BestCombination = @LowestBounds;
    my $BestHeight =  &MaximumHeight($LowBound, $HighBound,
                                     @Combination);
    &IncrementCombination($LowBound, $HighBound, 
                          @Combination);

    while(@Combination)
        {
        my $CurrentMaxHeight = 
          &MaximumHeight($LowBound,$HighBound,@Combination);
        if ($CurrentMaxHeight<$BestHeight)
            {
            $BestHeight = $CurrentMaxHeight;
            @BestCombination = @Combination;
            }
        @Combination =  &IncrementCombination($LowBound, 
                                  $HighBound, @Combination);
        }
    return ($BestHeight,@BestCombination);
}

my @TargetCombination = &CrunchBestCombination(1, 
                            $ColumnCount-1, $CategoryCount);
my $BestHeight = shift(@TargetCombination);

# TODO: translate @TargetCombination into </td><td> 
# cell breaks