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Re: Puzzle: need a more general algorithm

by zaimoni (Beadle)
on Jul 09, 2002 at 09:55 UTC ( #180438=note: print w/ replies, xml ) Need Help??


in reply to Puzzle: need a more general algorithm

Thinking vaguely...

Notation:

  • Columns: N
  • Category count: Q

Assuming the problem makes sense (Q>N), yet another way to think about the problem space is to enumerate the cells like this:

1 3 5 7
2 4 6 8

Then, if we index the categories by 0...Q-1 (fine because the category order must be preserved), we can consider the location of a category to be L(category index). The nature of this location is yet to be determined. I see that the categories always fill up from the top down.

Now, implementation details are critical...so I may be off on a wild goose chase. I'm envisioning this target (X)HTML table as a single row with a single cell for each column. Then the top-down ordering is automatic, and we're just calculating where to put the table cell delimiters. In this case, the location L corresponds simply to the column, and I can forget about rows entirely for the analysis. Then

L(0)<=L(1)<=...<=L(Q-2)<=L(Q-1)

In particular, L(0)=1 and L(Q-1)=N. In some way, we need to track the mininum category index for each column 2...N.

Merlyn pointed out the above, in his comment about binary representation.

Untested pseudo-code follows:

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

Obviously, all TODO comments should be implemented before the code has any chance of working. Also, a number of the example functions can be made more succinct.


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