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Solving Pi Day Sudoku 2009 with the global cardinality constraint

Here is an update of the Pi Day Sudoku 2009 problem.

After a suggestion from Mikael Zayenz Lagerkvist (of the Gecode team) the model now uses just the global cardinality constraint and is much faster than the model which used my own decomposition of all_different_except_0 (or rather all_different_except_Pi) and the count constraint.

Both of the new MiniZinc and Comet models solves the Pi Day Sudoku 2009 problem in less than a second (compared to couple of seconds before).

The Comet model has the following statistics for finding the first (and unique) solution.

time: 9
#choices = 6
#fail = 1
#propag = 1306

For Gecode/FlatZinc:

runtime: 20
solutions: 1
propagators: 35
propagations: 255
nodes: 3
failures: 0
peak depth: 2
peak memory: 158 KB

Compared to the former model, there are now much less failures.

Some more about the global cardinality constraint
Global cardinality (in Sudoku) is a way of stating how many occurrences there should be of each number per row, column or region. For the Pi Day Sudoku problem the following occurrences are needed:

Pi: 3 occurrences (coded as -1 in the model)
0: 0 occurrences (0 is a dummy value for handling the indata)
1..9: exactly one occurrence each

I MiniZinc the this is coded as

array[1..n, 1..n] of var -1..9: x;
array[-1..9] of 0..3: occ = array1d(-1..9, [3, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1]);
forall(i in 1..n) ( % rows
check([x[i,j] | j in 1..n], occ)

In Comet the function cardinality is used, i,e, m.post(cardinality(occ, all(j in 1..n) x[i,j]));.

For more information about the global cardinality constraint, see the entry in Global Constraint Catalog.

A comment
Introductions to constraint programming nowadays tend to start with the Sudoko problem (or SEND + MORE = MONEY). These models often use the alldifferent constraint, and seldom, if ever, have I seen any presentation using global cardinality. One reason may be that alldifferent is easier to understand in a introduction, another reason may be that it is general a better constraint to use for these problems.

"Vanilla" Sudoko solver using global cardinality
I have compared some few "vanilla" Sudoku problems with either alldifferent or global cardinality but haven't done any systematic comparison. Sometimes the global cardinality approach is better, sometimes it is somewhat worse.

sudoku_gcc.mzn is a "vanilla" Sudoku solver in MiniZinc using the global cardinality constraint. And the same approach in Comet: sudoku_gcc.co. Note: These models includes just a few problems.

Also see: Gecode's Sudoku solver is very fast, and includes 88 different problems.


The Pi Day Sudoku is very interesting. I'd also like to share a math puzzle that I recently came across: http://www.chycho.com/?q=Puzzle .

Calvin: Nice puzzle.

I did a MiniZinc model that solve n=5 (instead of your n=10) in just a blink, and for n=6 in 15 seconds.

This is modelled as a variant of the Hidato puzzle (the MiniZinc model is hidato.mzn) with just a few extra constraints.

That's interesting, hakank. Did you try n=10 with MiniZinc?

Calvin: Yes, I tried n = 10, but with my current MiniZinc model it simply takes too long. I have to think of a different approach.

By the way, the current model is here: calvin_puzzle.mzn. (Sorry for the name. :-)

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