My Constraint Programming Blog

The series Learning Constraint Programming is a preparation for My talk at SweConsNet Workshop 2009: "Learning Constraint Programming (MiniZinc, JaCoP, Choco, Gecode/R, Comet, Gecode): Some Lessons Learned". Confusingly, the entries is not numbered in any logical order. Sorry about that.

Here are the previous entries:

• Learning constraint programming (languages) - part I
• Learning constraint programming - part II: Modeling with the Element constraint

Introduction

The global constraint alldifferent_except_0 (or the more general variant alldifferent_except_c) is one of my favorite global constraints. It is very handy to use when 0 (or any constant c) is coded as an unknown or irrelevant value. Then we can constraint the rest to be all distinct.

The great Global Constraint Catalog entry alldifferent_except_0) explains this constraint as:

Purpose
Enforce all variables of the collection VARIABLES to take distinct values, except those variables that are assigned to 0.

Example
​(<5,​0,​1,​9,​0,​3>)​
The alldifferent_except_0 constraint holds since all the values (that are different from 0) 5, 1, 9 and 3 are distinct.

Models

I have modeled a decomposition of alldifferent_except_0 in the following models, where the constraint is just tested perhaps combined with some other constraint, e.g. sorted or that there must be at least some zeros:

- MiniZinc alldifferent_except_0.mzn
- Comet: alldifferent_except_0.co
- Gecode/R: all_different_except_0.rb
- Choco: AllDifferentExcept0_test.java
- JaCoP: AllDifferentExcept0_test.java
- Gecode: alldifferent_except_0.cpp

Some models using alldifferent_except_0

And here is some real use of the constraint:

- Nonogram (Comet): nonogram.co. (A faster model using the regular constraint, nonogram_regular.co, is described here and here).
- I wrote about alldifferent_except_0 in Pi Day Sudoku 2009. However, as faster way of solving the problem was found and is described in Solving Pi Day Sudoku 2009 with the global cardinality constraint). Note: the competition is still on, so there is no link to any model.
- Sudoku generate (Comet): sudoku_generate.co
- all paths graph (MiniZinc): all_paths_graph.mzn
- Cube sum (MiniZinc): cube_sum.mzn
- Message sending (MiniZinc): message_sending.mzn

As the first two entries indicates, there may be faster solutions than using (a decomposition) of alldifferent_except_0, but even as a decomposition is a great conceptual tool when modeling a problem.

Implementations

In the implementations below we also see how to define a function (predicate) in the constraint programming systems.

For the Gecode/R model there are different approaches:
- "standard" ("direct") approach where we loop over all different pairs of elements and ensures that if both values are different from 0 then they should be different
- using count
- using global cardinality ("simulated" in Gecode/R, see below)

Also, in some models we use the slighly more general version alldifferent_except_c where c is any constant (e.g. "Pi" in the Pi Day Sudoku puzzle mentioned above.

MiniZinc:

Model: alldifferent_except_0.mzn.

predicate all_different_except_0(array[int] of var int: x) =
   let {
      int: n = length(x)
   }
   in
   forall(i,j in 1..n where i != j) (
        (x[i] > 0 /\ x[j] > 0) 
        -> 
        x[i] != x[j]
   )
;

// usage:
constraint all_different_except_0(x);

Comet:

Model: alldifferent_except_0.co.

function void alldifferent_except_0(Solver m, var{int}[] x) {
  int n = x.getSize();
  forall(i in 1..n, j in i+1..n) {
    m.post(
           x[i] > 0 && x[j] > 0 
           => 
           x[i] != x[j]
           );
  }
}

// usage
exploreall {
  // ...
  alldifferent_except_0(m, x);
}

Gecode/R

Model:all_different_except_0.rb.

When modeling the constraint in Gecode/R, I experimented with different approaches. The reification variant all_different_except_0_reif is actually quite fast.

# The simplest and the fastest implementation 
# using count for 1..max (poor man's global cardinality)
def all_different_except_0
  (1..self[0].domain.max).each{|i|
    self.count(i).must <= 1
  }
end

# global cardinality version using an extra array with the counts
def global_cardinality(xgcc)
  (self[0].domain.max+1).times{|i|
    xgcc[i].must == self.count(i)
  }
end

#
# The standard approach using reification.
#
def all_different_except_0_reif(x)
  n = x.length
  b1_is_an bool_var_matrix(n,n)
  b2_is_an bool_var_matrix(n,n)
  b3_is_an bool_var_matrix(n,n)
  n.times{|i|
    n.times{|j|
      if i != j then 
        x[i].must_not.equal(0, :reify => b1[i,j]) 
        x[i].must_not.equal(0, :reify => b2[i,j]) 
        x[i].must_not.equal(x[j], :reify => b3[i,j])
        (b1[i,j] & b2[i,j]).must.imply(b3[i,j])
      else 
        b1[i,j].must.true
        b2[i,j].must.true
        b3[i,j].must.true
      end
    }
  }
end

    # ...
    # usage:
    x.all_different_except_0
    # all_different_except_0_gcc(x)
    # all_different_except_0_reif(x)

Choco

Model: AllDifferentExcept0_test.java.

Note that here alldifferent_except_0 is derived from the more general version alldifferent_except_c.

//
// decomposition of alldifferent except 0
//
public void allDifferentExcept0(CPModel m, IntegerVariable[] v) {
    allDifferentExceptC(m, v, 0);
}

//
// slightly more general: alldifferent except c
//
public void allDifferentExceptC(CPModel m, IntegerVariable[] v, int c) {
    int len = v.length;
    for(int i = 0; i < v.length; i++) {
        for(int j = i+1; j < v.length; j++) {
            m.addConstraint(ifThenElse(
                                       and(
                                           gt(v[i], c), 
                                           gt(v[j], c)
                                           ), 
                                       neq(v[i], v[j]),
                                       TRUE
                                       )
                            );
        }
    }    
}

// ...
// usage: 

    m.addConstraint(allDifferent(queens));

JaCoP

Model: AllDifferentExcept0_test.java

This is exactly the same approach as the Choco version.

//
// decomposition of alldifferent except 0
//
public void allDifferentExcept0(FDstore m, FDV[] v) {
    allDifferentExceptC(m, v, 0);
} // end allDifferentExcept0


//
// slightly more general: alldifferent except c
//
public void allDifferentExceptC(FDstore m, FDV[] v, int c) {
    int len = v.length;
    for(int i = 0; i < v.length; i++) {
        for(int j = i+1; j < v.length; j++) {
            m.impose(new IfThen(
                                       new And(
                                           new XneqC(v[i], c), 
                                           new XneqC(v[j], c)
                                           ), 
                                       new XneqY(v[i], v[j])
                                       )
                            );
        }
    }        
} // end allDifferentExceptC


        // ...
        // usage:
        allDifferentExcept0(store, x);

Gecode

alldifferent_except_0.cpp

The Gecode version is very succint since it use overloaded boolean operators. Very nice.

void alldifferent_except_0(Space& space, IntVarArray x, IntConLevel icl = ICL_BND) {
  for(int i = 0; i < x.size(); i++) {
    for(int j = i+1; j < x.size(); j++) {
      post(space,
           tt(
           imp(x[i] != 0 && x[j] != 0, 
           // =>
           x[i] != x[j])),
           icl
           );
    }
  }
} // alldifferent_except_0


// ...
// usage:
    alldifferent_except_0(*this, x, opt.icl());