N-ary Recursion & Permutations

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The most general form of recursion is n-ary recursion where n is not a constant but some parameter of a routine. Routines of this kind are useful in generating combinatorial objects such as permutations. The permutations of the integers 1..3 are:

1 2 3
1 3 2
2 1 3
2 3 1
3 1 2
3 2 1

Permutations of 1..3
There are three choices for the value in each column above (except that no integer can be repeated in a permutation) and three nested loops could be used to iterate through all 3! permutations:
for i1 :1..3
   for i2 :1..3
      if i2 ~= i1 then
         for i3 :1..3
            if i3 ~= i1 and i3 ~= i2 then
               print i1, i2, i3
            end if
         end for
      end if
   end for
end for
However this is ugly and cannot be generalised to produce permutations of 1..n where n is not a constant. The solution is to write a recursive procedure which contains just one loop. Within the loop, the procedure calls itself recursively. This gives the effect of a variable number of nested loops.

Permute(unused, p, L, N)
   if(L > N)
      process the permutation p
   else
      for each i in unused do
         p[L]=i;
         Permute(unused-{i}, p, L+1, N)    -- recursion
      end for
   end if
end Permute;

Permute({1..n}, p, 1, n)   -- initial call

A set is used to hold the free `unused' choices and to ensure that all elements of a permutation differ. Initially all marks are unused. Unused is reduced as choices are made and it is passed on through recursive calls. (If your programming language does not have set as a built in data type, you can implement it as a bit-mask, or as an array of boolean or of integer.)

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There are a great many ways to visualise the calls that Permute makes on itself:

-: Permute({1,2,3}, [-,-,-], 1, 3)
   1: Permute({2,3}, [1,-,-], 2, 3)
      1.1: Permute({3}, [1,2,-], 3,3)
           1.1.1: Permute({}, [1,2,3], 4, 3) --> 1,2,3
      1.2: Permute({2}, [1,3,-], 3,3)
           1.1.1: Permute({}, [1,3,2], 4, 3) --> 1,3,2
   2: Permute({1,3}, [2,-,-],  2, 3)
      2.1: Permute({3}, [2,1,-], 3,3)
           1.1.1: Permute({}, [2,1,3], 4, 3) --> 2,1,3
      2.2: Permute({1}, [2,3,-], 3,3)
           1.1.1: Permute({}, [2,3,1], 4, 3) --> 2,3,1
   3: Permute({1,2}, [3,-,-], 2, 3)
      3.1: Permute({2}, [3,1,-], 3,3)
           1.1.1: Permute({}, [3,1,2], 4, 3) --> 3,1,2
      3.2: Permute({1}, [3,2,-], 3,3)
           1.1.1: Permute({}, [3,2,1], 4, 3) --> 3,2,1

--- Call Trace of Permute ---


                      -: Permute({1,2,3}, [-,-,-], 1, 3)
                   .     .
                .           .
             .                 .
          .                       .
         1: Permute({2,3}, [1,-,-], 2, 3)
        .   .                            .
       .       .                            .
      .           .                          2: Permute({1,3}, [2,-,-], 2, 3)
     .               .                          etc.
    .                   .                      3: Permute({1,2}, [3,-,-], 2, 3)
   .                       .                       etc.
1.1: Permute({3}, [1,2,-], 3, 3)
  . .                              .
  .    .                            1.2: Permute({2}, [1,3,-], 3, 3)
  .       .
1.1.1: Permute({}, [1,2,3], 4, 3)
                .
                   .
                   1.2.1: Permute({}, [1,3,2], 4, 3)

--- Partial Tree of Calls for Permute({1,2,3}, p, 1, 3) ---

See also the later section on tree traversal.

Notes

  • There are iterative, non-recursive routines to generate permutations.
  • Many other `combinatorial objects' can be generated by similar routines.
  • Permutations figure in group theory.
  • An application of permutations is to test parallel programs that take inputs from different devices, simulating input from these devices in many different orders (permutations) to test for dead-lock situations.
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