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:
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.)
L . A l l i s o n
There are a great many ways to visualise the calls that Permute
makes on itself:
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.