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## -*- texinfo -*-
## @deftypefn {} {@var{P} =} perms (@var{v})
## @deftypefnx {} {@var{P} =} perms (@var{v}, "unique")
## Generate all permutations of vector @var{v} with one row per permutation.
##
## Results are returned in inverse lexicographic order. The result has size
## @code{factorial (@var{n}) * @var{n}}, where @var{n} is the length of
## @var{v}. Any repeated elements are included in the output.
##
## If the optional argument @qcode{"unique"} is given, then only unique
## permutations are returned, using less memory and generally taking less time
## than calling @code{unique (perms (@var{v}), "rows")}.
##
## Example 1
##
## @example
## @group
## perms ([1, 2, 3])
## @result{}
## 3 2 1
## 3 1 2
## 2 3 1
## 2 1 3
## 1 3 2
## 1 2 3
## @end group
## @end example
##
## Example 2
##
## @example
## @group
## perms ([1, 1, 2, 2], "unique")
## @result{}
## 2 2 1 1
## 2 1 2 1
## 2 1 1 2
## 1 2 2 1
## 1 2 1 2
## 1 1 2 2
## @end group
## @end example
##
## Programming Note: If the @qcode{"unique"} option is not used, the length of
## @var{v} should be no more than 10-12 to limit memory consumption. Even with
## @qcode{"unique"}, there should be no more than 10-12 unique elements in @var{v}.
## @seealso{permute, randperm, nchoosek}
## @end deftypefn
## FIXME: In principle it should be more efficient to do indexing using uint8
## type. However, benchmarking shows doubles are faster. If this changes in
## a later version of Octave the index variables here can be made uint8.
function P = perms (v, opt)
if (nargin < 1)
print_usage ();
endif
unique_v=0;
if (nargin == 2)
if (! strcmpi (opt, "unique"))
error ('perms: option must be the string "unique".');
else
unique_v=1;
endif
endif
v = v(:).'; # convert to row vector
if (isnumeric (v) || ischar (v))
## Order of output is only dependent on the actual values for
## character and numeric arrays.
v = sort (v, "ascend");
endif
n = numel (v);
if (n < 4) # special cases for small n
switch (n)
case 0
P = reshape (v, 1, 0);
case 1
P = v;
case 2
P = [v([2 1]);v];
case 3
P = v([3 2 1; 3 1 2; 2 3 1; 2 1 3; 1 3 2; 1 2 3]);
endswitch
if unique_v==1
P=unique(P,"rows");
end
else
if unique_v==0
v = v(end:-1:1);
n-= 1;
idx = zeros (factorial (n), n);
idx(1:6, n-2:n) = [1, 2, 3;1, 3, 2;2, 1, 3;2, 3, 1;3, 1, 2;3, 2, 1]+(n-3);
f = 2; # jump-start for efficiency with medium n
for j = 3:n-1
b = 1:n;
f *= j;
perm = idx(1:f, n-(j-1):n);
idx(1:(j+1)*f, n-j) = (n-j:n)(ones (f, 1),:)(:);
for i=0:j
b(i+n-j) -= 1;
idx((1:f)+i*f, n-(j-1):n) = b(perm);
endfor
endfor
n += 1;
f *= n-1;
P = v(1)(ones (factorial (n), n));
P(:,1) = v(ones (f, 1),:)(:);
for i = 1:n
b = v([1:i-1 i+1:n]);
P((1:f)+(i-1)*f, 2:end) = b(idx);
endfor
else
[v,i,j]=unique(v);
h=accumarray(j',1)';
idx = m_perms (h);
P = v(sortrows (idx'));
end
endif
endfunction
function out_perms=m_perms(multis)
l=length(multis);
if l==1
out_perms=uint8(ones(multis,1));
else
p1=m_perms(multis(1:floor(l/2)));
p2=m_perms(multis(floor(l/2)+1:l))+max(p1(:,1));
l1=size(p1,1);
l2=size(p2,1);
p=nchoosek(1:l1+l2,l1);
i=false(l1+l2,size(p,1));
i(p+(0:size(p,1)-1)'*(l1+l2))=true;
out_perms=zeros(l1+l2,size(p1,2)*size(p2,2)*size(p,1),"uint8");
out_perms(repmat(i,size(p1,2)*size(p2,2),1)(:))=repmat(p1,size(p2,2),size(p,1))(:);
out_perms(repmat(!i,size(p1,2)*size(p2,2),1)(:))=repmat(p2,1,size(p1,2)*size(p,1))(:);
end
endfunction
%!assert (rows (perms (1:6)), factorial (6))
%!assert (perms (pi), pi)
%!assert (perms ([pi, e]), [pi, e; e, pi])
%!assert (perms ([1,2,3]), [3,2,1;3,1,2;2,3,1;2,1,3;1,3,2;1,2,3])
%!assert (perms (1:5), perms ([2 5 4 1 3]'))
%!assert (perms ("abc"), char ("cba", "cab", "bca", "bac", "acb", "abc"))
%!assert (perms ("fobar"), sortrows (unique (perms ("fobar"), "rows"), -(1:5)))
%!assert (unique (perms (1:5)(:))', 1:5)
%!assert (perms (int8 (1:4)), int8 (perms (1:4)))
%!assert (sortrows (perms ("abb", "unique")), ["abb"; "bab"; "bba"]);
%!assert (size (perms ([1 1 1 1 2 2 2 3 3], "unique")), [1260 9])
%!assert (size (perms (int8([1 1 1 1 1 2 2 2 2 3 3 3]), "unique")), [27720 12])
## Should work for any array type, such as cells and structs, and not
## only for numeric data.
%!assert <*52431> (perms ({1}), {1})
%!assert <*52431> (perms ({0.1, "foo"}), {"foo", 0.1; 0.1, "foo"})
%!assert <*52431> (perms ({"foo", 0.1}), {0.1, "foo"; "foo", 0.1})
%!assert <*52431> (perms ({"foo"; 0.1}), {0.1, "foo"; "foo", 0.1})
%!assert <*52431> (perms ({0.1; "foo"}), {"foo", 0.1; 0.1, "foo"})
%!assert <*52431> (perms ({"foo", "bar"}), {"bar", "foo"; "foo", "bar"})
%!assert <*52431> (perms ({"bar", "foo"}), {"foo", "bar"; "bar", "foo"})
%!
%!assert <*52431> (perms (struct ()), struct ())
%!assert <*52431> (perms (struct ("foo", {1, 2})),
%! struct ("foo", {2, 1; 1, 2}))
%!assert <*52431> (perms (struct ("foo", {1, 2}, "bar", {3, 4})),
%! struct ("foo", {2, 1; 1, 2}, "bar", {4, 3; 3, 4}))
## Also sort logical input with order dependent on the input order and
## not their values.
%!assert <*52431> (perms (logical ([1 0])), logical ([0 1;, 1 0]))
%!assert <*52431> (perms (logical ([0 1])), logical ([1 0; 0 1]))
%!assert <*52431> (perms (logical ([0 1 0])),
%! logical ([0 1 0; 0 0 1; 1 0 0; 1 0 0; 0 0 1; 0 1 0]))
%!assert <*52431> (perms (logical ([0 1 1])),
%! logical ([1 1 0; 1 0 1; 1 1 0; 1 0 1; 0 1 1; 0 1 1]))
%!assert <*52432> (perms ([]), reshape ([], 1, 0))
%!assert <*52432> (perms (single ([])), reshape (single ([]), 1, 0))
%!assert <*52432> (perms (int8 ([])), reshape (int8 ([]), 1, 0))
%!assert <*52432> (perms ({}), cell (1, 0))
%!test <*52432>
%! s = struct ();
%! s(1) = [];
%! assert (perms (reshape (s, 0, 0)), reshape (s, 1, 0));
%! assert (perms (reshape (s, 0, 1)), reshape (s, 1, 0));
## Test input validation
%!error perms ()
%!error