######################################################################## ## ## Copyright (C) 2001-2022 The Octave Project Developers ## ## See the file COPYRIGHT.md in the top-level directory of this ## distribution or . ## ## This file is part of Octave. ## ## Octave is free software: you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## ######################################################################## ## -*- 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