wolffd@0: % MATCH - Solves the weighted bipartite matching (or assignment)
wolffd@0: %         problem.
wolffd@0: %
wolffd@0: % Usage:  a = match(C);
wolffd@0: %
wolffd@0: % Arguments:   
wolffd@0: %         C     - an m x n cost matrix; the sets are taken to be
wolffd@0: %                 1:m and 1:n; C(i, j) gives the cost of matching
wolffd@0: %                 items i (of the first set) and j (of the second set)
wolffd@0: %
wolffd@0: % Returns:
wolffd@0: %
wolffd@0: %         a     - an m x 1 assignment vector, which gives the
wolffd@0: %                 minimum cost assignment.  a(i) is the index of
wolffd@0: %                 the item of 1:n that was matched to item i of
wolffd@0: %                 1:m.  If item i (of 1:m) was not matched to any 
wolffd@0: %                 item of 1:n, then a(i) is zero.
wolffd@0: 
wolffd@0: % Copyright (C) 2002 Mark A. Paskin
wolffd@0: %
wolffd@0: % This program is free software; you can redistribute it and/or modify
wolffd@0: % it under the terms of the GNU General Public License as published by
wolffd@0: % the Free Software Foundation; either version 2 of the License, or
wolffd@0: % (at your option) any later version.
wolffd@0: %
wolffd@0: % This program is distributed in the hope that it will be useful, but
wolffd@0: % WITHOUT ANY WARRANTY; without even the implied warranty of
wolffd@0: % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
wolffd@0: % General Public License for more details.
wolffd@0: %
wolffd@0: % You should have received a copy of the GNU General Public License
wolffd@0: % along with this program; if not, write to the Free Software
wolffd@0: % Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307
wolffd@0: % USA.
wolffd@0: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
wolffd@0: 
wolffd@0: function [a] = optimalMatching(C)
wolffd@0: 
wolffd@0: % Trivial cases:
wolffd@0: [p, q] = size(C);
wolffd@0: if (p == 0)
wolffd@0:   a = []; 
wolffd@0:   return;
wolffd@0: elseif (q == 0)
wolffd@0:   a = zeros(p, 1);
wolffd@0:   return;
wolffd@0: end
wolffd@0: 
wolffd@0: 
wolffd@0: if  0
wolffd@0: % First, reduce the problem by making easy optimal matches.  If two
wolffd@0: % elements agree that they are the best match, then match them up.
wolffd@0: [x, a] = min(C, [], 2);
wolffd@0: [y, b] = min(C, [], 1);
wolffd@0: u = find(1:p ~= b(a(:)));
wolffd@0: a(u) = 0;
wolffd@0: v = find(1:q ~= a(b(:))');
wolffd@0: C = C(u, v);
wolffd@0: if (isempty(C)) return; end
wolffd@0: end
wolffd@0: 
wolffd@0: % Get the (new) size of the two sets, u and v.
wolffd@0: [m, n] = size(C);
wolffd@0: 
wolffd@0: %mx = realmax;
wolffd@0: mx = 2*max(C(:));
wolffd@0: mn = -2*min(C(:));
wolffd@0: % Pad the affinity matrix to be square
wolffd@0: if (m < n)
wolffd@0:   C = [C; mx * ones(n - m, n)];
wolffd@0: elseif (n < m)
wolffd@0:   C = [C, mx * ones(m, m - n)];
wolffd@0: end
wolffd@0: 
wolffd@0: % Run the Hungarian method.  First replace infinite values by the
wolffd@0: % largest (or smallest) finite values.
wolffd@0: C(find(isinf(C) & (C > 0))) = mx;
wolffd@0: C(find(isinf(C) & (C < 0))) = mn;
wolffd@0: %fprintf('running hungarian\n');
wolffd@0: [b, cost] = hungarian(C');
wolffd@0: 
wolffd@0: % Extract only the real assignments
wolffd@0: ap = b(1:m)';
wolffd@0: ap(find(ap > n)) = 0;
wolffd@0: 
wolffd@0: a = ap; 
wolffd@0: %% Incorporate this sub-assignment into the complete assignment
wolffd@0: %  k = find(ap);
wolffd@0: %  a(u(k)) = v(ap(k));
wolffd@0: