Algorithm::Munkres(3pm) 				User Contributed Perl Documentation				   Algorithm::Munkres(3pm)

NAME

	   Algorithm::Munkres - Perl extension for Munkres' solution to
	   classical Assignment problem for square and rectangular matrices
	   This module extends the solution of Assignment problem for square
	   matrices to rectangular matrices by padding zeros. Thus a rectangular
	   matrix is converted to square matrix by padding necessary zeros.

SYNOPSIS

       use Algorithm::Munkres;

	   @mat = (
		[2, 4, 7, 9],
		[3, 9, 5, 1],
		[8, 2, 9, 7],
		);

       assign(@mat,@out_mat);

	   Then the @out_mat array will have the output as: (0,3,1,2),
	   where
	   0th element indicates that 0th row is assigned 0th column i.e value=2
	   1st element indicates that 1st row is assigned 3rd column i.e.value=1
	   2nd element indicates that 2nd row is assigned 1st column.i.e.value=2
	   3rd element indicates that 3rd row is assigned 2nd column.i.e.value=0

DESCRIPTION

	   Assignment Problem: Given N jobs, N workers and the time taken by
	   each worker to complete a job then how should the assignment of a
	   Worker to a Job be done, so as to minimize the time taken.

	       Thus if we have 3 jobs p,q,r and 3 workers x,y,z such that:
		   x  y  z
		p  2  4  7
		q  3  9  5
		r  8  2  9

	       where the cell values of the above matrix give the time required
	       for the worker(given by column name) to complete the job(given by
	       the row name)

	       then possible solutions are:
				Total
		1. 2, 9, 9	 20
		2. 2, 2, 5	  9
		3. 3, 4, 9	 16
		4. 3, 2, 7	 12
		5. 8, 9, 7	 24
		6. 8, 4, 5	 17

	   Thus(2) is the optimal solution for the above problem.
	   This kind of brute-force approach of solving Assignment problem
	   quickly becomes slow and bulky as N grows, because the number of
	   possible solution are N! and thus the task is to evaluate each
	   and then find the optimal solution.(If N=10, number of possible
	   solutions: 3628800 !)
	   Munkres' gives us a solution to this problem, which is implemented
	   in this module.

	   This module also solves Assignment problem for rectangular matrices
	   (M x N) by converting them to square matrices by padding zeros. ex:
	   If input matrix is:
		[2, 4, 7, 9],
		[3, 9, 5, 1],
		[8, 2, 9, 7]
	   i.e 3 x 4 then we will convert it to 4 x 4 and the modified input
	   matrix will be:
		[2, 4, 7, 9],
		[3, 9, 5, 1],
		[8, 2, 9, 7],
		[0, 0, 0, 0]

EXPORT

	   "assign" function by default.

INPUT

	   The input matrix should be in a two dimensional array(array of
	   array) and the 'assign' subroutine expects a reference to this
	   array and not the complete array.
	   eg:assign(@inp_mat, @out_mat);
	   The second argument to the assign subroutine is the reference
	   to the output array.

OUTPUT

	   The assign subroutine expects references to two arrays as its
	   input paramenters. The second parameter is the reference to the
	   output array. This array is populated by assign subroutine. This
	   array is single dimensional Nx1 matrix.
	   For above example the output array returned will be:
	    (0,
	    2,
	    1)

	   where
	   0th element indicates that 0th row is assigned 0th column i.e value=2
	   1st element indicates that 1st row is assigned 2nd column i.e.value=5
	   2nd element indicates that 2nd row is assigned 1st column.i.e.value=2

SEE ALSO

	   1. http://216.249.163.93/bob.pilgrim/445/munkres.html

	   2. Munkres, J. Algorithms for the assignment and transportation
	      Problems. J. Siam 5 (Mar. 1957), 32-38

	   3. FranA~Xois Bourgeois and Jean-Claude Lassalle. 1971.
	      An extension of the Munkres algorithm for the assignment
	      problem to rectangular matrices.
	      Communication ACM, 14(12):802-804

AUTHOR

	   Anagha Kulkarni, University of Minnesota Duluth
	   kulka020 <at> d.umn.edu

	   Ted Pedersen, University of Minnesota Duluth
	   tpederse <at> d.umn.edu

COPYRIGHT AND LICENSE

       Copyright (C) 2007-2008, Ted Pedersen and Anagha Kulkarni

       This program 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 2 of the License, or (at your option) any later version.  This program 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 this program; if not, write to the Free Software Foundation,
       Inc., 59 Temple Place - Suite 330, Boston, MA  02111-1307, USA.

perl v5.10.0							    2008-10-22						   Algorithm::Munkres(3pm)