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. François 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.

