Algorithm::Munkres







    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.

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


    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 !)









                             ‐2‐


    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]


    "assign" function by default.


    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.


    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


    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











                             ‐3‐



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

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

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

     This program is free software; you can redistribute it
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License as published by the Free Software Foundation; either
version 2 of the License, or (at your option) any later
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