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Module munkres
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<h1 class="epydoc">Module munkres</h1><p class="nomargin-top"><span class="codelink"><a href="munkres-pysrc.html">source code</a></span></p>
<div class="rst-section" id="rst-introduction">
<h1 class="heading">Introduction</h1>
<p>The Munkres module provides an implementation of the Munkres algorithm
(also called the Hungarian algorithm or the Kuhn-Munkres algorithm),
useful for solving the Assignment Problem.</p>
</div>
<div class="rst-section" id="rst-assignment-problem">
<h1 class="heading">Assignment Problem</h1>
<p>Let <em>C</em> be an <em>n</em>x<em>n</em> matrix representing the costs of each of <em>n</em> workers
to perform any of <em>n</em> jobs. The assignment problem is to assign jobs to
workers in a way that minimizes the total cost. Since each worker can perform
only one job and each job can be assigned to only one worker the assignments
represent an independent set of the matrix <em>C</em>.</p>
<p>One way to generate the optimal set is to create all permutations of
the indexes necessary to traverse the matrix so that no row and column
are used more than once. For instance, given this matrix (expressed in
Python):</p>
<pre class="rst-literal-block">
matrix = [[5, 9, 1],
[10, 3, 2],
[8, 7, 4]]
</pre>
<p>You could use this code to generate the traversal indexes:</p>
<pre class="rst-literal-block">
def permute(a, results):
if len(a) == 1:
results.insert(len(results), a)
else:
for i in range(0, len(a)):
element = a[i]
a_copy = [a[j] for j in range(0, len(a)) if j != i]
subresults = []
permute(a_copy, subresults)
for subresult in subresults:
result = [element] + subresult
results.insert(len(results), result)
results = []
permute(range(len(matrix)), results) # [0, 1, 2] for a 3x3 matrix
</pre>
<p>After the call to permute(), the results matrix would look like this:</p>
<pre class="rst-literal-block">
[[0, 1, 2],
[0, 2, 1],
[1, 0, 2],
[1, 2, 0],
[2, 0, 1],
[2, 1, 0]]
</pre>
<p>You could then use that index matrix to loop over the original cost matrix
and calculate the smallest cost of the combinations:</p>
<pre class="rst-literal-block">
n = len(matrix)
minval = sys.maxsize
for row in range(n):
cost = 0
for col in range(n):
cost += matrix[row][col]
minval = min(cost, minval)
print minval
</pre>
<p>While this approach works fine for small matrices, it does not scale. It
executes in O(<em>n</em>!) time: Calculating the permutations for an <em>n</em>x<em>n</em>
matrix requires <em>n</em>! operations. For a 12x12 matrix, that's 479,001,600
traversals. Even if you could manage to perform each traversal in just one
millisecond, it would still take more than 133 hours to perform the entire
traversal. A 20x20 matrix would take 2,432,902,008,176,640,000 operations. At
an optimistic millisecond per operation, that's more than 77 million years.</p>
<p>The Munkres algorithm runs in O(<em>n</em>^3) time, rather than O(<em>n</em>!). This
package provides an implementation of that algorithm.</p>
<p>This version is based on
<a class="rst-reference external" href="http://www.public.iastate.edu/~ddoty/HungarianAlgorithm.html" target="_top">http://www.public.iastate.edu/~ddoty/HungarianAlgorithm.html</a>.</p>
<p>This version was written for Python by Brian Clapper from the (Ada) algorithm
at the above web site. (The <tt class="rst-docutils literal"><span class="pre">Algorithm::Munkres</span></tt> Perl version, in CPAN, was
clearly adapted from the same web site.)</p>
</div>
<div class="rst-section" id="rst-usage">
<h1 class="heading">Usage</h1>
<p>Construct a Munkres object:</p>
<pre class="rst-literal-block">
from munkres import Munkres
m = Munkres()
</pre>
<p>Then use it to compute the lowest cost assignment from a cost matrix. Here's
a sample program:</p>
<pre class="rst-literal-block">
from munkres import Munkres, print_matrix
matrix = [[5, 9, 1],
[10, 3, 2],
[8, 7, 4]]
m = Munkres()
indexes = m.compute(matrix)
print_matrix(matrix, msg='Lowest cost through this matrix:')
total = 0
for row, column in indexes:
value = matrix[row][column]
total += value
print '(%d, %d) -> %d' % (row, column, value)
print 'total cost: %d' % total
</pre>
<p>Running that program produces:</p>
<pre class="rst-literal-block">
Lowest cost through this matrix:
[5, 9, 1]
[10, 3, 2]
[8, 7, 4]
(0, 0) -> 5
(1, 1) -> 3
(2, 2) -> 4
total cost=12
</pre>
<p>The instantiated Munkres object can be used multiple times on different
matrices.</p>
</div>
<div class="rst-section" id="rst-non-square-cost-matrices">
<h1 class="heading">Non-square Cost Matrices</h1>
<p>The Munkres algorithm assumes that the cost matrix is square. However, it's
possible to use a rectangular matrix if you first pad it with 0 values to make
it square. This module automatically pads rectangular cost matrices to make
them square.</p>
<p>Notes:</p>
<ul class="rst-simple">
<li>The module operates on a <em>copy</em> of the caller's matrix, so any padding will
not be seen by the caller.</li>
<li>The cost matrix must be rectangular or square. An irregular matrix will
<em>not</em> work.</li>
</ul>
</div>
<div class="rst-section" id="rst-calculating-profit-rather-than-cost">
<h1 class="heading">Calculating Profit, Rather than Cost</h1>
<p>The cost matrix is just that: A cost matrix. The Munkres algorithm finds
the combination of elements (one from each row and column) that results in
the smallest cost. It's also possible to use the algorithm to maximize
profit. To do that, however, you have to convert your profit matrix to a
cost matrix. The simplest way to do that is to subtract all elements from a
large value. For example:</p>
<pre class="rst-literal-block">
from munkres import Munkres, print_matrix
matrix = [[5, 9, 1],
[10, 3, 2],
[8, 7, 4]]
cost_matrix = []
for row in matrix:
cost_row = []
for col in row:
cost_row += [sys.maxsize - col]
cost_matrix += [cost_row]
m = Munkres()
indexes = m.compute(cost_matrix)
print_matrix(matrix, msg='Highest profit through this matrix:')
total = 0
for row, column in indexes:
value = matrix[row][column]
total += value
print '(%d, %d) -> %d' % (row, column, value)
print 'total profit=%d' % total
</pre>
<p>Running that program produces:</p>
<pre class="rst-literal-block">
Highest profit through this matrix:
[5, 9, 1]
[10, 3, 2]
[8, 7, 4]
(0, 1) -> 9
(1, 0) -> 10
(2, 2) -> 4
total profit=23
</pre>
<p>The <tt class="rst-docutils literal">munkres</tt> module provides a convenience method for creating a cost
matrix from a profit matrix. Since it doesn't know whether the matrix contains
floating point numbers, decimals, or integers, you have to provide the
conversion function; but the convenience method takes care of the actual
creation of the cost matrix:</p>
<pre class="rst-literal-block">
import munkres
cost_matrix = munkres.make_cost_matrix(matrix,
lambda cost: sys.maxsize - cost)
</pre>
<p>So, the above profit-calculation program can be recast as:</p>
<pre class="rst-literal-block">
from munkres import Munkres, print_matrix, make_cost_matrix
matrix = [[5, 9, 1],
[10, 3, 2],
[8, 7, 4]]
cost_matrix = make_cost_matrix(matrix, lambda cost: sys.maxsize - cost)
m = Munkres()
indexes = m.compute(cost_matrix)
print_matrix(matrix, msg='Lowest cost through this matrix:')
total = 0
for row, column in indexes:
value = matrix[row][column]
total += value
print '(%d, %d) -> %d' % (row, column, value)
print 'total profit=%d' % total
</pre>
</div>
<div class="rst-section" id="rst-references">
<h1 class="heading">References</h1>
<ol class="rst-arabic simple">
<li><a class="rst-reference external" href="http://www.public.iastate.edu/~ddoty/HungarianAlgorithm.html" target="_top">http://www.public.iastate.edu/~ddoty/HungarianAlgorithm.html</a></li>
<li>Harold W. Kuhn. The Hungarian Method for the assignment problem.
<em>Naval Research Logistics Quarterly</em>, 2:83-97, 1955.</li>
<li>Harold W. Kuhn. Variants of the Hungarian method for assignment
problems. <em>Naval Research Logistics Quarterly</em>, 3: 253-258, 1956.</li>
<li>Munkres, J. Algorithms for the Assignment and Transportation Problems.
<em>Journal of the Society of Industrial and Applied Mathematics</em>,
5(1):32-38, March, 1957.</li>
<li><a class="rst-reference external" href="http://en.wikipedia.org/wiki/Hungarian_algorithm" target="_top">http://en.wikipedia.org/wiki/Hungarian_algorithm</a></li>
</ol>
</div>
<div class="rst-section" id="rst-copyright-and-license">
<h1 class="heading">Copyright and License</h1>
<p>This software is released under a BSD license, adapted from
<<a class="rst-reference external" href="http://opensource.org/licenses/bsd-license.php" target="_top">http://opensource.org/licenses/bsd-license.php</a>></p>
<p>Copyright (c) 2008 Brian M. Clapper
All rights reserved.</p>
<p>Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:</p>
<ul class="rst-simple">
<li>Redistributions of source code must retain the above copyright notice,
this list of conditions and the following disclaimer.</li>
<li>Redistributions in binary form must reproduce the above copyright notice,
this list of conditions and the following disclaimer in the documentation
and/or other materials provided with the distribution.</li>
<li>Neither the name "clapper.org" nor the names of its contributors may be
used to endorse or promote products derived from this software without
specific prior written permission.</li>
</ul>
<p>THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
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SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
POSSIBILITY OF SUCH DAMAGE.</p>
</div>
<hr />
<div class="fields"> <p><strong>Version:</strong>
1.0.7
</p>
<p><strong>Author:</strong>
Brian Clapper, bmc@clapper.org
</p>
<p><strong>Copyright:</strong>
(c) 2008 Brian M. Clapper
</p>
<p><strong>License:</strong>
BSD-style license
</p>
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<a href="munkres.Munkres-class.html" class="summary-name">Munkres</a><br />
Calculate the Munkres solution to the classical assignment problem.
See the module documentation for usage.
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<td><span class="summary-sig"><a href="munkres-module.html#make_cost_matrix" class="summary-sig-name">make_cost_matrix</a>(<span class="summary-sig-arg">profit_matrix</span>,
<span class="summary-sig-arg">inversion_function</span>)</span><br />
Create a cost matrix from a profit matrix by calling
'inversion_function' to invert each value. The inversion
function must take one numeric argument (of any type) and return
another numeric argument which is presumed to be the cost inverse
of the original profit.</td>
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<span class="summary-sig-arg">msg</span>=<span class="summary-sig-default">None</span>)</span><br />
Convenience function: Displays the contents of a matrix of integers.</td>
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<h3 class="epydoc"><span class="sig"><span class="sig-name">make_cost_matrix</span>(<span class="sig-arg">profit_matrix</span>,
<span class="sig-arg">inversion_function</span>)</span>
</h3>
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<p>Create a cost matrix from a profit matrix by calling
'inversion_function' to invert each value. The inversion
function must take one numeric argument (of any type) and return
another numeric argument which is presumed to be the cost inverse
of the original profit.</p>
<p>This is a static method. Call it like this:</p>
<pre class="py-doctest">
cost_matrix = Munkres.make_cost_matrix(matrix, inversion_func)</pre>
<p>For example:</p>
<pre class="py-doctest">
cost_matrix = Munkres.make_cost_matrix(matrix, <span class="py-keyword">lambda</span> x : sys.maxsize - x)</pre>
<dl class="fields">
<dt>Parameters:</dt>
<dd><ul class="nomargin-top">
<li><strong class="pname"><code>profit_matrix</code></strong> (list of lists) - The matrix to convert from a profit to a cost matrix</li>
<li><strong class="pname"><code>inversion_function</code></strong> (function) - The function to use to invert each entry in the profit matrix</li>
</ul></dd>
<dt>Returns: list of lists</dt>
<dd>The converted matrix</dd>
</dl>
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<span class="sig-arg">msg</span>=<span class="sig-default">None</span>)</span>
</h3>
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><span class="codelink"><a href="munkres-pysrc.html#print_matrix">source code</a></span>
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Convenience function: Displays the contents of a matrix of integers.
<dl class="fields">
<dt>Parameters:</dt>
<dd><ul class="nomargin-top">
<li><strong class="pname"><code>matrix</code></strong> (list of lists) - Matrix to print</li>
<li><strong class="pname"><code>msg</code></strong> (str) - Optional message to print before displaying the matrix</li>
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