"""
Introduction
============
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.
For complete usage documentation, see: https://software.clapper.org/munkres/
"""
__docformat__ = 'markdown'
# ---------------------------------------------------------------------------
# Imports
# ---------------------------------------------------------------------------
import sys
import copy
from typing import Union, NewType, Sequence, Tuple, Optional, Callable
# ---------------------------------------------------------------------------
# Exports
# ---------------------------------------------------------------------------
__all__ = ['Munkres', 'make_cost_matrix', 'DISALLOWED']
# ---------------------------------------------------------------------------
# Globals
# ---------------------------------------------------------------------------
AnyNum = NewType('AnyNum', Union[int, float])
Matrix = NewType('Matrix', Sequence[Sequence[AnyNum]])
# Info about the module
__version__ = "1.1.4"
__author__ = "Brian Clapper, bmc@clapper.org"
__url__ = "https://software.clapper.org/munkres/"
__copyright__ = "(c) 2008-2020 Brian M. Clapper"
__license__ = "Apache Software License"
# Constants
class DISALLOWED_OBJ(object):
pass
DISALLOWED = DISALLOWED_OBJ()
DISALLOWED_PRINTVAL = "D"
# ---------------------------------------------------------------------------
# Exceptions
# ---------------------------------------------------------------------------
class UnsolvableMatrix(Exception):
"""
Exception raised for unsolvable matrices
"""
pass
# ---------------------------------------------------------------------------
# Classes
# ---------------------------------------------------------------------------
class Munkres:
"""
Calculate the Munkres solution to the classical assignment problem.
See the module documentation for usage.
"""
def __init__(self):
"""Create a new instance"""
self.C = None
self.row_covered = []
self.col_covered = []
self.n = 0
self.Z0_r = 0
self.Z0_c = 0
self.marked = None
self.path = None
def pad_matrix(self, matrix: Matrix, pad_value: int=0) -> Matrix:
"""
Pad a possibly non-square matrix to make it square.
**Parameters**
- `matrix` (list of lists of numbers): matrix to pad
- `pad_value` (`int`): value to use to pad the matrix
**Returns**
a new, possibly padded, matrix
"""
max_columns = 0
total_rows = len(matrix)
for row in matrix:
max_columns = max(max_columns, len(row))
total_rows = max(max_columns, total_rows)
new_matrix = []
for row in matrix:
row_len = len(row)
new_row = row[:]
if total_rows > row_len:
# Row too short. Pad it.
new_row += [pad_value] * (total_rows - row_len)
new_matrix += [new_row]
while len(new_matrix) < total_rows:
new_matrix += [[pad_value] * total_rows]
return new_matrix
def compute(self, cost_matrix: Matrix) -> Sequence[Tuple[int, int]]:
"""
Compute the indexes for the lowest-cost pairings between rows and
columns in the database. Returns a list of `(row, column)` tuples
that can be used to traverse the matrix.
**WARNING**: This code handles square and rectangular matrices. It
does *not* handle irregular matrices.
**Parameters**
- `cost_matrix` (list of lists of numbers): The cost matrix. If this
cost matrix is not square, it will be padded with zeros, via a call
to `pad_matrix()`. (This method does *not* modify the caller's
matrix. It operates on a copy of the matrix.)
**Returns**
A list of `(row, column)` tuples that describe the lowest cost path
through the matrix
"""
self.C = self.pad_matrix(cost_matrix)
self.n = len(self.C)
self.original_length = len(cost_matrix)
self.original_width = len(cost_matrix[0])
self.row_covered = [False for i in range(self.n)]
self.col_covered = [False for i in range(self.n)]
self.Z0_r = 0
self.Z0_c = 0
self.path = self.__make_matrix(self.n * 2, 0)
self.marked = self.__make_matrix(self.n, 0)
done = False
step = 1
steps = { 1 : self.__step1,
2 : self.__step2,
3 : self.__step3,
4 : self.__step4,
5 : self.__step5,
6 : self.__step6 }
while not done:
try:
func = steps[step]
step = func()
except KeyError:
done = True
# Look for the starred columns
results = []
for i in range(self.original_length):
for j in range(self.original_width):
if self.marked[i][j] == 1:
results += [(i, j)]
return results
def __copy_matrix(self, matrix: Matrix) -> Matrix:
"""Return an exact copy of the supplied matrix"""
return copy.deepcopy(matrix)
def __make_matrix(self, n: int, val: AnyNum) -> Matrix:
"""Create an *n*x*n* matrix, populating it with the specific value."""
matrix = []
for i in range(n):
matrix += [[val for j in range(n)]]
return matrix
def __step1(self) -> int:
"""
For each row of the matrix, find the smallest element and
subtract it from every element in its row. Go to Step 2.
"""
C = self.C
n = self.n
for i in range(n):
vals = [x for x in self.C[i] if x is not DISALLOWED]
if len(vals) == 0:
# All values in this row are DISALLOWED. This matrix is
# unsolvable.
raise UnsolvableMatrix(
"Row {0} is entirely DISALLOWED.".format(i)
)
minval = min(vals)
# Find the minimum value for this row and subtract that minimum
# from every element in the row.
for j in range(n):
if self.C[i][j] is not DISALLOWED:
self.C[i][j] -= minval
return 2
def __step2(self) -> int:
"""
Find a zero (Z) in the resulting matrix. If there is no starred
zero in its row or column, star Z. Repeat for each element in the
matrix. Go to Step 3.
"""
n = self.n
for i in range(n):
for j in range(n):
if (self.C[i][j] == 0) and \
(not self.col_covered[j]) and \
(not self.row_covered[i]):
self.marked[i][j] = 1
self.col_covered[j] = True
self.row_covered[i] = True
break
self.__clear_covers()
return 3
def __step3(self) -> int:
"""
Cover each column containing a starred zero. If K columns are
covered, the starred zeros describe a complete set of unique
assignments. In this case, Go to DONE, otherwise, Go to Step 4.
"""
n = self.n
count = 0
for i in range(n):
for j in range(n):
if self.marked[i][j] == 1 and not self.col_covered[j]:
self.col_covered[j] = True
count += 1
if count >= n:
step = 7 # done
else:
step = 4
return step
def __step4(self) -> int:
"""
Find a noncovered zero and prime it. If there is no starred zero
in the row containing this primed zero, Go to Step 5. Otherwise,
cover this row and uncover the column containing the starred
zero. Continue in this manner until there are no uncovered zeros
left. Save the smallest uncovered value and Go to Step 6.
"""
step = 0
done = False
row = 0
col = 0
star_col = -1
while not done:
(row, col) = self.__find_a_zero(row, col)
if row < 0:
done = True
step = 6
else:
self.marked[row][col] = 2
star_col = self.__find_star_in_row(row)
if star_col >= 0:
col = star_col
self.row_covered[row] = True
self.col_covered[col] = False
else:
done = True
self.Z0_r = row
self.Z0_c = col
step = 5
return step
def __step5(self) -> int:
"""
Construct a series of alternating primed and starred zeros as
follows. Let Z0 represent the uncovered primed zero found in Step 4.
Let Z1 denote the starred zero in the column of Z0 (if any).
Let Z2 denote the primed zero in the row of Z1 (there will always
be one). Continue until the series terminates at a primed zero
that has no starred zero in its column. Unstar each starred zero
of the series, star each primed zero of the series, erase all
primes and uncover every line in the matrix. Return to Step 3
"""
count = 0
path = self.path
path[count][0] = self.Z0_r
path[count][1] = self.Z0_c
done = False
while not done:
row = self.__find_star_in_col(path[count][1])
if row >= 0:
count += 1
path[count][0] = row
path[count][1] = path[count-1][1]
else:
done = True
if not done:
col = self.__find_prime_in_row(path[count][0])
count += 1
path[count][0] = path[count-1][0]
path[count][1] = col
self.__convert_path(path, count)
self.__clear_covers()
self.__erase_primes()
return 3
def __step6(self) -> int:
"""
Add the value found in Step 4 to every element of each covered
row, and subtract it from every element of each uncovered column.
Return to Step 4 without altering any stars, primes, or covered
lines.
"""
minval = self.__find_smallest()
events = 0 # track actual changes to matrix
for i in range(self.n):
for j in range(self.n):
if self.C[i][j] is DISALLOWED:
continue
if self.row_covered[i]:
self.C[i][j] += minval
events += 1
if not self.col_covered[j]:
self.C[i][j] -= minval
events += 1
if self.row_covered[i] and not self.col_covered[j]:
events -= 2 # change reversed, no real difference
if (events == 0):
raise UnsolvableMatrix("Matrix cannot be solved!")
return 4
def __find_smallest(self) -> AnyNum:
"""Find the smallest uncovered value in the matrix."""
minval = sys.maxsize
for i in range(self.n):
for j in range(self.n):
if (not self.row_covered[i]) and (not self.col_covered[j]):
if self.C[i][j] is not DISALLOWED and minval > self.C[i][j]:
minval = self.C[i][j]
return minval
def __find_a_zero(self, i0: int = 0, j0: int = 0) -> Tuple[int, int]:
"""Find the first uncovered element with value 0"""
row = -1
col = -1
i = i0
n = self.n
done = False
while not done:
j = j0
while True:
if (self.C[i][j] == 0) and \
(not self.row_covered[i]) and \
(not self.col_covered[j]):
row = i
col = j
done = True
j = (j + 1) % n
if j == j0:
break
i = (i + 1) % n
if i == i0:
done = True
return (row, col)
def __find_star_in_row(self, row: Sequence[AnyNum]) -> int:
"""
Find the first starred element in the specified row. Returns
the column index, or -1 if no starred element was found.
"""
col = -1
for j in range(self.n):
if self.marked[row][j] == 1:
col = j
break
return col
def __find_star_in_col(self, col: Sequence[AnyNum]) -> int:
"""
Find the first starred element in the specified row. Returns
the row index, or -1 if no starred element was found.
"""
row = -1
for i in range(self.n):
if self.marked[i][col] == 1:
row = i
break
return row
def __find_prime_in_row(self, row) -> int:
"""
Find the first prime element in the specified row. Returns
the column index, or -1 if no starred element was found.
"""
col = -1
for j in range(self.n):
if self.marked[row][j] == 2:
col = j
break
return col
def __convert_path(self,
path: Sequence[Sequence[int]],
count: int) -> None:
for i in range(count+1):
if self.marked[path[i][0]][path[i][1]] == 1:
self.marked[path[i][0]][path[i][1]] = 0
else:
self.marked[path[i][0]][path[i][1]] = 1
def __clear_covers(self) -> None:
"""Clear all covered matrix cells"""
for i in range(self.n):
self.row_covered[i] = False
self.col_covered[i] = False
def __erase_primes(self) -> None:
"""Erase all prime markings"""
for i in range(self.n):
for j in range(self.n):
if self.marked[i][j] == 2:
self.marked[i][j] = 0
# ---------------------------------------------------------------------------
# Functions
# ---------------------------------------------------------------------------
def make_cost_matrix(
profit_matrix: Matrix,
inversion_function: Optional[Callable[[AnyNum], AnyNum]] = None
) -> Matrix:
"""
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 value. If the inversion function
is not provided, a given cell's inverted value is calculated as
`max(matrix) - value`.
This is a static method. Call it like this:
from munkres import Munkres
cost_matrix = Munkres.make_cost_matrix(matrix, inversion_func)
For example:
from munkres import Munkres
cost_matrix = Munkres.make_cost_matrix(matrix, lambda x : sys.maxsize - x)
**Parameters**
- `profit_matrix` (list of lists of numbers): The matrix to convert from
profit to cost values.
- `inversion_function` (`function`): The function to use to invert each
entry in the profit matrix.
**Returns**
A new matrix representing the inversion of `profix_matrix`.
"""
if not inversion_function:
maximum = max(max(row) for row in profit_matrix)
inversion_function = lambda x: maximum - x
cost_matrix = []
for row in profit_matrix:
cost_matrix.append([inversion_function(value) for value in row])
return cost_matrix
def print_matrix(matrix: Matrix, msg: Optional[str] = None) -> None:
"""
Convenience function: Displays the contents of a matrix.
**Parameters**
- `matrix` (list of lists of numbers): The matrix to print
- `msg` (`str`): Optional message to print before displaying the matrix
"""
import math
if msg is not None:
print(msg)
# Calculate the appropriate format width.
width = 0
for row in matrix:
for val in row:
if val is DISALLOWED:
val = DISALLOWED_PRINTVAL
width = max(width, len(str(val)))
# Make the format string
format = ('%%%d' % width)
# Print the matrix
for row in matrix:
sep = '['
for val in row:
if val is DISALLOWED:
val = DISALLOWED_PRINTVAL
formatted = ((format + 's') % val)
sys.stdout.write(sep + formatted)
sep = ', '
sys.stdout.write(']\n')
# ---------------------------------------------------------------------------
# Main
# ---------------------------------------------------------------------------
if __name__ == '__main__':
matrices = [
# Square
([[400, 150, 400],
[400, 450, 600],
[300, 225, 300]],
850), # expected cost
# Rectangular variant
([[400, 150, 400, 1],
[400, 450, 600, 2],
[300, 225, 300, 3]],
452), # expected cost
# Square
([[10, 10, 8],
[9, 8, 1],
[9, 7, 4]],
18),
# Square variant with floating point value
([[10.1, 10.2, 8.3],
[9.4, 8.5, 1.6],
[9.7, 7.8, 4.9]],
19.5),
# Rectangular variant
([[10, 10, 8, 11],
[9, 8, 1, 1],
[9, 7, 4, 10]],
15),
# Rectangular variant with floating point value
([[10.01, 10.02, 8.03, 11.04],
[9.05, 8.06, 1.07, 1.08],
[9.09, 7.1, 4.11, 10.12]],
15.2),
# Rectangular with DISALLOWED
([[4, 5, 6, DISALLOWED],
[1, 9, 12, 11],
[DISALLOWED, 5, 4, DISALLOWED],
[12, 12, 12, 10]],
20),
# Rectangular variant with DISALLOWED and floating point value
([[4.001, 5.002, 6.003, DISALLOWED],
[1.004, 9.005, 12.006, 11.007],
[DISALLOWED, 5.008, 4.009, DISALLOWED],
[12.01, 12.011, 12.012, 10.013]],
20.028),
# DISALLOWED to force pairings
([[1, DISALLOWED, DISALLOWED, DISALLOWED],
[DISALLOWED, 2, DISALLOWED, DISALLOWED],
[DISALLOWED, DISALLOWED, 3, DISALLOWED],
[DISALLOWED, DISALLOWED, DISALLOWED, 4]],
10),
# DISALLOWED to force pairings with floating point value
([[1.1, DISALLOWED, DISALLOWED, DISALLOWED],
[DISALLOWED, 2.2, DISALLOWED, DISALLOWED],
[DISALLOWED, DISALLOWED, 3.3, DISALLOWED],
[DISALLOWED, DISALLOWED, DISALLOWED, 4.4]],
11.0)]
m = Munkres()
for cost_matrix, expected_total in matrices:
print_matrix(cost_matrix, msg='cost matrix')
indexes = m.compute(cost_matrix)
total_cost = 0
for r, c in indexes:
x = cost_matrix[r][c]
total_cost += x
print(('(%d, %d) -> %s' % (r, c, x)))
print(('lowest cost=%s' % total_cost))
assert expected_total == total_cost