1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578
|
"""
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: http://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__ = "1.1.2"
__author__ = "Brian Clapper, bmc@clapper.org"
__url__ = "http://software.clapper.org/munkres/"
__copyright__ = "(c) 2008-2019 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 of integers.
**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:
formatted = ((format + 's') % DISALLOWED_PRINTVAL)
else: formatted = ((format + 'd') % 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),
# Rectangular variant
([[10, 10, 8, 11],
[9, 8, 1, 1],
[9, 7, 4, 10]],
15),
# Rectangular with DISALLOWED
([[4, 5, 6, DISALLOWED],
[1, 9, 12, 11],
[DISALLOWED, 5, 4, DISALLOWED],
[12, 12, 12, 10]],
20),
# DISALLOWED to force pairings
([[1, DISALLOWED, DISALLOWED, DISALLOWED],
[DISALLOWED, 2, DISALLOWED, DISALLOWED],
[DISALLOWED, DISALLOWED, 3, DISALLOWED],
[DISALLOWED, DISALLOWED, DISALLOWED, 4]],
10)]
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) -> %d' % (r, c, x)))
print(('lowest cost=%d' % total_cost))
assert expected_total == total_cost
|