199 lines
6.3 KiB
Python
199 lines
6.3 KiB
Python
# Copyright 2010 Hakan Kjellerstrand hakank@gmail.com
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#
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# Licensed under the Apache License, Version 2.0 (the "License");
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# you may not use this file except in compliance with the License.
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# You may obtain a copy of the License at
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#
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# http://www.apache.org/licenses/LICENSE-2.0
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#
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# Unless required by applicable law or agreed to in writing, software
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# distributed under the License is distributed on an "AS IS" BASIS,
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# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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# See the License for the specific language governing permissions and
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# limitations under the License.
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"""
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A programming puzzle from Einav in Google CP Solver.
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From
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'A programming puzzle from Einav'
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http://gcanyon.wordpress.com/2009/10/28/a-programming-puzzle-from-einav/
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'''
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My friend Einav gave me this programming puzzle to work on. Given
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this array of positive and negative numbers:
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33 30 -10 -6 18 7 -11 -23 6
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...
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-25 4 16 30 33 -23 -4 4 -23
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You can flip the sign of entire rows and columns, as many of them
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as you like. The goal is to make all the rows and columns sum to positive
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numbers (or zero), and then to find the solution (there are more than one)
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that has the smallest overall sum. So for example, for this array:
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33 30 -10
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-16 19 9
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-17 -12 -14
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You could flip the sign for the bottom row to get this array:
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33 30 -10
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-16 19 9
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17 12 14
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Now all the rows and columns have positive sums, and the overall total is
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108.
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But you could instead flip the second and third columns, and the second
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row, to get this array:
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33 -30 10
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16 19 9
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-17 12 14
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All the rows and columns still total positive, and the overall sum is just
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66. So this solution is better (I don't know if it's the best)
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A pure brute force solution would have to try over 30 billion solutions.
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I wrote code to solve this in J. I'll post that separately.
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'''
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Compare with the following models:
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* MiniZinc http://www.hakank.org/minizinc/einav_puzzle.mzn
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* SICStus: http://hakank.org/sicstus/einav_puzzle.pl
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Note:
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einav_puzzle2.py is Laurent Perron version, which don't use as many
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decision variables as this version.
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This model was created by Hakan Kjellerstrand (hakank@gmail.com)
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Also see my other Google CP Solver models:
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http://www.hakank.org/google_or_tools/
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"""
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from ortools.constraint_solver import pywrapcp
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def main():
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# Create the solver.
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solver = pywrapcp.Solver('Einav puzzle')
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#
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# data
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#
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# small problem
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# rows = 3;
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# cols = 3;
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# data = [
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# [ 33, 30, -10],
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# [-16, 19, 9],
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# [-17, -12, -14]
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# ]
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# Full problem
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rows = 27
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cols = 9
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data = [[33, 30, 10, -6, 18, -7, -11, 23, -6],
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[16, -19, 9, -26, -8, -19, -8, -21, -14],
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[17, 12, -14, 31, -30, 13, -13, 19, 16],
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[-6, -11, 1, 17, -12, -4, -7, 14, -21],
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[18, -31, 34, -22, 17, -19, 20, 24, 6],
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[33, -18, 17, -15, 31, -5, 3, 27, -3],
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[-18, -20, -18, 31, 6, 4, -2, -12, 24],
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[27, 14, 4, -29, -3, 5, -29, 8, -12],
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[-15, -7, -23, 23, -9, -8, 6, 8, -12],
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[33, -23, -19, -4, -8, -7, 11, -12, 31],
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[-20, 19, -15, -30, 11, 32, 7, 14, -5],
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[-23, 18, -32, -2, -31, -7, 8, 24, 16],
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[32, -4, -10, -14, -6, -1, 0, 23, 23],
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[25, 0, -23, 22, 12, 28, -27, 15, 4],
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[-30, -13, -16, -3, -3, -32, -3, 27, -31],
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[22, 1, 26, 4, -2, -13, 26, 17, 14],
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[-9, -18, 3, -20, -27, -32, -11, 27, 13],
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[-17, 33, -7, 19, -32, 13, -31, -2, -24],
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[-31, 27, -31, -29, 15, 2, 29, -15, 33],
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[-18, -23, 15, 28, 0, 30, -4, 12, -32],
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[-3, 34, 27, -25, -18, 26, 1, 34, 26],
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[-21, -31, -10, -13, -30, -17, -12, -26, 31],
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[23, -31, -19, 21, -17, -10, 2, -23, 23],
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[-3, 6, 0, -3, -32, 0, -10, -25, 14],
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[-19, 9, 14, -27, 20, 15, -5, -27, 18],
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[11, -6, 24, 7, -17, 26, 20, -31, -25],
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[-25, 4, -16, 30, 33, 23, -4, -4, 23]]
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#
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# variables
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#
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x = {}
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for i in range(rows):
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for j in range(cols):
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x[i, j] = solver.IntVar(-100, 100, 'x[%i,%i]' % (i, j))
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x_flat = [x[i, j] for i in range(rows) for j in range(cols)]
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row_sums = [solver.IntVar(0, 300, 'row_sums(%i)' % i) for i in range(rows)]
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col_sums = [solver.IntVar(0, 300, 'col_sums(%i)' % j) for j in range(cols)]
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row_signs = [solver.IntVar([-1, 1], 'row_signs(%i)' % i) for i in range(rows)]
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col_signs = [solver.IntVar([-1, 1], 'col_signs(%i)' % j) for j in range(cols)]
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# total sum: to be minimized
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total_sum = solver.IntVar(0, 1000, 'total_sum')
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#
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# constraints
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#
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for i in range(rows):
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for j in range(cols):
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solver.Add(x[i, j] == data[i][j] * row_signs[i] * col_signs[j])
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total_sum_a = [
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data[i][j] * row_signs[i] * col_signs[j]
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for i in range(rows)
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for j in range(cols)
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]
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solver.Add(total_sum == solver.Sum(total_sum_a))
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# row sums
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for i in range(rows):
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s = [row_signs[i] * col_signs[j] * data[i][j] for j in range(cols)]
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solver.Add(row_sums[i] == solver.Sum(s))
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# column sums
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for j in range(cols):
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s = [row_signs[i] * col_signs[j] * data[i][j] for i in range(rows)]
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solver.Add(col_sums[j] == solver.Sum(s))
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# objective
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objective = solver.Minimize(total_sum, 1)
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#
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# search and result
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#
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# Note: The order of the variables makes a big difference.
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# If row_signs are before col_sign it is much slower.
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db = solver.Phase(col_signs + row_signs, solver.CHOOSE_MIN_SIZE_LOWEST_MIN,
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solver.ASSIGN_MAX_VALUE)
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solver.NewSearch(db, [objective])
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num_solutions = 0
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while solver.NextSolution():
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num_solutions += 1
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print('total_sum:', total_sum.Value())
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print('row_sums:', [row_sums[i].Value() for i in range(rows)])
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print('col_sums:', [col_sums[j].Value() for j in range(cols)])
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print('row_signs:', [row_signs[i].Value() for i in range(rows)])
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print('col_signs:', [col_signs[j].Value() for j in range(cols)])
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print('x:')
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for i in range(rows):
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for j in range(cols):
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print('%3i' % x[i, j].Value(), end=' ')
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print()
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print()
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solver.EndSearch()
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print()
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print('num_solutions:', num_solutions)
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print('failures:', solver.Failures())
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print('branches:', solver.Branches())
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print('WallTime:', solver.WallTime(), 'ms')
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if __name__ == '__main__':
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main()
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