broken_weights.cs

//
// Copyright 2012 Hakan Kjellerstrand
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//     http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

using System;
using System.Linq;
using Google.OrTools.ConstraintSolver;


public class BrokenWeights
{

  /**
   *
   * Broken weights problem.
   *
   * From http://www.mathlesstraveled.com/?p=701
   * """
   * Here's a fantastic problem I recently heard. Apparently it was first
   * posed by Claude Gaspard Bachet de Meziriac in a book of arithmetic problems
   * published in 1612, and can also be found in Heinrich Dorrie's 100
   * Great Problems of Elementary Mathematics.
   *
   *   A merchant had a forty pound measuring weight that broke
   *   into four pieces as the result of a fall. When the pieces were
   *   subsequently weighed, it was found that the weight of each piece
   *   was a whole number of pounds and that the four pieces could be
   *   used to weigh every integral weight between 1 and 40 pounds. What
   *   were the weights of the pieces?
   *
   * Note that since this was a 17th-century merchant, he of course used a
   * balance scale to weigh things. So, for example, he could use a 1-pound
   * weight and a 4-pound weight to weigh a 3-pound object, by placing the
   * 3-pound object and 1-pound weight on one side of the scale, and
   * the 4-pound weight on the other side.
   * """
   *
   * Also see http://www.hakank.org/or-tools/broken_weights.py
   *
   */
  private static void Solve(int m=40, int n=4)
  {
    Solver solver = new Solver("BrokenWeights");

    Console.WriteLine("Total weight (m): {0}", m);
    Console.WriteLine("Number of pieces (n): {0}", n);
    Console.WriteLine();


    //
    // Decision variables
    //

    IntVar[] weights = solver.MakeIntVarArray(n, 1, m , "weights");
    IntVar[,] x = new IntVar[m, n];
    // Note: in x_flat we insert the weights array before x
    IntVar[] x_flat = new IntVar[m*n + n];
    for(int j = 0; j < n; j++) {
      x_flat[j] = weights[j];
    }
    for(int i = 0; i < m; i++) {
      for(int j = 0; j < n; j++) {
        x[i,j] = solver.MakeIntVar(-1, 1, "x["+i+","+j+"]");
        x_flat[n+i*n+j] = x[i,j];
      }
    }



    //
    // Constraints
    //


    // symmetry breaking
    for(int j = 1; j < n; j++) {
      solver.Add(weights[j-1] < weights[j]);
    }


    solver.Add(weights.Sum() == m);

    // Check that all weights from 1 to n (default 40) can be made.
    //
    // Since all weights can be on either side
    // of the side of the scale we allow either
    // -1, 0, or 1 of the weights, assuming that
    // -1 is the weights on the left and 1 is on the right.
    //
    for(int i = 0; i < m; i++) {
      solver.Add( (from j in Enumerable.Range(0, n)
                   select weights[j] * x[i,j]).ToArray().Sum() == i+1);
    }


    //
    // The objective is to minimize the last weight.
    //
    OptimizeVar obj = weights[n-1].Minimize(1);


    //
    // Search
    //
    DecisionBuilder db = solver.MakePhase(x_flat,
                                          Solver.CHOOSE_FIRST_UNBOUND,
                                          Solver.ASSIGN_MIN_VALUE);

    solver.NewSearch(db, obj);

    while (solver.NextSolution()) {
      Console.Write("weights:  ");
      for(int i = 0; i < n; i++) {
        Console.Write("{0,3} ", weights[i].Value());
      }
      Console.WriteLine();
      for(int i = 0; i < 10+n*4; i++) {
        Console.Write("-");
      }
      Console.WriteLine();
      for(int i = 0; i < m; i++) {
        Console.Write("weight {0,2}:", i+1);
        for(int j = 0; j < n; j++) {
          Console.Write("{0,3} ", x[i,j].Value());
        }
        Console.WriteLine();
      }
      Console.WriteLine();
    }

    Console.WriteLine("\nSolutions: {0}", solver.Solutions());
    Console.WriteLine("WallTime: {0}ms", solver.WallTime());
    Console.WriteLine("Failures: {0}", solver.Failures());
    Console.WriteLine("Branches: {0} ", solver.Branches());

    solver.EndSearch();

  }

  public static void Main(String[] args)
  {

    int m = 40;
    int n = 4;

    if (args.Length > 0) {
      m = Convert.ToInt32(args[0]);
    }

    if (args.Length > 1) {
      n = Convert.ToInt32(args[1]);
    }

    Solve(m, n);
  }
}