/*
    * Licensed to the Apache Software Foundation (ASF) under one or more
    * contributor license agreements.  See the NOTICE file distributed with
    * this work for additional information regarding copyright ownership.
    * The ASF licenses this file to You 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.
   */
  package org.apache.commons.math3.optim.linear;
  
  import java.io.IOException;
  import java.io.ObjectInputStream;
  import java.io.ObjectOutputStream;
  import java.io.Serializable;
  import java.util.ArrayList;
  import java.util.Arrays;
  import java.util.Collection;
  import java.util.HashSet;
  import java.util.List;
  import java.util.Set;
  import java.util.TreeSet;
  
  import org.apache.commons.math3.linear.Array2DRowRealMatrix;
  import org.apache.commons.math3.linear.MatrixUtils;
  import org.apache.commons.math3.linear.RealVector;
  import org.apache.commons.math3.optim.nonlinear.scalar.GoalType;
  import org.apache.commons.math3.optim.PointValuePair;
  import org.apache.commons.math3.util.Precision;
  
  /**
   * A tableau for use in the Simplex method.
   *
   * <p>
   * Example:
   * <pre>
   *   W |  Z |  x1 |  x2 |  x- | s1 |  s2 |  a1 |  RHS
   * ---------------------------------------------------
   *  -1    0    0     0     0     0     0     1     0   &lt;= phase 1 objective
   *   0    1   -15   -10    0     0     0     0     0   &lt;= phase 2 objective
   *   0    0    1     0     0     1     0     0     2   &lt;= constraint 1
   *   0    0    0     1     0     0     1     0     3   &lt;= constraint 2
   *   0    0    1     1     0     0     0     1     4   &lt;= constraint 3
   * </pre>
   * W: Phase 1 objective function</br>
   * Z: Phase 2 objective function</br>
   * x1 &amp; x2: Decision variables</br>
   * x-: Extra decision variable to allow for negative values</br>
   * s1 &amp; s2: Slack/Surplus variables</br>
   * a1: Artificial variable</br>
   * RHS: Right hand side</br>
   * </p>
   * @since 2.0
   */
  class SimplexTableau implements Serializable {
  
      /** Column label for negative vars. */
      private static final String NEGATIVE_VAR_COLUMN_LABEL = "x-";
  
      /** Serializable version identifier. */
      private static final long serialVersionUID = -1369660067587938365L;
  
      /** Linear objective function. */
      private final LinearObjectiveFunction f;
  
      /** Linear constraints. */
      private final List<LinearConstraint> constraints;
  
      /** Whether to restrict the variables to non-negative values. */
      private final boolean restrictToNonNegative;
  
      /** The variables each column represents */
      private final List<String> columnLabels = new ArrayList<String>();
  
      /** Simple tableau. */
      private transient Array2DRowRealMatrix tableau;
  
      /** Number of decision variables. */
      private final int numDecisionVariables;
  
      /** Number of slack variables. */
      private final int numSlackVariables;
  
      /** Number of artificial variables. */
      private int numArtificialVariables;
  
      /** Amount of error to accept when checking for optimality. */
      private final double epsilon;
  
      /** Amount of error to accept in floating point comparisons. */
      private final int maxUlps;
  
     /** Maps basic variables to row they are basic in. */
     private int[] basicVariables;
 
     /** Maps rows to their corresponding basic variables. */
     private int[] basicRows;
 
     /**
      * Builds a tableau for a linear problem.
      *
      * @param f Linear objective function.
      * @param constraints Linear constraints.
      * @param goalType Optimization goal: either {@link GoalType#MAXIMIZE}
      * or {@link GoalType#MINIMIZE}.
      * @param restrictToNonNegative Whether to restrict the variables to non-negative values.
      * @param epsilon Amount of error to accept when checking for optimality.
      */
     SimplexTableau(final LinearObjectiveFunction f,
                    final Collection<LinearConstraint> constraints,
                    final GoalType goalType,
                    final boolean restrictToNonNegative,
                    final double epsilon) {
         this(f, constraints, goalType, restrictToNonNegative, epsilon, SimplexSolver.DEFAULT_ULPS);
     }
 
     /**
      * Build a tableau for a linear problem.
      * @param f linear objective function
      * @param constraints linear constraints
      * @param goalType type of optimization goal: either {@link GoalType#MAXIMIZE} or {@link GoalType#MINIMIZE}
      * @param restrictToNonNegative whether to restrict the variables to non-negative values
      * @param epsilon amount of error to accept when checking for optimality
      * @param maxUlps amount of error to accept in floating point comparisons
      */
     SimplexTableau(final LinearObjectiveFunction f,
                    final Collection<LinearConstraint> constraints,
                    final GoalType goalType,
                    final boolean restrictToNonNegative,
                    final double epsilon,
                    final int maxUlps) {
         this.f                      = f;
         this.constraints            = normalizeConstraints(constraints);
         this.restrictToNonNegative  = restrictToNonNegative;
         this.epsilon                = epsilon;
         this.maxUlps                = maxUlps;
         this.numDecisionVariables   = f.getCoefficients().getDimension() + (restrictToNonNegative ? 0 : 1);
         this.numSlackVariables      = getConstraintTypeCounts(Relationship.LEQ) +
                                       getConstraintTypeCounts(Relationship.GEQ);
         this.numArtificialVariables = getConstraintTypeCounts(Relationship.EQ) +
                                       getConstraintTypeCounts(Relationship.GEQ);
         this.tableau = createTableau(goalType == GoalType.MAXIMIZE);
         // initialize the basic variables for phase 1:
         //   we know that only slack or artificial variables can be basic
         initializeBasicVariables(getSlackVariableOffset());
         initializeColumnLabels();
     }
 
     /**
      * Initialize the labels for the columns.
      */
     protected void initializeColumnLabels() {
       if (getNumObjectiveFunctions() == 2) {
         columnLabels.add("W");
       }
       columnLabels.add("Z");
       for (int i = 0; i < getOriginalNumDecisionVariables(); i++) {
         columnLabels.add("x" + i);
       }
       if (!restrictToNonNegative) {
         columnLabels.add(NEGATIVE_VAR_COLUMN_LABEL);
       }
       for (int i = 0; i < getNumSlackVariables(); i++) {
         columnLabels.add("s" + i);
       }
       for (int i = 0; i < getNumArtificialVariables(); i++) {
         columnLabels.add("a" + i);
       }
       columnLabels.add("RHS");
     }
 
     /**
      * Create the tableau by itself.
      * @param maximize if true, goal is to maximize the objective function
      * @return created tableau
      */
     protected Array2DRowRealMatrix createTableau(final boolean maximize) {
 
         // create a matrix of the correct size
         int width = numDecisionVariables + numSlackVariables +
         numArtificialVariables + getNumObjectiveFunctions() + 1; // + 1 is for RHS
         int height = constraints.size() + getNumObjectiveFunctions();
         Array2DRowRealMatrix matrix = new Array2DRowRealMatrix(height, width);
 
         // initialize the objective function rows
         if (getNumObjectiveFunctions() == 2) {
             matrix.setEntry(0, 0, -1);
         }
 
         int zIndex = (getNumObjectiveFunctions() == 1) ? 0 : 1;
         matrix.setEntry(zIndex, zIndex, maximize ? 1 : -1);
         RealVector objectiveCoefficients = maximize ? f.getCoefficients().mapMultiply(-1) : f.getCoefficients();
         copyArray(objectiveCoefficients.toArray(), matrix.getDataRef()[zIndex]);
         matrix.setEntry(zIndex, width - 1, maximize ? f.getConstantTerm() : -1 * f.getConstantTerm());
 
         if (!restrictToNonNegative) {
             matrix.setEntry(zIndex, getSlackVariableOffset() - 1,
                             getInvertedCoefficientSum(objectiveCoefficients));
         }
 
         // initialize the constraint rows
         int slackVar = 0;
         int artificialVar = 0;
         for (int i = 0; i < constraints.size(); i++) {
             LinearConstraint constraint = constraints.get(i);
             int row = getNumObjectiveFunctions() + i;
 
             // decision variable coefficients
             copyArray(constraint.getCoefficients().toArray(), matrix.getDataRef()[row]);
 
             // x-
             if (!restrictToNonNegative) {
                 matrix.setEntry(row, getSlackVariableOffset() - 1,
                                 getInvertedCoefficientSum(constraint.getCoefficients()));
             }
 
             // RHS
             matrix.setEntry(row, width - 1, constraint.getValue());
 
             // slack variables
             if (constraint.getRelationship() == Relationship.LEQ) {
                 matrix.setEntry(row, getSlackVariableOffset() + slackVar++, 1);  // slack
             } else if (constraint.getRelationship() == Relationship.GEQ) {
                 matrix.setEntry(row, getSlackVariableOffset() + slackVar++, -1); // excess
             }
 
             // artificial variables
             if ((constraint.getRelationship() == Relationship.EQ) ||
                 (constraint.getRelationship() == Relationship.GEQ)) {
                 matrix.setEntry(0, getArtificialVariableOffset() + artificialVar, 1);
                 matrix.setEntry(row, getArtificialVariableOffset() + artificialVar++, 1);
                 matrix.setRowVector(0, matrix.getRowVector(0).subtract(matrix.getRowVector(row)));
             }
         }
 
         return matrix;
     }
 
     /**
      * Get new versions of the constraints which have positive right hand sides.
      * @param originalConstraints original (not normalized) constraints
      * @return new versions of the constraints
      */
     public List<LinearConstraint> normalizeConstraints(Collection<LinearConstraint> originalConstraints) {
         List<LinearConstraint> normalized = new ArrayList<LinearConstraint>(originalConstraints.size());
         for (LinearConstraint constraint : originalConstraints) {
             normalized.add(normalize(constraint));
         }
         return normalized;
     }
 
     /**
      * Get a new equation equivalent to this one with a positive right hand side.
      * @param constraint reference constraint
      * @return new equation
      */
     private LinearConstraint normalize(final LinearConstraint constraint) {
         if (constraint.getValue() < 0) {
             return new LinearConstraint(constraint.getCoefficients().mapMultiply(-1),
                                         constraint.getRelationship().oppositeRelationship(),
                                         -1 * constraint.getValue());
         }
         return new LinearConstraint(constraint.getCoefficients(),
                                     constraint.getRelationship(), constraint.getValue());
     }
 
     /**
      * Get the number of objective functions in this tableau.
      * @return 2 for Phase 1.  1 for Phase 2.
      */
     protected final int getNumObjectiveFunctions() {
         return this.numArtificialVariables > 0 ? 2 : 1;
     }
 
     /**
      * Get a count of constraints corresponding to a specified relationship.
      * @param relationship relationship to count
      * @return number of constraint with the specified relationship
      */
     private int getConstraintTypeCounts(final Relationship relationship) {
         int count = 0;
         for (final LinearConstraint constraint : constraints) {
             if (constraint.getRelationship() == relationship) {
                 ++count;
             }
         }
         return count;
     }
 
     /**
      * Get the -1 times the sum of all coefficients in the given array.
      * @param coefficients coefficients to sum
      * @return the -1 times the sum of all coefficients in the given array.
      */
     protected static double getInvertedCoefficientSum(final RealVector coefficients) {
         double sum = 0;
         for (double coefficient : coefficients.toArray()) {
             sum -= coefficient;
         }
         return sum;
     }
 
     /**
      * Checks whether the given column is basic.
      * @param col index of the column to check
      * @return the row that the variable is basic in.  null if the column is not basic
      */
     protected Integer getBasicRow(final int col) {
         final int row = basicVariables[col];
         return row == -1 ? null : row;
     }
 
     /**
      * Returns the variable that is basic in this row.
      * @param row the index of the row to check
      * @return the variable that is basic for this row.
      */
     protected int getBasicVariable(final int row) {
         return basicRows[row];
     }
 
     /**
      * Initializes the basic variable / row mapping.
      * @param startColumn the column to start
      */
     private void initializeBasicVariables(final int startColumn) {
         basicVariables = new int[getWidth() - 1];
         basicRows = new int[getHeight()];
 
         Arrays.fill(basicVariables, -1);
 
         for (int i = startColumn; i < getWidth() - 1; i++) {
             Integer row = findBasicRow(i);
             if (row != null) {
                 basicVariables[i] = row;
                 basicRows[row] = i;
             }
         }
     }
 
     /**
      * Returns the row in which the given column is basic.
      * @param col index of the column
      * @return the row that the variable is basic in, or {@code null} if the variable is not basic.
      */
     private Integer findBasicRow(final int col) {
         Integer row = null;
         for (int i = 0; i < getHeight(); i++) {
             final double entry = getEntry(i, col);
             if (Precision.equals(entry, 1d, maxUlps) && (row == null)) {
                 row = i;
             } else if (!Precision.equals(entry, 0d, maxUlps)) {
                 return null;
             }
         }
         return row;
     }
 
     /**
      * Removes the phase 1 objective function, positive cost non-artificial variables,
      * and the non-basic artificial variables from this tableau.
      */
     protected void dropPhase1Objective() {
         if (getNumObjectiveFunctions() == 1) {
             return;
         }
 
         final Set<Integer> columnsToDrop = new TreeSet<Integer>();
         columnsToDrop.add(0);
 
         // positive cost non-artificial variables
         for (int i = getNumObjectiveFunctions(); i < getArtificialVariableOffset(); i++) {
             final double entry = getEntry(0, i);
             if (Precision.compareTo(entry, 0d, epsilon) > 0) {
                 columnsToDrop.add(i);
             }
         }
 
         // non-basic artificial variables
         for (int i = 0; i < getNumArtificialVariables(); i++) {
             int col = i + getArtificialVariableOffset();
             if (getBasicRow(col) == null) {
                 columnsToDrop.add(col);
             }
         }
 
         final double[][] matrix = new double[getHeight() - 1][getWidth() - columnsToDrop.size()];
         for (int i = 1; i < getHeight(); i++) {
             int col = 0;
             for (int j = 0; j < getWidth(); j++) {
                 if (!columnsToDrop.contains(j)) {
                     matrix[i - 1][col++] = getEntry(i, j);
                 }
             }
         }
 
         // remove the columns in reverse order so the indices are correct
         Integer[] drop = columnsToDrop.toArray(new Integer[columnsToDrop.size()]);
         for (int i = drop.length - 1; i >= 0; i--) {
             columnLabels.remove((int) drop[i]);
         }
 
         this.tableau = new Array2DRowRealMatrix(matrix);
         this.numArtificialVariables = 0;
         // need to update the basic variable mappings as row/columns have been dropped
         initializeBasicVariables(getNumObjectiveFunctions());
     }
 
     /**
      * @param src the source array
      * @param dest the destination array
      */
     private void copyArray(final double[] src, final double[] dest) {
         System.arraycopy(src, 0, dest, getNumObjectiveFunctions(), src.length);
     }
 
     /**
      * Returns whether the problem is at an optimal state.
      * @return whether the model has been solved
      */
     boolean isOptimal() {
         final double[] objectiveFunctionRow = getRow(0);
         final int end = getRhsOffset();
         for (int i = getNumObjectiveFunctions(); i < end; i++) {
             final double entry = objectiveFunctionRow[i];
             if (Precision.compareTo(entry, 0d, epsilon) < 0) {
                 return false;
             }
         }
         return true;
     }
 
     /**
      * Get the current solution.
      * @return current solution
      */
     protected PointValuePair getSolution() {
         int negativeVarColumn = columnLabels.indexOf(NEGATIVE_VAR_COLUMN_LABEL);
         Integer negativeVarBasicRow = negativeVarColumn > 0 ? getBasicRow(negativeVarColumn) : null;
         double mostNegative = negativeVarBasicRow == null ? 0 : getEntry(negativeVarBasicRow, getRhsOffset());
 
         final Set<Integer> usedBasicRows = new HashSet<Integer>();
         final double[] coefficients = new double[getOriginalNumDecisionVariables()];
         for (int i = 0; i < coefficients.length; i++) {
             int colIndex = columnLabels.indexOf("x" + i);
             if (colIndex < 0) {
                 coefficients[i] = 0;
                 continue;
             }
             Integer basicRow = getBasicRow(colIndex);
             if (basicRow != null && basicRow == 0) {
                 // if the basic row is found to be the objective function row
                 // set the coefficient to 0 -> this case handles unconstrained
                 // variables that are still part of the objective function
                 coefficients[i] = 0;
             } else if (usedBasicRows.contains(basicRow)) {
                 // if multiple variables can take a given value
                 // then we choose the first and set the rest equal to 0
                 coefficients[i] = 0 - (restrictToNonNegative ? 0 : mostNegative);
             } else {
                 usedBasicRows.add(basicRow);
                 coefficients[i] =
                     (basicRow == null ? 0 : getEntry(basicRow, getRhsOffset())) -
                     (restrictToNonNegative ? 0 : mostNegative);
             }
         }
         return new PointValuePair(coefficients, f.value(coefficients));
     }
 
     /**
      * Perform the row operations of the simplex algorithm with the selected
      * pivot column and row.
      * @param pivotCol the pivot column
      * @param pivotRow the pivot row
      */
     protected void performRowOperations(int pivotCol, int pivotRow) {
         // set the pivot element to 1
         final double pivotVal = getEntry(pivotRow, pivotCol);
         divideRow(pivotRow, pivotVal);
 
         // set the rest of the pivot column to 0
         for (int i = 0; i < getHeight(); i++) {
             if (i != pivotRow) {
                 final double multiplier = getEntry(i, pivotCol);
                 if (multiplier != 0.0) {
                     subtractRow(i, pivotRow, multiplier);
                 }
             }
         }
 
         // update the basic variable mappings
         final int previousBasicVariable = getBasicVariable(pivotRow);
         basicVariables[previousBasicVariable] = -1;
         basicVariables[pivotCol] = pivotRow;
         basicRows[pivotRow] = pivotCol;
     }
 
     /**
      * Divides one row by a given divisor.
      * <p>
      * After application of this operation, the following will hold:
      * <pre>dividendRow = dividendRow / divisor</pre>
      *
      * @param dividendRowIndex index of the row
      * @param divisor value of the divisor
      */
     protected void divideRow(final int dividendRowIndex, final double divisor) {
         final double[] dividendRow = getRow(dividendRowIndex);
         for (int j = 0; j < getWidth(); j++) {
             dividendRow[j] /= divisor;
         }
     }
 
     /**
      * Subtracts a multiple of one row from another.
      * <p>
      * After application of this operation, the following will hold:
      * <pre>minuendRow = minuendRow - multiple * subtrahendRow</pre>
      *
      * @param minuendRowIndex row index
      * @param subtrahendRowIndex row index
      * @param multiplier multiplication factor
      */
     protected void subtractRow(final int minuendRowIndex, final int subtrahendRowIndex, final double multiplier) {
         final double[] minuendRow = getRow(minuendRowIndex);
         final double[] subtrahendRow = getRow(subtrahendRowIndex);
         for (int i = 0; i < getWidth(); i++) {
             minuendRow[i] -= subtrahendRow[i] * multiplier;
         }
     }
 
     /**
      * Get the width of the tableau.
      * @return width of the tableau
      */
     protected final int getWidth() {
         return tableau.getColumnDimension();
     }
 
     /**
      * Get the height of the tableau.
      * @return height of the tableau
      */
     protected final int getHeight() {
         return tableau.getRowDimension();
     }
 
     /**
      * Get an entry of the tableau.
      * @param row row index
      * @param column column index
      * @return entry at (row, column)
      */
     protected final double getEntry(final int row, final int column) {
         return tableau.getEntry(row, column);
     }
 
     /**
      * Set an entry of the tableau.
      * @param row row index
      * @param column column index
      * @param value for the entry
      */
     protected final void setEntry(final int row, final int column, final double value) {
         tableau.setEntry(row, column, value);
     }
 
     /**
      * Get the offset of the first slack variable.
      * @return offset of the first slack variable
      */
     protected final int getSlackVariableOffset() {
         return getNumObjectiveFunctions() + numDecisionVariables;
     }
 
     /**
      * Get the offset of the first artificial variable.
      * @return offset of the first artificial variable
      */
     protected final int getArtificialVariableOffset() {
         return getNumObjectiveFunctions() + numDecisionVariables + numSlackVariables;
     }
 
     /**
      * Get the offset of the right hand side.
      * @return offset of the right hand side
      */
     protected final int getRhsOffset() {
         return getWidth() - 1;
     }
 
     /**
      * Get the number of decision variables.
      * <p>
      * If variables are not restricted to positive values, this will include 1 extra decision variable to represent
      * the absolute value of the most negative variable.
      *
      * @return number of decision variables
      * @see #getOriginalNumDecisionVariables()
      */
     protected final int getNumDecisionVariables() {
         return numDecisionVariables;
     }
 
     /**
      * Get the original number of decision variables.
      * @return original number of decision variables
      * @see #getNumDecisionVariables()
      */
     protected final int getOriginalNumDecisionVariables() {
         return f.getCoefficients().getDimension();
     }
 
     /**
      * Get the number of slack variables.
      * @return number of slack variables
      */
     protected final int getNumSlackVariables() {
         return numSlackVariables;
     }
 
     /**
      * Get the number of artificial variables.
      * @return number of artificial variables
      */
     protected final int getNumArtificialVariables() {
         return numArtificialVariables;
     }
 
     /**
      * Get the row from the tableau.
      * @param row the row index
      * @return the reference to the underlying row data
      */
     protected final double[] getRow(int row) {
         return tableau.getDataRef()[row];
     }
 
     /**
      * Get the tableau data.
      * @return tableau data
      */
     protected final double[][] getData() {
         return tableau.getData();
     }
 
     @Override
     public boolean equals(Object other) {
 
       if (this == other) {
         return true;
       }
 
       if (other instanceof SimplexTableau) {
           SimplexTableau rhs = (SimplexTableau) other;
           return (restrictToNonNegative  == rhs.restrictToNonNegative) &&
                  (numDecisionVariables   == rhs.numDecisionVariables) &&
                  (numSlackVariables      == rhs.numSlackVariables) &&
                  (numArtificialVariables == rhs.numArtificialVariables) &&
                  (epsilon                == rhs.epsilon) &&
                  (maxUlps                == rhs.maxUlps) &&
                  f.equals(rhs.f) &&
                  constraints.equals(rhs.constraints) &&
                  tableau.equals(rhs.tableau);
       }
       return false;
     }
 
     @Override
     public int hashCode() {
         return Boolean.valueOf(restrictToNonNegative).hashCode() ^
                numDecisionVariables ^
                numSlackVariables ^
                numArtificialVariables ^
                Double.valueOf(epsilon).hashCode() ^
                maxUlps ^
                f.hashCode() ^
                constraints.hashCode() ^
                tableau.hashCode();
     }
 
     /**
      * Serialize the instance.
      * @param oos stream where object should be written
      * @throws IOException if object cannot be written to stream
      */
     private void writeObject(ObjectOutputStream oos)
         throws IOException {
         oos.defaultWriteObject();
         MatrixUtils.serializeRealMatrix(tableau, oos);
     }
 
     /**
      * Deserialize the instance.
      * @param ois stream from which the object should be read
      * @throws ClassNotFoundException if a class in the stream cannot be found
      * @throws IOException if object cannot be read from the stream
      */
     private void readObject(ObjectInputStream ois)
       throws ClassNotFoundException, IOException {
         ois.defaultReadObject();
         MatrixUtils.deserializeRealMatrix(this, "tableau", ois);
     }
 }