/*
    * 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
    *
    *      https://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.
   */
  
  /*
   * This is not the original file distributed by the Apache Software Foundation
   * It has been modified by the Hipparchus project
   */
  package org.hipparchus.util;
  
  import java.lang.reflect.Array;
  import java.util.Iterator;
  import java.util.List;
  import java.util.Spliterator;
  import java.util.Spliterators;
  import java.util.concurrent.atomic.AtomicReference;
  import java.util.stream.Stream;
  import java.util.stream.StreamSupport;
  
  import org.hipparchus.exception.LocalizedCoreFormats;
  import org.hipparchus.exception.MathIllegalArgumentException;
  import org.hipparchus.exception.MathRuntimeException;
  import org.hipparchus.special.Gamma;
  
  /**
   * Combinatorial utilities.
   */
  public final class CombinatoricsUtils {
  
      /** Maximum index of Bell number that fits into a long.
       * @since 2.2
       */
      public static final int MAX_BELL = 25;
  
      /** All long-representable factorials */
      static final long[] FACTORIALS = {
                         1l,                  1l,                   2l,
                         6l,                 24l,                 120l,
                       720l,               5040l,               40320l,
                    362880l,            3628800l,            39916800l,
                 479001600l,         6227020800l,         87178291200l,
             1307674368000l,     20922789888000l,     355687428096000l,
          6402373705728000l, 121645100408832000l, 2432902008176640000l };
  
      /** Stirling numbers of the second kind. */
      static final AtomicReference<long[][]> STIRLING_S2 = new AtomicReference<> (null);
  
      /**
       * Default implementation of {@link #factorialLog(int)} method:
       * <ul>
       *  <li>No pre-computation</li>
       *  <li>No cache allocation</li>
       * </ul>
       */
      private static final FactorialLog FACTORIAL_LOG_NO_CACHE = FactorialLog.create();
  
      /** Bell numbers.
       * @since 2.2
       */
      private static final AtomicReference<long[]> BELL = new AtomicReference<> (null);
  
      /** Private constructor (class contains only static methods). */
      private CombinatoricsUtils() {}
  
  
      /**
       * Returns an exact representation of the <a
       * href="http://mathworld.wolfram.com/BinomialCoefficient.html"> Binomial
       * Coefficient</a>, "{@code n choose k}", the number of
       * {@code k}-element subsets that can be selected from an
       * {@code n}-element set.
       * <p><Strong>Preconditions</strong>:</p>
       * <ul>
       * <li> {@code 0 <= k <= n } (otherwise
       * {@code MathIllegalArgumentException} is thrown)</li>
       * <li> The result is small enough to fit into a {@code long}. The
       * largest value of {@code n} for which all coefficients are
       * {@code  < Long.MAX_VALUE} is 66. If the computed value exceeds
       * {@code Long.MAX_VALUE} a {@code MathRuntimeException} is
       * thrown.</li>
       * </ul>
       *
       * @param n the size of the set
       * @param k the size of the subsets to be counted
       * @return {@code n choose k}
       * @throws MathIllegalArgumentException if {@code n < 0}.
      * @throws MathIllegalArgumentException if {@code k > n}.
      * @throws MathRuntimeException if the result is too large to be
      * represented by a long integer.
      */
     public static long binomialCoefficient(final int n, final int k)
         throws MathIllegalArgumentException, MathRuntimeException {
         CombinatoricsUtils.checkBinomial(n, k);
         if ((n == k) || (k == 0)) {
             return 1;
         }
         if ((k == 1) || (k == n - 1)) {
             return n;
         }
         // Use symmetry for large k
         if (k > n / 2) {
             return binomialCoefficient(n, n - k);
         }
 
         // We use the formula
         // (n choose k) = n! / (n-k)! / k!
         // (n choose k) == ((n-k+1)*...*n) / (1*...*k)
         // which could be written
         // (n choose k) == (n-1 choose k-1) * n / k
         long result = 1;
         if (n <= 61) {
             // For n <= 61, the naive implementation cannot overflow.
             int i = n - k + 1;
             for (int j = 1; j <= k; j++) {
                 result = result * i / j;
                 i++;
             }
         } else if (n <= 66) {
             // For n > 61 but n <= 66, the result cannot overflow,
             // but we must take care not to overflow intermediate values.
             int i = n - k + 1;
             for (int j = 1; j <= k; j++) {
                 // We know that (result * i) is divisible by j,
                 // but (result * i) may overflow, so we split j:
                 // Filter out the gcd, d, so j/d and i/d are integer.
                 // result is divisible by (j/d) because (j/d)
                 // is relative prime to (i/d) and is a divisor of
                 // result * (i/d).
                 final long d = ArithmeticUtils.gcd(i, j);
                 result = (result / (j / d)) * (i / d);
                 i++;
             }
         } else {
             // For n > 66, a result overflow might occur, so we check
             // the multiplication, taking care to not overflow
             // unnecessary.
             int i = n - k + 1;
             for (int j = 1; j <= k; j++) {
                 final long d = ArithmeticUtils.gcd(i, j);
                 result = ArithmeticUtils.mulAndCheck(result / (j / d), i / d);
                 i++;
             }
         }
         return result;
     }
 
     /**
      * Returns a {@code double} representation of the <a
      * href="http://mathworld.wolfram.com/BinomialCoefficient.html"> Binomial
      * Coefficient</a>, "{@code n choose k}", the number of
      * {@code k}-element subsets that can be selected from an
      * {@code n}-element set.
      * <p>* <Strong>Preconditions</strong>:</p>
      * <ul>
      * <li> {@code 0 <= k <= n } (otherwise
      * {@code IllegalArgumentException} is thrown)</li>
      * <li> The result is small enough to fit into a {@code double}. The
      * largest value of {@code n} for which all coefficients are &lt;
      * Double.MAX_VALUE is 1029. If the computed value exceeds Double.MAX_VALUE,
      * Double.POSITIVE_INFINITY is returned</li>
      * </ul>
      *
      * @param n the size of the set
      * @param k the size of the subsets to be counted
      * @return {@code n choose k}
      * @throws MathIllegalArgumentException if {@code n < 0}.
      * @throws MathIllegalArgumentException if {@code k > n}.
      * @throws MathRuntimeException if the result is too large to be
      * represented by a long integer.
      */
     public static double binomialCoefficientDouble(final int n, final int k)
         throws MathIllegalArgumentException, MathRuntimeException {
         CombinatoricsUtils.checkBinomial(n, k);
         if ((n == k) || (k == 0)) {
             return 1d;
         }
         if ((k == 1) || (k == n - 1)) {
             return n;
         }
         if (k > n/2) {
             return binomialCoefficientDouble(n, n - k);
         }
         if (n < 67) {
             return binomialCoefficient(n,k);
         }
 
         double result = 1d;
         for (int i = 1; i <= k; i++) {
              result *= (double)(n - k + i) / (double)i;
         }
 
         return FastMath.floor(result + 0.5);
     }
 
     /**
      * Returns the natural {@code log} of the <a
      * href="http://mathworld.wolfram.com/BinomialCoefficient.html"> Binomial
      * Coefficient</a>, "{@code n choose k}", the number of
      * {@code k}-element subsets that can be selected from an
      * {@code n}-element set.
      * <p>* <Strong>Preconditions</strong>:</p>
      * <ul>
      * <li> {@code 0 <= k <= n } (otherwise
      * {@code MathIllegalArgumentException} is thrown)</li>
      * </ul>
      *
      * @param n the size of the set
      * @param k the size of the subsets to be counted
      * @return {@code n choose k}
      * @throws MathIllegalArgumentException if {@code n < 0}.
      * @throws MathIllegalArgumentException if {@code k > n}.
      * @throws MathRuntimeException if the result is too large to be
      * represented by a long integer.
      */
     public static double binomialCoefficientLog(final int n, final int k)
         throws MathIllegalArgumentException, MathRuntimeException {
         CombinatoricsUtils.checkBinomial(n, k);
         if ((n == k) || (k == 0)) {
             return 0;
         }
         if ((k == 1) || (k == n - 1)) {
             return FastMath.log(n);
         }
 
         /*
          * For values small enough to do exact integer computation,
          * return the log of the exact value
          */
         if (n < 67) {
             return FastMath.log(binomialCoefficient(n,k));
         }
 
         /*
          * Return the log of binomialCoefficientDouble for values that will not
          * overflow binomialCoefficientDouble
          */
         if (n < 1030) {
             return FastMath.log(binomialCoefficientDouble(n, k));
         }
 
         if (k > n / 2) {
             return binomialCoefficientLog(n, n - k);
         }
 
         /*
          * Sum logs for values that could overflow
          */
         double logSum = 0;
 
         // n!/(n-k)!
         for (int i = n - k + 1; i <= n; i++) {
             logSum += FastMath.log(i);
         }
 
         // divide by k!
         for (int i = 2; i <= k; i++) {
             logSum -= FastMath.log(i);
         }
 
         return logSum;
     }
 
     /**
      * Returns n!. Shorthand for {@code n} <a
      * href="http://mathworld.wolfram.com/Factorial.html"> Factorial</a>, the
      * product of the numbers {@code 1,...,n}.
      * <p>* <Strong>Preconditions</strong>:</p>
      * <ul>
      * <li> {@code n >= 0} (otherwise
      * {@code MathIllegalArgumentException} is thrown)</li>
      * <li> The result is small enough to fit into a {@code long}. The
      * largest value of {@code n} for which {@code n!} does not exceed
      * Long.MAX_VALUE} is 20. If the computed value exceeds {@code Long.MAX_VALUE}
      * an {@code MathRuntimeException } is thrown.</li>
      * </ul>
      *
      * @param n argument
      * @return {@code n!}
      * @throws MathRuntimeException if the result is too large to be represented
      * by a {@code long}.
      * @throws MathIllegalArgumentException if {@code n < 0}.
      * @throws MathIllegalArgumentException if {@code n > 20}: The factorial value is too
      * large to fit in a {@code long}.
      */
     public static long factorial(final int n) throws MathIllegalArgumentException {
         if (n < 0) {
             throw new MathIllegalArgumentException(LocalizedCoreFormats.FACTORIAL_NEGATIVE_PARAMETER, n);
         }
         if (n > 20) {
             throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE, n, 20);
         }
         return FACTORIALS[n];
     }
 
     /**
      * Compute n!, the<a href="http://mathworld.wolfram.com/Factorial.html">
      * factorial</a> of {@code n} (the product of the numbers 1 to n), as a
      * {@code double}.
      * The result should be small enough to fit into a {@code double}: The
      * largest {@code n} for which {@code n!} does not exceed
      * {@code Double.MAX_VALUE} is 170. If the computed value exceeds
      * {@code Double.MAX_VALUE}, {@code Double.POSITIVE_INFINITY} is returned.
      *
      * @param n Argument.
      * @return {@code n!}
      * @throws MathIllegalArgumentException if {@code n < 0}.
      */
     public static double factorialDouble(final int n) throws MathIllegalArgumentException {
         if (n < 0) {
             throw new MathIllegalArgumentException(LocalizedCoreFormats.FACTORIAL_NEGATIVE_PARAMETER,
                                            n);
         }
         if (n < 21) {
             return FACTORIALS[n];
         }
         return FastMath.floor(FastMath.exp(CombinatoricsUtils.factorialLog(n)) + 0.5);
     }
 
     /**
      * Compute the natural logarithm of the factorial of {@code n}.
      *
      * @param n Argument.
      * @return {@code log(n!)}
      * @throws MathIllegalArgumentException if {@code n < 0}.
      */
     public static double factorialLog(final int n) throws MathIllegalArgumentException {
         return FACTORIAL_LOG_NO_CACHE.value(n);
     }
 
     /**
      * Returns the <a
      * href="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html">
      * Stirling number of the second kind</a>, "{@code S(n,k)}", the number of
      * ways of partitioning an {@code n}-element set into {@code k} non-empty
      * subsets.
      * <p>
      * The preconditions are {@code 0 <= k <= n } (otherwise
      * {@code MathIllegalArgumentException} is thrown)
      * </p>
      * @param n the size of the set
      * @param k the number of non-empty subsets
      * @return {@code S(n,k)}
      * @throws MathIllegalArgumentException if {@code k < 0}.
      * @throws MathIllegalArgumentException if {@code k > n}.
      * @throws MathRuntimeException if some overflow happens, typically for n exceeding 25 and
      * k between 20 and n-2 (S(n,n-1) is handled specifically and does not overflow)
      */
     public static long stirlingS2(final int n, final int k)
         throws MathIllegalArgumentException, MathRuntimeException {
         if (k < 0) {
             throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_SMALL, k, 0);
         }
         if (k > n) {
             throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE,
                                                    k, n);
         }
 
         long[][] stirlingS2 = STIRLING_S2.get();
 
         if (stirlingS2 == null) {
             // the cache has never been initialized, compute the first numbers
             // by direct recurrence relation
 
             // as S(26,9) = 11201516780955125625 is larger than Long.MAX_VALUE
             // we must stop computation at row 26
             final int maxIndex = 26;
             stirlingS2 = new long[maxIndex][];
             stirlingS2[0] = new long[] { 1l };
             for (int i = 1; i < stirlingS2.length; ++i) {
                 stirlingS2[i] = new long[i + 1];
                 stirlingS2[i][0] = 0;
                 stirlingS2[i][1] = 1;
                 stirlingS2[i][i] = 1;
                 for (int j = 2; j < i; ++j) {
                     stirlingS2[i][j] = j * stirlingS2[i - 1][j] + stirlingS2[i - 1][j - 1];
                 }
             }
 
             // atomically save the cache
             STIRLING_S2.compareAndSet(null, stirlingS2);
 
         }
 
         if (n < stirlingS2.length) {
             // the number is in the small cache
             return stirlingS2[n][k];
         } else {
             // use explicit formula to compute the number without caching it
             if (k == 0) {
                 return 0;
             } else if (k == 1 || k == n) {
                 return 1;
             } else if (k == 2) {
                 return (1l << (n - 1)) - 1l;
             } else if (k == n - 1) {
                 return binomialCoefficient(n, 2);
             } else {
                 // definition formula: note that this may trigger some overflow
                 long sum = 0;
                 long sign = ((k & 0x1) == 0) ? 1 : -1;
                 for (int j = 1; j <= k; ++j) {
                     sign = -sign;
                     sum += sign * binomialCoefficient(k, j) * ArithmeticUtils.pow(j, n);
                     if (sum < 0) {
                         // there was an overflow somewhere
                         throw new MathRuntimeException(LocalizedCoreFormats.OUT_OF_RANGE_SIMPLE,
                                                           n, 0, stirlingS2.length - 1);
                     }
                 }
                 return sum / factorial(k);
             }
         }
 
     }
 
     /**
      * Returns an iterator whose range is the k-element subsets of {0, ..., n - 1}
      * represented as {@code int[]} arrays.
      * <p>
      * The arrays returned by the iterator are sorted in descending order and
      * they are visited in lexicographic order with significance from right to
      * left. For example, combinationsIterator(4, 2) returns an Iterator that
      * will generate the following sequence of arrays on successive calls to
      * {@code next()}:</p><p>
      * {@code [0, 1], [0, 2], [1, 2], [0, 3], [1, 3], [2, 3]}
      * </p><p>
      * If {@code k == 0} an Iterator containing an empty array is returned and
      * if {@code k == n} an Iterator containing [0, ..., n -1] is returned.</p>
      *
      * @param n Size of the set from which subsets are selected.
      * @param k Size of the subsets to be enumerated.
      * @return an {@link Iterator iterator} over the k-sets in n.
      * @throws MathIllegalArgumentException if {@code n < 0}.
      * @throws MathIllegalArgumentException if {@code k > n}.
      */
     public static Iterator<int[]> combinationsIterator(int n, int k) {
         return new Combinations(n, k).iterator();
     }
 
     /**
      * Check binomial preconditions.
      *
      * @param n Size of the set.
      * @param k Size of the subsets to be counted.
      * @throws MathIllegalArgumentException if {@code n < 0}.
      * @throws MathIllegalArgumentException if {@code k > n}.
      */
     public static void checkBinomial(final int n,
                                      final int k)
         throws MathIllegalArgumentException {
         if (n < k) {
             throw new MathIllegalArgumentException(LocalizedCoreFormats.BINOMIAL_INVALID_PARAMETERS_ORDER,
                                                 k, n, true);
         }
         if (n < 0) {
             throw new MathIllegalArgumentException(LocalizedCoreFormats.BINOMIAL_NEGATIVE_PARAMETER, n);
         }
     }
 
     /** Compute the Bell number (number of partitions of a set).
      * @param n number of elements of the set
      * @return Bell number Bₙ
      * @since 2.2
      */
     public static long bellNumber(final int n) {
 
         if (n < 0) {
             throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_SMALL, n, 0);
         }
         if (n > MAX_BELL) {
             throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_LARGE, n, MAX_BELL);
         }
 
         long[] bell = BELL.get();
 
         if (bell == null) {
 
             // the cache has never been initialized, compute the numbers using the Bell triangle
             // storage for one line of the Bell triangle
             bell = new long[MAX_BELL];
             bell[0] = 1l;
 
             final long[] row = new long[bell.length];
             for (int k = 1; k < row.length; ++k) {
 
                 // first row, with one element
                 row[0] = 1l;
 
                 // iterative computation of rows
                 for (int i = 1; i < k; ++i) {
                     long previous = row[0];
                     row[0] = row[i - 1];
                     for (int j = 1; j <= i; ++j) {
                         long rj = row[j - 1] + previous;
                         previous = row[j];
                         row[j] = rj;
                     }
                 }
 
                 bell[k] = row[k - 1];
 
             }
 
             BELL.compareAndSet(null, bell);
 
         }
 
         return bell[n];
 
     }
 
     /** Generate a stream of partitions of a list.
      * <p>
      * This method implements the iterative algorithm described in
      * <a href="https://academic.oup.com/comjnl/article/32/3/281/331557">Short Note:
      * A Fast Iterative Algorithm for Generating Set Partitions</a>
      * by B. Djokić, M. Miyakawa, S. Sekiguchi, I. Semba, and I. Stojmenović
      * (The Computer Journal, Volume 32, Issue 3, 1989, Pages 281–282,
      * <a href="https://doi.org/10.1093/comjnl/32.3.281">https://doi.org/10.1093/comjnl/32.3.281</a>
      * </p>
      * @param <T> type of the list elements
      * @param list list to partition
      * @return stream of partitions of the list, each partition is an array or parts
      * and each part is a list of elements
      * @since 2.2
      */
     public static <T> Stream<List<T>[]> partitions(final List<T> list) {
 
         // handle special cases of empty and singleton lists
         if (list.size() < 2) {
             @SuppressWarnings("unchecked")
             final List<T>[] partition = (List<T>[]) Array.newInstance(List.class, 1);
             partition[0] = list;
             final Stream.Builder<List<T>[]> builder = Stream.builder();
             return builder.add(partition).build();
         }
 
         return StreamSupport.stream(Spliterators.spliteratorUnknownSize(new PartitionsIterator<T>(list),
                                                                         Spliterator.DISTINCT | Spliterator.NONNULL |
                                                                         Spliterator.IMMUTABLE | Spliterator.ORDERED),
                                     false);
 
     }
 
     /** Generate a stream of permutations of a list.
      * <p>
      * This method implements the Steinhaus–Johnson–Trotter algorithm
      * with Even's speedup
      * <a href="https://en.wikipedia.org/wiki/Steinhaus%E2%80%93Johnson%E2%80%93Trotter_algorithm">Steinhaus–Johnson–Trotter algorithm</a>
      * @param <T> type of the list elements
      * @param list list to permute
      * @return stream of permutations of the list
      * @since 2.2
      */
     public static <T> Stream<List<T>> permutations(final List<T> list) {
 
         // handle special cases of empty and singleton lists
         if (list.size() < 2) {
             return Stream.of(list);
         }
 
         return StreamSupport.stream(Spliterators.spliteratorUnknownSize(new PermutationsIterator<T>(list),
                                                                         Spliterator.DISTINCT | Spliterator.NONNULL |
                                                                         Spliterator.IMMUTABLE | Spliterator.ORDERED),
                                     false);
 
     }
 
     /**
      * Class for computing the natural logarithm of the factorial of {@code n}.
      * It allows to allocate a cache of precomputed values.
      * In case of cache miss, computation is preformed by a call to
      * {@link Gamma#logGamma(double)}.
      */
     public static final class FactorialLog {
         /**
          * Precomputed values of the function:
          * {@code LOG_FACTORIALS[i] = log(i!)}.
          */
         private final double[] LOG_FACTORIALS;
 
         /**
          * Creates an instance, reusing the already computed values if available.
          *
          * @param numValues Number of values of the function to compute.
          * @param cache Existing cache.
          * @throw MathIllegalArgumentException if {@code n < 0}.
          */
         private FactorialLog(int numValues,
                              double[] cache) {
             if (numValues < 0) {
                 throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_SMALL, numValues, 0);
             }
 
             LOG_FACTORIALS = new double[numValues];
 
             final int beginCopy = 2;
             final int endCopy = cache == null || cache.length <= beginCopy ?
                 beginCopy : cache.length <= numValues ?
                 cache.length : numValues;
 
             // Copy available values.
             for (int i = beginCopy; i < endCopy; i++) {
                 LOG_FACTORIALS[i] = cache[i];
             }
 
             // Precompute.
             for (int i = endCopy; i < numValues; i++) {
                 LOG_FACTORIALS[i] = LOG_FACTORIALS[i - 1] + FastMath.log(i);
             }
         }
 
         /**
          * Creates an instance with no precomputed values.
          * @return instance with no precomputed values
          */
         public static FactorialLog create() {
             return new FactorialLog(0, null);
         }
 
         /**
          * Creates an instance with the specified cache size.
          *
          * @param cacheSize Number of precomputed values of the function.
          * @return a new instance where {@code cacheSize} values have been
          * precomputed.
          * @throws MathIllegalArgumentException if {@code n < 0}.
          */
         public FactorialLog withCache(final int cacheSize) {
             return new FactorialLog(cacheSize, LOG_FACTORIALS);
         }
 
         /**
          * Computes {@code log(n!)}.
          *
          * @param n Argument.
          * @return {@code log(n!)}.
          * @throws MathIllegalArgumentException if {@code n < 0}.
          */
         public double value(final int n) {
             if (n < 0) {
                 throw new MathIllegalArgumentException(LocalizedCoreFormats.FACTORIAL_NEGATIVE_PARAMETER,
                                                n);
             }
 
             // Use cache of precomputed values.
             if (n < LOG_FACTORIALS.length) {
                 return LOG_FACTORIALS[n];
             }
 
             // Use cache of precomputed factorial values.
             if (n < FACTORIALS.length) {
                 return FastMath.log(FACTORIALS[n]);
             }
 
             // Delegate.
             return Gamma.logGamma(n + 1);
         }
     }
 }