1 | /*
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2 | * Licensed to the Apache Software Foundation (ASF) under one or more
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3 | * contributor license agreements. See the NOTICE file distributed with
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4 | * this work for additional information regarding copyright ownership.
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5 | * The ASF licenses this file to You under the Apache License, Version 2.0
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6 | * (the "License"); you may not use this file except in compliance with
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7 | * the License. You may obtain a copy of the License at
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8 | *
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9 | * http://www.apache.org/licenses/LICENSE-2.0
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10 | *
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11 | * Unless required by applicable law or agreed to in writing, software
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12 | * distributed under the License is distributed on an "AS IS" BASIS,
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13 | * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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14 | * See the License for the specific language governing permissions and
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15 | * limitations under the License.
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16 | */
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17 | package agents.anac.y2019.harddealer.math3.util;
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18 |
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19 | import java.util.Iterator;
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20 | import java.util.concurrent.atomic.AtomicReference;
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21 |
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22 | import agents.anac.y2019.harddealer.math3.exception.MathArithmeticException;
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23 | import agents.anac.y2019.harddealer.math3.exception.NotPositiveException;
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24 | import agents.anac.y2019.harddealer.math3.exception.NumberIsTooLargeException;
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25 | import agents.anac.y2019.harddealer.math3.exception.util.LocalizedFormats;
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26 |
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27 | /**
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28 | * Combinatorial utilities.
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29 | *
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30 | * @since 3.3
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31 | */
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32 | public final class CombinatoricsUtils {
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33 |
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34 | /** All long-representable factorials */
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35 | static final long[] FACTORIALS = new long[] {
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36 | 1l, 1l, 2l,
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37 | 6l, 24l, 120l,
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38 | 720l, 5040l, 40320l,
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39 | 362880l, 3628800l, 39916800l,
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40 | 479001600l, 6227020800l, 87178291200l,
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41 | 1307674368000l, 20922789888000l, 355687428096000l,
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42 | 6402373705728000l, 121645100408832000l, 2432902008176640000l };
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43 |
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44 | /** Stirling numbers of the second kind. */
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45 | static final AtomicReference<long[][]> STIRLING_S2 = new AtomicReference<long[][]> (null);
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46 |
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47 | /** Private constructor (class contains only static methods). */
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48 | private CombinatoricsUtils() {}
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49 |
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50 |
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51 | /**
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52 | * Returns an exact representation of the <a
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53 | * href="http://mathworld.wolfram.com/BinomialCoefficient.html"> Binomial
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54 | * Coefficient</a>, "{@code n choose k}", the number of
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55 | * {@code k}-element subsets that can be selected from an
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56 | * {@code n}-element set.
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57 | * <p>
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58 | * <Strong>Preconditions</strong>:
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59 | * <ul>
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60 | * <li> {@code 0 <= k <= n } (otherwise
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61 | * {@code MathIllegalArgumentException} is thrown)</li>
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62 | * <li> The result is small enough to fit into a {@code long}. The
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63 | * largest value of {@code n} for which all coefficients are
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64 | * {@code < Long.MAX_VALUE} is 66. If the computed value exceeds
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65 | * {@code Long.MAX_VALUE} a {@code MathArithMeticException} is
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66 | * thrown.</li>
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67 | * </ul></p>
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68 | *
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69 | * @param n the size of the set
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70 | * @param k the size of the subsets to be counted
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71 | * @return {@code n choose k}
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72 | * @throws NotPositiveException if {@code n < 0}.
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73 | * @throws NumberIsTooLargeException if {@code k > n}.
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74 | * @throws MathArithmeticException if the result is too large to be
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75 | * represented by a long integer.
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76 | */
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77 | public static long binomialCoefficient(final int n, final int k)
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78 | throws NotPositiveException, NumberIsTooLargeException, MathArithmeticException {
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79 | CombinatoricsUtils.checkBinomial(n, k);
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80 | if ((n == k) || (k == 0)) {
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81 | return 1;
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82 | }
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83 | if ((k == 1) || (k == n - 1)) {
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84 | return n;
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85 | }
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86 | // Use symmetry for large k
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87 | if (k > n / 2) {
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88 | return binomialCoefficient(n, n - k);
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89 | }
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90 |
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91 | // We use the formula
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92 | // (n choose k) = n! / (n-k)! / k!
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93 | // (n choose k) == ((n-k+1)*...*n) / (1*...*k)
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94 | // which could be written
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95 | // (n choose k) == (n-1 choose k-1) * n / k
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96 | long result = 1;
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97 | if (n <= 61) {
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98 | // For n <= 61, the naive implementation cannot overflow.
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99 | int i = n - k + 1;
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100 | for (int j = 1; j <= k; j++) {
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101 | result = result * i / j;
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102 | i++;
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103 | }
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104 | } else if (n <= 66) {
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105 | // For n > 61 but n <= 66, the result cannot overflow,
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106 | // but we must take care not to overflow intermediate values.
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107 | int i = n - k + 1;
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108 | for (int j = 1; j <= k; j++) {
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109 | // We know that (result * i) is divisible by j,
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110 | // but (result * i) may overflow, so we split j:
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111 | // Filter out the gcd, d, so j/d and i/d are integer.
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112 | // result is divisible by (j/d) because (j/d)
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113 | // is relative prime to (i/d) and is a divisor of
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114 | // result * (i/d).
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115 | final long d = ArithmeticUtils.gcd(i, j);
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116 | result = (result / (j / d)) * (i / d);
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117 | i++;
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118 | }
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119 | } else {
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120 | // For n > 66, a result overflow might occur, so we check
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121 | // the multiplication, taking care to not overflow
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122 | // unnecessary.
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123 | int i = n - k + 1;
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124 | for (int j = 1; j <= k; j++) {
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125 | final long d = ArithmeticUtils.gcd(i, j);
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126 | result = ArithmeticUtils.mulAndCheck(result / (j / d), i / d);
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127 | i++;
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128 | }
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129 | }
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130 | return result;
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131 | }
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132 |
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133 | /**
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134 | * Returns a {@code double} representation of the <a
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135 | * href="http://mathworld.wolfram.com/BinomialCoefficient.html"> Binomial
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136 | * Coefficient</a>, "{@code n choose k}", the number of
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137 | * {@code k}-element subsets that can be selected from an
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138 | * {@code n}-element set.
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139 | * <p>
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140 | * <Strong>Preconditions</strong>:
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141 | * <ul>
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142 | * <li> {@code 0 <= k <= n } (otherwise
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143 | * {@code IllegalArgumentException} is thrown)</li>
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144 | * <li> The result is small enough to fit into a {@code double}. The
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145 | * largest value of {@code n} for which all coefficients are less than
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146 | * Double.MAX_VALUE is 1029. If the computed value exceeds Double.MAX_VALUE,
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147 | * Double.POSITIVE_INFINITY is returned</li>
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148 | * </ul></p>
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149 | *
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150 | * @param n the size of the set
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151 | * @param k the size of the subsets to be counted
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152 | * @return {@code n choose k}
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153 | * @throws NotPositiveException if {@code n < 0}.
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154 | * @throws NumberIsTooLargeException if {@code k > n}.
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155 | * @throws MathArithmeticException if the result is too large to be
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156 | * represented by a long integer.
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157 | */
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158 | public static double binomialCoefficientDouble(final int n, final int k)
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159 | throws NotPositiveException, NumberIsTooLargeException, MathArithmeticException {
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160 | CombinatoricsUtils.checkBinomial(n, k);
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161 | if ((n == k) || (k == 0)) {
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162 | return 1d;
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163 | }
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164 | if ((k == 1) || (k == n - 1)) {
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165 | return n;
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166 | }
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167 | if (k > n/2) {
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168 | return binomialCoefficientDouble(n, n - k);
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169 | }
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170 | if (n < 67) {
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171 | return binomialCoefficient(n,k);
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172 | }
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173 |
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174 | double result = 1d;
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175 | for (int i = 1; i <= k; i++) {
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176 | result *= (double)(n - k + i) / (double)i;
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177 | }
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178 |
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179 | return FastMath.floor(result + 0.5);
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180 | }
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181 |
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182 | /**
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183 | * Returns the natural {@code log} of the <a
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184 | * href="http://mathworld.wolfram.com/BinomialCoefficient.html"> Binomial
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185 | * Coefficient</a>, "{@code n choose k}", the number of
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186 | * {@code k}-element subsets that can be selected from an
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187 | * {@code n}-element set.
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188 | * <p>
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189 | * <Strong>Preconditions</strong>:
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190 | * <ul>
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191 | * <li> {@code 0 <= k <= n } (otherwise
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192 | * {@code MathIllegalArgumentException} is thrown)</li>
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193 | * </ul></p>
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194 | *
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195 | * @param n the size of the set
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196 | * @param k the size of the subsets to be counted
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197 | * @return {@code n choose k}
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198 | * @throws NotPositiveException if {@code n < 0}.
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199 | * @throws NumberIsTooLargeException if {@code k > n}.
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200 | * @throws MathArithmeticException if the result is too large to be
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201 | * represented by a long integer.
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202 | */
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203 | public static double binomialCoefficientLog(final int n, final int k)
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204 | throws NotPositiveException, NumberIsTooLargeException, MathArithmeticException {
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205 | CombinatoricsUtils.checkBinomial(n, k);
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206 | if ((n == k) || (k == 0)) {
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207 | return 0;
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208 | }
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209 | if ((k == 1) || (k == n - 1)) {
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210 | return FastMath.log(n);
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211 | }
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212 |
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213 | /*
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214 | * For values small enough to do exact integer computation,
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215 | * return the log of the exact value
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216 | */
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217 | if (n < 67) {
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218 | return FastMath.log(binomialCoefficient(n,k));
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219 | }
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220 |
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221 | /*
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222 | * Return the log of binomialCoefficientDouble for values that will not
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223 | * overflow binomialCoefficientDouble
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224 | */
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225 | if (n < 1030) {
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226 | return FastMath.log(binomialCoefficientDouble(n, k));
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227 | }
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228 |
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229 | if (k > n / 2) {
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230 | return binomialCoefficientLog(n, n - k);
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231 | }
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232 |
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233 | /*
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234 | * Sum logs for values that could overflow
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235 | */
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236 | double logSum = 0;
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237 |
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238 | // n!/(n-k)!
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239 | for (int i = n - k + 1; i <= n; i++) {
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240 | logSum += FastMath.log(i);
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241 | }
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242 |
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243 | // divide by k!
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244 | for (int i = 2; i <= k; i++) {
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245 | logSum -= FastMath.log(i);
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246 | }
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247 |
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248 | return logSum;
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249 | }
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250 |
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251 | /**
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252 | * Returns n!. Shorthand for {@code n} <a
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253 | * href="http://mathworld.wolfram.com/Factorial.html"> Factorial</a>, the
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254 | * product of the numbers {@code 1,...,n}.
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255 | * <p>
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256 | * <Strong>Preconditions</strong>:
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257 | * <ul>
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258 | * <li> {@code n >= 0} (otherwise
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259 | * {@code MathIllegalArgumentException} is thrown)</li>
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260 | * <li> The result is small enough to fit into a {@code long}. The
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261 | * largest value of {@code n} for which {@code n!} does not exceed
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262 | * Long.MAX_VALUE} is 20. If the computed value exceeds {@code Long.MAX_VALUE}
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263 | * an {@code MathArithMeticException } is thrown.</li>
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264 | * </ul>
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265 | * </p>
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266 | *
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267 | * @param n argument
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268 | * @return {@code n!}
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269 | * @throws MathArithmeticException if the result is too large to be represented
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270 | * by a {@code long}.
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271 | * @throws NotPositiveException if {@code n < 0}.
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272 | * @throws MathArithmeticException if {@code n > 20}: The factorial value is too
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273 | * large to fit in a {@code long}.
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274 | */
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275 | public static long factorial(final int n) throws NotPositiveException, MathArithmeticException {
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276 | if (n < 0) {
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277 | throw new NotPositiveException(LocalizedFormats.FACTORIAL_NEGATIVE_PARAMETER,
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278 | n);
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279 | }
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280 | if (n > 20) {
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281 | throw new MathArithmeticException();
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282 | }
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283 | return FACTORIALS[n];
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284 | }
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285 |
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286 | /**
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287 | * Compute n!, the<a href="http://mathworld.wolfram.com/Factorial.html">
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288 | * factorial</a> of {@code n} (the product of the numbers 1 to n), as a
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289 | * {@code double}.
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290 | * The result should be small enough to fit into a {@code double}: The
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291 | * largest {@code n} for which {@code n!} does not exceed
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292 | * {@code Double.MAX_VALUE} is 170. If the computed value exceeds
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293 | * {@code Double.MAX_VALUE}, {@code Double.POSITIVE_INFINITY} is returned.
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294 | *
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295 | * @param n Argument.
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296 | * @return {@code n!}
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297 | * @throws NotPositiveException if {@code n < 0}.
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298 | */
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299 | public static double factorialDouble(final int n) throws NotPositiveException {
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300 | if (n < 0) {
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301 | throw new NotPositiveException(LocalizedFormats.FACTORIAL_NEGATIVE_PARAMETER,
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302 | n);
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303 | }
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304 | if (n < 21) {
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305 | return FACTORIALS[n];
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306 | }
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307 | return FastMath.floor(FastMath.exp(CombinatoricsUtils.factorialLog(n)) + 0.5);
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308 | }
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309 |
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310 | /**
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311 | * Compute the natural logarithm of the factorial of {@code n}.
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312 | *
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313 | * @param n Argument.
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314 | * @return {@code n!}
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315 | * @throws NotPositiveException if {@code n < 0}.
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316 | */
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317 | public static double factorialLog(final int n) throws NotPositiveException {
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318 | if (n < 0) {
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319 | throw new NotPositiveException(LocalizedFormats.FACTORIAL_NEGATIVE_PARAMETER,
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320 | n);
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321 | }
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322 | if (n < 21) {
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323 | return FastMath.log(FACTORIALS[n]);
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324 | }
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325 | double logSum = 0;
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326 | for (int i = 2; i <= n; i++) {
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327 | logSum += FastMath.log(i);
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328 | }
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329 | return logSum;
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330 | }
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331 |
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332 | /**
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333 | * Returns the <a
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334 | * href="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html">
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335 | * Stirling number of the second kind</a>, "{@code S(n,k)}", the number of
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336 | * ways of partitioning an {@code n}-element set into {@code k} non-empty
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337 | * subsets.
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338 | * <p>
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339 | * The preconditions are {@code 0 <= k <= n } (otherwise
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340 | * {@code NotPositiveException} is thrown)
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341 | * </p>
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342 | * @param n the size of the set
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343 | * @param k the number of non-empty subsets
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344 | * @return {@code S(n,k)}
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345 | * @throws NotPositiveException if {@code k < 0}.
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346 | * @throws NumberIsTooLargeException if {@code k > n}.
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347 | * @throws MathArithmeticException if some overflow happens, typically for n exceeding 25 and
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348 | * k between 20 and n-2 (S(n,n-1) is handled specifically and does not overflow)
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349 | * @since 3.1
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350 | */
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351 | public static long stirlingS2(final int n, final int k)
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352 | throws NotPositiveException, NumberIsTooLargeException, MathArithmeticException {
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353 | if (k < 0) {
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354 | throw new NotPositiveException(k);
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355 | }
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356 | if (k > n) {
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357 | throw new NumberIsTooLargeException(k, n, true);
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358 | }
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359 |
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360 | long[][] stirlingS2 = STIRLING_S2.get();
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361 |
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362 | if (stirlingS2 == null) {
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363 | // the cache has never been initialized, compute the first numbers
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364 | // by direct recurrence relation
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365 |
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366 | // as S(26,9) = 11201516780955125625 is larger than Long.MAX_VALUE
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367 | // we must stop computation at row 26
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368 | final int maxIndex = 26;
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369 | stirlingS2 = new long[maxIndex][];
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370 | stirlingS2[0] = new long[] { 1l };
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371 | for (int i = 1; i < stirlingS2.length; ++i) {
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372 | stirlingS2[i] = new long[i + 1];
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373 | stirlingS2[i][0] = 0;
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374 | stirlingS2[i][1] = 1;
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375 | stirlingS2[i][i] = 1;
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376 | for (int j = 2; j < i; ++j) {
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377 | stirlingS2[i][j] = j * stirlingS2[i - 1][j] + stirlingS2[i - 1][j - 1];
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378 | }
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379 | }
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380 |
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381 | // atomically save the cache
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382 | STIRLING_S2.compareAndSet(null, stirlingS2);
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383 |
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384 | }
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385 |
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386 | if (n < stirlingS2.length) {
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387 | // the number is in the small cache
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388 | return stirlingS2[n][k];
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389 | } else {
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390 | // use explicit formula to compute the number without caching it
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391 | if (k == 0) {
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392 | return 0;
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393 | } else if (k == 1 || k == n) {
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394 | return 1;
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395 | } else if (k == 2) {
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396 | return (1l << (n - 1)) - 1l;
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397 | } else if (k == n - 1) {
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398 | return binomialCoefficient(n, 2);
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399 | } else {
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400 | // definition formula: note that this may trigger some overflow
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401 | long sum = 0;
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402 | long sign = ((k & 0x1) == 0) ? 1 : -1;
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403 | for (int j = 1; j <= k; ++j) {
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404 | sign = -sign;
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405 | sum += sign * binomialCoefficient(k, j) * ArithmeticUtils.pow(j, n);
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406 | if (sum < 0) {
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407 | // there was an overflow somewhere
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408 | throw new MathArithmeticException(LocalizedFormats.ARGUMENT_OUTSIDE_DOMAIN,
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409 | n, 0, stirlingS2.length - 1);
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410 | }
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411 | }
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412 | return sum / factorial(k);
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413 | }
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414 | }
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415 |
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416 | }
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417 |
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418 | /**
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419 | * Returns an iterator whose range is the k-element subsets of {0, ..., n - 1}
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420 | * represented as {@code int[]} arrays.
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421 | * <p>
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422 | * The arrays returned by the iterator are sorted in descending order and
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423 | * they are visited in lexicographic order with significance from right to
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424 | * left. For example, combinationsIterator(4, 2) returns an Iterator that
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425 | * will generate the following sequence of arrays on successive calls to
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426 | * {@code next()}:</p><p>
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427 | * {@code [0, 1], [0, 2], [1, 2], [0, 3], [1, 3], [2, 3]}
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428 | * </p><p>
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429 | * If {@code k == 0} an Iterator containing an empty array is returned and
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430 | * if {@code k == n} an Iterator containing [0, ..., n -1] is returned.</p>
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431 | *
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432 | * @param n Size of the set from which subsets are selected.
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433 | * @param k Size of the subsets to be enumerated.
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434 | * @return an {@link Iterator iterator} over the k-sets in n.
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435 | * @throws NotPositiveException if {@code n < 0}.
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436 | * @throws NumberIsTooLargeException if {@code k > n}.
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437 | */
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438 | public static Iterator<int[]> combinationsIterator(int n, int k) {
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439 | return new Combinations(n, k).iterator();
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440 | }
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441 |
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442 | /**
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443 | * Check binomial preconditions.
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444 | *
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445 | * @param n Size of the set.
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446 | * @param k Size of the subsets to be counted.
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447 | * @throws NotPositiveException if {@code n < 0}.
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448 | * @throws NumberIsTooLargeException if {@code k > n}.
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449 | */
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450 | public static void checkBinomial(final int n,
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451 | final int k)
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452 | throws NumberIsTooLargeException,
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453 | NotPositiveException {
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454 | if (n < k) {
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455 | throw new NumberIsTooLargeException(LocalizedFormats.BINOMIAL_INVALID_PARAMETERS_ORDER,
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456 | k, n, true);
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457 | }
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458 | if (n < 0) {
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459 | throw new NotPositiveException(LocalizedFormats.BINOMIAL_NEGATIVE_PARAMETER, n);
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460 | }
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461 | }
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462 | }
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