/*
* 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.util;
import java.util.Iterator;
import java.util.concurrent.atomic.AtomicReference;
import org.apache.commons.math3.exception.MathArithmeticException;
import org.apache.commons.math3.exception.NotPositiveException;
import org.apache.commons.math3.exception.NumberIsTooLargeException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
/**
* Combinatorial utilities.
*
* @since 3.3
*/
public final class CombinatoricsUtils {
/** All long-representable factorials */
static final long[] FACTORIALS = new long[] {
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<long[][]> (null);
/** Private constructor. */
private CombinatoricsUtils() {
super();
}
/**
* 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>:
* <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} an {@code ArithMeticException} is
* thrown.</li>
* </ul></p>
*
* @param n the size of the set
* @param k the size of the subsets to be counted
* @return {@code n choose k}
* @throws NotPositiveException if {@code n < 0}.
* @throws NumberIsTooLargeException if {@code k > n}.
* @throws MathArithmeticException if the result is too large to be
* represented by a long integer.
*/
public static long binomialCoefficient(final int n, final int k)
throws NotPositiveException, NumberIsTooLargeException, MathArithmeticException {
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>:
* <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 <
* Double.MAX_VALUE is 1029. If the computed value exceeds Double.MAX_VALUE,
* Double.POSITIVE_INFINITY is returned</li>
* </ul></p>
*
* @param n the size of the set
* @param k the size of the subsets to be counted
* @return {@code n choose k}
* @throws NotPositiveException if {@code n < 0}.
* @throws NumberIsTooLargeException if {@code k > n}.
* @throws MathArithmeticException if the result is too large to be
* represented by a long integer.
*/
public static double binomialCoefficientDouble(final int n, final int k)
throws NotPositiveException, NumberIsTooLargeException, MathArithmeticException {
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>:
* <ul>
* <li> {@code 0 <= k <= n } (otherwise
* {@code IllegalArgumentException} is thrown)</li>
* </ul></p>
*
* @param n the size of the set
* @param k the size of the subsets to be counted
* @return {@code n choose k}
* @throws NotPositiveException if {@code n < 0}.
* @throws NumberIsTooLargeException if {@code k > n}.
* @throws MathArithmeticException if the result is too large to be
* represented by a long integer.
*/
public static double binomialCoefficientLog(final int n, final int k)
throws NotPositiveException, NumberIsTooLargeException, MathArithmeticException {
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>:
* <ul>
* <li> {@code n >= 0} (otherwise
* {@code IllegalArgumentException} 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!} <
* Long.MAX_VALUE} is 20. If the computed value exceeds {@code Long.MAX_VALUE}
* an {@code ArithMeticException } is thrown.</li>
* </ul>
* </p>
*
* @param n argument
* @return {@code n!}
* @throws MathArithmeticException if the result is too large to be represented
* by a {@code long}.
* @throws NotPositiveException if {@code n < 0}.
* @throws MathArithmeticException if {@code n > 20}: The factorial value is too
* large to fit in a {@code long}.
*/
public static long factorial(final int n) throws NotPositiveException, MathArithmeticException {
if (n < 0) {
throw new NotPositiveException(LocalizedFormats.FACTORIAL_NEGATIVE_PARAMETER,
n);
}
if (n > 20) {
throw new MathArithmeticException();
}
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! < 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 NotPositiveException if {@code n < 0}.
*/
public static double factorialDouble(final int n) throws NotPositiveException {
if (n < 0) {
throw new NotPositiveException(LocalizedFormats.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 n!}
* @throws NotPositiveException if {@code n < 0}.
*/
public static double factorialLog(final int n) throws NotPositiveException {
if (n < 0) {
throw new NotPositiveException(LocalizedFormats.FACTORIAL_NEGATIVE_PARAMETER,
n);
}
if (n < 21) {
return FastMath.log(FACTORIALS[n]);
}
double logSum = 0;
for (int i = 2; i <= n; i++) {
logSum += FastMath.log(i);
}
return logSum;
}
/**
* 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 NotPositiveException} is thrown)
* </p>
* @param n the size of the set
* @param k the number of non-empty subsets
* @return {@code S(n,k)}
* @throws NotPositiveException if {@code k < 0}.
* @throws NumberIsTooLargeException if {@code k > n}.
* @throws MathArithmeticException 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)
* @since 3.1
*/
public static long stirlingS2(final int n, final int k)
throws NotPositiveException, NumberIsTooLargeException, MathArithmeticException {
if (k < 0) {
throw new NotPositiveException(k);
}
if (k > n) {
throw new NumberIsTooLargeException(k, n, true);
}
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 MathArithmeticException(LocalizedFormats.ARGUMENT_OUTSIDE_DOMAIN,
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()}:<br/>
* {@code [0, 1], [0, 2], [1, 2], [0, 3], [1, 3], [2, 3]}
* </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.
*
* @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 NotPositiveException if {@code n < 0}.
* @throws NumberIsTooLargeException 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 NotPositiveException if {@code n < 0}.
* @throws NumberIsTooLargeException if {@code k > n}.
*/
public static void checkBinomial(final int n,
final int k)
throws NumberIsTooLargeException,
NotPositiveException {
if (n < k) {
throw new NumberIsTooLargeException(LocalizedFormats.BINOMIAL_INVALID_PARAMETERS_ORDER,
k, n, true);
}
if (n < 0) {
throw new NotPositiveException(LocalizedFormats.BINOMIAL_NEGATIVE_PARAMETER, n);
}
}
}