CombinatoricsUtils.java
/*
* 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 <
* 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);
}
}
}