Returns an exact representation of the Binomial Coefficient, "n choose k", the number of k-element subsets that can be selected from an n-element set. - Java java.lang

Description

Returns an exact representation of the Binomial Coefficient, "n choose k", the number of k-element subsets that can be selected from an n-element set.

Demo Code

/*//from   w  w  w.j  a v a 2  s.c  o m
 * 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 com.java2s;
import java.math.BigDecimal;

public class Main {
    /** -1.0 cast as a byte. */
    private static final byte NB = (byte) -1;
    /** -1.0 cast as a short. */
    private static final short NS = (short) -1;
    /** 1.0 cast as a byte. */
    private static final byte PB = (byte) 1;
    /** 1.0 cast as a short. */
    private static final short PS = (short) 1;
    /** 0.0 cast as a byte. */
    private static final byte ZB = (byte) 0;
    /** 0.0 cast as a short. */
    private static final short ZS = (short) 0;

    /**
     * Returns an exact representation of the <a
     * href="http://mathworld.wolfram.com/BinomialCoefficient.html"> Binomial
     * Coefficient</a>, "<code>n choose k</code>", the number of
     * <code>k</code>-element subsets that can be selected from an
     * <code>n</code>-element set.
     * <p>
     * <Strong>Preconditions</strong>:
     * <ul>
     * <li> <code>0 <= k <= n </code> (otherwise
     * <code>IllegalArgumentException</code> is thrown)</li>
     * <li> The result is small enough to fit into a <code>long</code>. The
     * largest value of <code>n</code> for which all coefficients are
     * <code> < Long.MAX_VALUE</code> is 66. If the computed value exceeds
     * <code>Long.MAX_VALUE</code> an <code>ArithMeticException
     *      </code> 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</code>
     * @throws IllegalArgumentException if preconditions are not met.
     * @throws ArithmeticException if the result is too large to be represented
     *         by a long integer.
     */
    public static long binomialCoefficient(final int n, final int k) {
        if (n < k) {
            throw new IllegalArgumentException(
                    "must have n >= k for binomial coefficient (n,k)");
        }
        if (n < 0) {
            throw new IllegalArgumentException(
                    "must have n >= 0 for binomial coefficient (n,k)");
        }
        if ((n == k) || (k == 0)) {
            return 1;
        }
        if ((k == 1) || (k == n - 1)) {
            return n;
        }

        long result = Math.round(binomialCoefficientDouble(n, k));
        if (result == Long.MAX_VALUE) {
            throw new ArithmeticException(
                    "result too large to represent in a long integer");
        }
        return result;
    }

    /**
     * Round the given value to the specified number of decimal places. The
     * value is rounded using the {@link BigDecimal#ROUND_HALF_UP} method.
     * 
     * @param x the value to round.
     * @param scale the number of digits to the right of the decimal point.
     * @return the rounded value.
     * @since 1.1
     */
    public static double round(double x, int scale) {
        return round(x, scale, BigDecimal.ROUND_HALF_UP);
    }

    /**
     * Round the given value to the specified number of decimal places. The
     * value is rounded using the given method which is any method defined in
     * {@link BigDecimal}.
     * 
     * @param x the value to round.
     * @param scale the number of digits to the right of the decimal point.
     * @param roundingMethod the rounding method as defined in
     *        {@link BigDecimal}.
     * @return the rounded value.
     * @since 1.1
     */
    public static double round(double x, int scale, int roundingMethod) {
        try {
            return (new BigDecimal(Double.toString(x)).setScale(scale,
                    roundingMethod)).doubleValue();
        } catch (NumberFormatException ex) {
            if (Double.isInfinite(x)) {
                return x;
            } else {
                return Double.NaN;
            }
        }
    }

    /**
     * Round the given value to the specified number of decimal places. The
     * value is rounding using the {@link BigDecimal#ROUND_HALF_UP} method.
     * 
     * @param x the value to round.
     * @param scale the number of digits to the right of the decimal point.
     * @return the rounded value.
     * @since 1.1
     */
    public static float round(float x, int scale) {
        return round(x, scale, BigDecimal.ROUND_HALF_UP);
    }

    /**
     * Round the given value to the specified number of decimal places. The
     * value is rounded using the given method which is any method defined in
     * {@link BigDecimal}.
     * 
     * @param x the value to round.
     * @param scale the number of digits to the right of the decimal point.
     * @param roundingMethod the rounding method as defined in
     *        {@link BigDecimal}.
     * @return the rounded value.
     * @since 1.1
     */
    public static float round(float x, int scale, int roundingMethod) {
        float sign = indicator(x);
        float factor = (float) Math.pow(10.0f, scale) * sign;
        return (float) roundUnscaled(x * factor, sign, roundingMethod)
                / factor;
    }

    /**
     * Returns a <code>double</code> representation of the <a
     * href="http://mathworld.wolfram.com/BinomialCoefficient.html"> Binomial
     * Coefficient</a>, "<code>n choose k</code>", the number of
     * <code>k</code>-element subsets that can be selected from an
     * <code>n</code>-element set.
     * <p>
     * <Strong>Preconditions</strong>:
     * <ul>
     * <li> <code>0 <= k <= n </code> (otherwise
     * <code>IllegalArgumentException</code> is thrown)</li>
     * <li> The result is small enough to fit into a <code>double</code>. The
     * largest value of <code>n</code> 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</code>
     * @throws IllegalArgumentException if preconditions are not met.
     */
    public static double binomialCoefficientDouble(final int n, final int k) {
        return Math.floor(Math.exp(binomialCoefficientLog(n, k)) + 0.5);
    }

    /**
     * For a byte value x, this method returns (byte)(+1) if x >= 0 and
     * (byte)(-1) if x < 0.
     * 
     * @param x the value, a byte
     * @return (byte)(+1) or (byte)(-1), depending on the sign of x
     */
    public static byte indicator(final byte x) {
        return (x >= ZB) ? PB : NB;
    }

    /**
     * For a double precision value x, this method returns +1.0 if x >= 0 and
     * -1.0 if x < 0. Returns <code>NaN</code> if <code>x</code> is
     * <code>NaN</code>.
     * 
     * @param x the value, a double
     * @return +1.0 or -1.0, depending on the sign of x
     */
    public static double indicator(final double x) {
        if (Double.isNaN(x)) {
            return Double.NaN;
        }
        return (x >= 0.0) ? 1.0 : -1.0;
    }

    /**
     * For a float value x, this method returns +1.0F if x >= 0 and -1.0F if x <
     * 0. Returns <code>NaN</code> if <code>x</code> is <code>NaN</code>.
     * 
     * @param x the value, a float
     * @return +1.0F or -1.0F, depending on the sign of x
     */
    public static float indicator(final float x) {
        if (Float.isNaN(x)) {
            return Float.NaN;
        }
        return (x >= 0.0F) ? 1.0F : -1.0F;
    }

    /**
     * For an int value x, this method returns +1 if x >= 0 and -1 if x < 0.
     * 
     * @param x the value, an int
     * @return +1 or -1, depending on the sign of x
     */
    public static int indicator(final int x) {
        return (x >= 0) ? 1 : -1;
    }

    /**
     * For a long value x, this method returns +1L if x >= 0 and -1L if x < 0.
     * 
     * @param x the value, a long
     * @return +1L or -1L, depending on the sign of x
     */
    public static long indicator(final long x) {
        return (x >= 0L) ? 1L : -1L;
    }

    /**
     * For a short value x, this method returns (short)(+1) if x >= 0 and
     * (short)(-1) if x < 0.
     * 
     * @param x the value, a short
     * @return (short)(+1) or (short)(-1), depending on the sign of x
     */
    public static short indicator(final short x) {
        return (x >= ZS) ? PS : NS;
    }

    /**
     * Round the given non-negative, value to the "nearest" integer. Nearest is
     * determined by the rounding method specified. Rounding methods are defined
     * in {@link BigDecimal}.
     * 
     * @param unscaled the value to round.
     * @param sign the sign of the original, scaled value.
     * @param roundingMethod the rounding method as defined in
     *        {@link BigDecimal}.
     * @return the rounded value.
     * @since 1.1
     */
    private static double roundUnscaled(double unscaled, double sign,
            int roundingMethod) {
        switch (roundingMethod) {
        case BigDecimal.ROUND_CEILING:
            if (sign == -1) {
                unscaled = Math.floor(nextAfter(unscaled,
                        Double.NEGATIVE_INFINITY));
            } else {
                unscaled = Math.ceil(nextAfter(unscaled,
                        Double.POSITIVE_INFINITY));
            }
            break;
        case BigDecimal.ROUND_DOWN:
            unscaled = Math.floor(nextAfter(unscaled,
                    Double.NEGATIVE_INFINITY));
            break;
        case BigDecimal.ROUND_FLOOR:
            if (sign == -1) {
                unscaled = Math.ceil(nextAfter(unscaled,
                        Double.POSITIVE_INFINITY));
            } else {
                unscaled = Math.floor(nextAfter(unscaled,
                        Double.NEGATIVE_INFINITY));
            }
            break;
        case BigDecimal.ROUND_HALF_DOWN: {
            unscaled = nextAfter(unscaled, Double.NEGATIVE_INFINITY);
            double fraction = unscaled - Math.floor(unscaled);
            if (fraction > 0.5) {
                unscaled = Math.ceil(unscaled);
            } else {
                unscaled = Math.floor(unscaled);
            }
            break;
        }
        case BigDecimal.ROUND_HALF_EVEN: {
            double fraction = unscaled - Math.floor(unscaled);
            if (fraction > 0.5) {
                unscaled = Math.ceil(unscaled);
            } else if (fraction < 0.5) {
                unscaled = Math.floor(unscaled);
            } else {
                // The following equality test is intentional and needed for rounding purposes
                if (Math.floor(unscaled) / 2.0 == Math.floor(Math
                        .floor(unscaled) / 2.0)) { // even
                    unscaled = Math.floor(unscaled);
                } else { // odd
                    unscaled = Math.ceil(unscaled);
                }
            }
            break;
        }
        case BigDecimal.ROUND_HALF_UP: {
            unscaled = nextAfter(unscaled, Double.POSITIVE_INFINITY);
            double fraction = unscaled - Math.floor(unscaled);
            if (fraction >= 0.5) {
                unscaled = Math.ceil(unscaled);
            } else {
                unscaled = Math.floor(unscaled);
            }
            break;
        }
        case BigDecimal.ROUND_UNNECESSARY:
            if (unscaled != Math.floor(unscaled)) {
                throw new ArithmeticException(
                        "Inexact result from rounding");
            }
            break;
        case BigDecimal.ROUND_UP:
            unscaled = Math.ceil(nextAfter(unscaled,
                    Double.POSITIVE_INFINITY));
            break;
        default:
            throw new IllegalArgumentException("Invalid rounding method.");
        }
        return unscaled;
    }

    /**
     * Returns the natural <code>log</code> of the <a
     * href="http://mathworld.wolfram.com/BinomialCoefficient.html"> Binomial
     * Coefficient</a>, "<code>n choose k</code>", the number of
     * <code>k</code>-element subsets that can be selected from an
     * <code>n</code>-element set.
     * <p>
     * <Strong>Preconditions</strong>:
     * <ul>
     * <li> <code>0 <= k <= n </code> (otherwise
     * <code>IllegalArgumentException</code> 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</code>
     * @throws IllegalArgumentException if preconditions are not met.
     */
    public static double binomialCoefficientLog(final int n, final int k) {
        if (n < k) {
            throw new IllegalArgumentException(
                    "must have n >= k for binomial coefficient (n,k)");
        }
        if (n < 0) {
            throw new IllegalArgumentException(
                    "must have n >= 0 for binomial coefficient (n,k)");
        }
        if ((n == k) || (k == 0)) {
            return 0;
        }
        if ((k == 1) || (k == n - 1)) {
            return Math.log((double) n);
        }
        double logSum = 0;

        // n!/k!
        for (int i = k + 1; i <= n; i++) {
            logSum += Math.log((double) i);
        }

        // divide by (n-k)!
        for (int i = 2; i <= n - k; i++) {
            logSum -= Math.log((double) i);
        }

        return logSum;
    }

    /**
     * Get the next machine representable number after a number, moving
     * in the direction of another number.
     * <p>
     * If <code>direction</code> is greater than or equal to<code>d</code>,
     * the smallest machine representable number strictly greater than
     * <code>d</code> is returned; otherwise the largest representable number
     * strictly less than <code>d</code> is returned.</p>
     * <p>
     * If <code>d</code> is NaN or Infinite, it is returned unchanged.</p>
     * 
     * @param d base number
     * @param direction (the only important thing is whether
     * direction is greater or smaller than d)
     * @return the next machine representable number in the specified direction
     * @since 1.2
     */
    public static double nextAfter(double d, double direction) {

        // handling of some important special cases
        if (Double.isNaN(d) || Double.isInfinite(d)) {
            return d;
        } else if (d == 0) {
            return (direction < 0) ? -Double.MIN_VALUE : Double.MIN_VALUE;
        }
        // special cases MAX_VALUE to infinity and  MIN_VALUE to 0
        // are handled just as normal numbers

        // split the double in raw components
        long bits = Double.doubleToLongBits(d);
        long sign = bits & 0x8000000000000000L;
        long exponent = bits & 0x7ff0000000000000L;
        long mantissa = bits & 0x000fffffffffffffL;

        if (d * (direction - d) >= 0) {
            // we should increase the mantissa
            if (mantissa == 0x000fffffffffffffL) {
                return Double.longBitsToDouble(sign
                        | (exponent + 0x0010000000000000L));
            } else {
                return Double.longBitsToDouble(sign | exponent
                        | (mantissa + 1));
            }
        } else {
            // we should decrease the mantissa
            if (mantissa == 0L) {
                return Double.longBitsToDouble(sign
                        | (exponent - 0x0010000000000000L)
                        | 0x000fffffffffffffL);
            } else {
                return Double.longBitsToDouble(sign | exponent
                        | (mantissa - 1));
            }
        }

    }

    /** 
     * <p>Returns the 
     * <a href="http://mathworld.wolfram.com/Logarithm.html">logarithm</a>
     * for base <code>b</code> of <code>x</code>.
     * </p>
     * <p>Returns <code>NaN<code> if either argument is negative.  If 
     * <code>base</code> is 0 and <code>x</code> is positive, 0 is returned.
     * If <code>base</code> is positive and <code>x</code> is 0, 
     * <code>Double.NEGATIVE_INFINITY</code> is returned.  If both arguments
     * are 0, the result is <code>NaN</code>.</p>
     * 
     * @param base the base of the logarithm, must be greater than 0
     * @param x argument, must be greater than 0
     * @return the value of the logarithm - the number y such that base^y = x.
     * @since 1.2
     */
    public static double log(double base, double x) {
        return Math.log(x) / Math.log(base);
    }
}

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