数据结构之BloomFilter

• BloomFilter是什么？

       BloomFilter主要提供两种操作: add()和contains()，作用分别是将元素加入其中以及判断一个元素是否在其中，类似于Java中的Set接口，它内部采用的byte数组来节省空间。其独特之处在于contains()方法，当我们需要查询某个元素是否包含在BloomFilter中时，如果返回true，结果可能是不正确的，也就是元素也有可能不在其中；但是如果返回的是false，那么元素一定不在其中。这也就是所谓的no false negative和small probability of  false  positive。通俗地来就是，“说了没有就是没有”，“说有但实际上有可能没有”。

 

• 为什么要使用BloomFilter？

       任何一种数据结构，特别是比较巧妙的，都有它的应用场景，BloomFilter也不例外。许多实际背景和应用这里不再啰嗦，举个简单例子：我们需要存储大量的URL，并且对于新得到的URL需要知道是否已经存储了。如果将这些URL全扔进Set，那就悲剧了，内存可吃不消。这个时候就要想到用byte来存储了，可以节省大量的空间。

 

• BloomFilter实现

        既然涉及到byte存储，那么必然需要一种映射关系了，也就是需要知道一个元素用哪些byte来表示，这也就是hash函数所干的事情。直观地想，一个元素一个byte基本上不可能，一般情况下这样的hash函数可以说不存在，所以这里假设用k位来表示，一般采用k次hash来确定这些byte。实际上，BloomFilter中一般包含m(byte数组的大小),k（hash次数）和n(需要存储的数据量)。 在元素加入实现add()操作时，连续k次hash，将得到的对应k位全置为1。当查询元素是否在集合中时，也是连续k次hash，如果这k次得到的位不是全为1，那么返回false；否则返回 true。传统的算法优化都是以时间换空间或者以空间换时间，BloomFilter则是以正确率换空间。

Class BloomFilter<E> {
	public BloomFilter(int m, int k){……}
	public void add(E obj){……}
	public boolean contains(E obj){……}
}

 

• BloomFilter相关结论

      根据上面的标记，可以采用数学方法推导出false positive的概率大约是p = (1-exp(-kn/m))^k，那么由此可得出，最优的k=ln(2)*(m/n)。我们可以计算一下上面提到的例子，假设共有10000000个URL(n=10000000)，如果采用m=80000000，k=8，得出p大约2%；但是所占用的内存只有10M，相同情况下使用Set则需要1G的内存。

具体的实现如下(来自项目http://code.google.com/p/java-bloomfilter/，代码实现不错，可直接使用):

/**
 * This program is free software: you can redistribute it and/or modify
 * it under the terms of the GNU Lesser General Public License as published by
 * the Free Software Foundation, either version 3 of the License, or
 * (at your option) any later version.
 *
 * This program is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU Lesser General Public License for more details.
 *
 * You should have received a copy of the GNU Lesser General Public License
 * along with this program.  If not, see <http://www.gnu.org/licenses/>.
 */

package com.skjegstad.utils;

import java.io.Serializable;
import java.nio.charset.Charset;
import java.security.MessageDigest;
import java.security.NoSuchAlgorithmException;
import java.util.BitSet;
import java.util.Collection;

/**
 * Implementation of a Bloom-filter, as described here:
 * http://en.wikipedia.org/wiki/Bloom_filter
 *
 * For updates and bugfixes, see http://github.com/magnuss/java-bloomfilter
 *
 * Inspired by the SimpleBloomFilter-class written by Ian Clarke. This
 * implementation provides a more evenly distributed Hash-function by
 * using a proper digest instead of the Java RNG. Many of the changes
 * were proposed in comments in his blog:
 * http://blog.locut.us/2008/01/12/a-decent-stand-alone-java-bloom-filter-implementation/
 *
 * @param <E> Object type that is to be inserted into the Bloom filter, e.g. String or Integer.
 * @author Magnus Skjegstad <[e-mail]>
 */
public class BloomFilter<E> implements Serializable {
    private BitSet bitset;
    private int bitSetSize;
    private double bitsPerElement;
    private int expectedNumberOfFilterElements; 
    private int numberOfAddedElements; 
    private int k; // number of hash functions

    static final Charset charset = Charset.forName("UTF-8"); 

    static final String hashName = "MD5"; 
    static final MessageDigest digestFunction;
    static { // The digest method is reused between instances
        MessageDigest tmp;
        try {
            tmp = java.security.MessageDigest.getInstance(hashName);
        } catch (NoSuchAlgorithmException e) {
            tmp = null;
        }
        digestFunction = tmp;
    }

    /**
      * Constructs an empty Bloom filter. The total length of the Bloom filter will be
      * c*n.
      *
      * @param c is the number of bits used per element.
      * @param n is the expected number of elements the filter will contain.
      * @param k is the number of hash functions used.
      */
    public BloomFilter(double c, int n, int k) {
      this.expectedNumberOfFilterElements = n;
      this.k = k;
      this.bitsPerElement = c;
      this.bitSetSize = (int)Math.ceil(c * n);
      numberOfAddedElements = 0;
      this.bitset = new BitSet(bitSetSize);
    }

    /**
     * Constructs an empty Bloom filter. The optimal number of hash functions (k) is estimated from
     * the total size of the Bloom
     * and the number of expected elements.
     *
     * @param bitSetSize defines how many bits should be used in total for the filter.
     * @param expectedNumberOElements defines the maximum number of elements the filter is 
     * expected to contain.
     */
    public BloomFilter(int bitSetSize, int expectedNumberOElements) {
        this(bitSetSize / (double)expectedNumberOElements,
             expectedNumberOElements,
             (int) Math.round((bitSetSize / (double)expectedNumberOElements) * Math.log(2.0)));
    }

    /**
     * Constructs an empty Bloom filter with a given false positive probability. The number of bits per
     * element and the number of hash functions is estimated
     * to match the false positive probability.
     *
     * @param falsePositiveProbability is the desired false positive probability.
     * @param expectedNumberOfElements is the expected number of elements in the Bloom filter.
     */
    public BloomFilter(double falsePositiveProbability, int expectedNumberOfElements) {
        this(Math.ceil(-(Math.log(falsePositiveProbability) / Math.log(2))) / Math.log(2), 
             expectedNumberOfElements,
             (int)Math.ceil(-(Math.log(falsePositiveProbability) / Math.log(2)))); 
    }

    /**
     * Construct a new Bloom filter based on existing Bloom filter data.
     *
     * @param bitSetSize defines how many bits should be used for the filter.
     * @param expectedNumberOfFilterElements defines the maximum number of elements the filter
     *    is expected to contain.
     * @param actualNumberOfFilterElements specifies how many elements have been inserted into 
     *    the <code>filterData</code> BitSet.
     * @param filterData a BitSet representing an existing Bloom filter.
     */
    public BloomFilter(int bitSetSize, int expectedNumberOfFilterElements, 
                                                       int actualNumberOfFilterElements, BitSet filterData) {
        this(bitSetSize, expectedNumberOfFilterElements);
        this.bitset = filterData;
        this.numberOfAddedElements = actualNumberOfFilterElements;
    }

    /**
     * Generates a digest based on the contents of a String.
     *
     * @param val specifies the input data.
     * @param charset specifies the encoding of the input data.
     * @return digest as long.
     */
    public static int createHash(String val, Charset charset) {
        return createHash(val.getBytes(charset));
    }

    /**
     * Generates a digest based on the contents of a String.
     *
     * @param val specifies the input data. The encoding is expected to be UTF-8.
     * @return digest as long.
     */
    public static int createHash(String val) {
        return createHash(val, charset);
    }

    /**
     * Generates a digest based on the contents of an array of bytes.
     *
     * @param data specifies input data.
     * @return digest as long.
     */
    public static int createHash(byte[] data) {
        return createHashes(data, 1)[0];
    }

    /**
     * Generates digests based on the contents of an array of bytes and splits the result into 
     * 4-byte int's and store them in an array. The
     * digest function is called until the required number of int's are produced. For each call to
     * digest a salt
     * is prepended to the data. The salt is increased by 1 for each call.
     *
     * @param data specifies input data.
     * @param hashes number of hashes/int's to produce.
     * @return array of int-sized hashes
     */
    public static int[] createHashes(byte[] data, int hashes) {
        int[] result = new int[hashes];

        int k = 0;
        byte salt = 0;
        while (k < hashes) {
            byte[] digest;
            synchronized (digestFunction) {
                digestFunction.update(salt);
                salt++;
                digest = digestFunction.digest(data);                
            }
        
            for (int i = 0; i < digest.length/4 && k < hashes; i++) {
                int h = 0;
                for (int j = (i*4); j < (i*4)+4; j++) {
                    h <<= 8;
                    h |= ((int) digest[j]) & 0xFF;
                }
                result[k] = h;
                k++;
            }
        }
        return result;
    }

    /**
     * Compares the contents of two instances to see if they are equal.
     *
     * @param obj is the object to compare to.
     * @return True if the contents of the objects are equal.
     */
    @Override
    public boolean equals(Object obj) {
        if (obj == null) {
            return false;
        }
        if (getClass() != obj.getClass()) {
            return false;
        }
        final BloomFilter<E> other = (BloomFilter<E>) obj;        
        if (this.expectedNumberOfFilterElements != other.expectedNumberOfFilterElements) {
            return false;
        }
        if (this.k != other.k) {
            return false;
        }
        if (this.bitSetSize != other.bitSetSize) {
            return false;
        }
        if (this.bitset != other.bitset && (this.bitset == null || !this.bitset.equals(other.bitset))) {
            return false;
        }
        return true;
    }

    /**
     * Calculates a hash code for this class.
     * @return hash code representing the contents of an instance of this class.
     */
    @Override
    public int hashCode() {
        int hash = 7;
        hash = 61 * hash + (this.bitset != null ? this.bitset.hashCode() : 0);
        hash = 61 * hash + this.expectedNumberOfFilterElements;
        hash = 61 * hash + this.bitSetSize;
        hash = 61 * hash + this.k;
        return hash;
    }


    /**
     * Calculates the expected probability of false positives based on
     * the number of expected filter elements and the size of the Bloom filter.
     * <br /><br />
     * The value returned by this method is the <i>expected</i> rate of false
     * positives, assuming the number of inserted elements equals the number of
     * expected elements. If the number of elements in the Bloom filter is less
     * than the expected value, the true probability of false positives will be lower.
     *
     * @return expected probability of false positives.
     */
    public double expectedFalsePositiveProbability() {
        return getFalsePositiveProbability(expectedNumberOfFilterElements);
    }

    /**
     * Calculate the probability of a false positive given the specified
     * number of inserted elements.
     *
     * @param numberOfElements number of inserted elements.
     * @return probability of a false positive.
     */
    public double getFalsePositiveProbability(double numberOfElements) {
        // (1 - e^(-k * n / m)) ^ k
        return Math.pow((1 - Math.exp(-k * (double) numberOfElements
                        / (double) bitSetSize)), k);

    }

    /**
     * Get the current probability of a false positive. The probability is calculated from
     * the size of the Bloom filter and the current number of elements added to it.
     *
     * @return probability of false positives.
     */
    public double getFalsePositiveProbability() {
        return getFalsePositiveProbability(numberOfAddedElements);
    }


    /**
     * Returns the value chosen for K.<br />
     * <br />
     * K is the optimal number of hash functions based on the size
     * of the Bloom filter and the expected number of inserted elements.
     *
     * @return optimal k.
     */
    public int getK() {
        return k;
    }

    /**
     * Sets all bits to false in the Bloom filter.
     */
    public void clear() {
        bitset.clear();
        numberOfAddedElements = 0;
    }

    /**
     * Adds an object to the Bloom filter. The output from the object's
     * toString() method is used as input to the hash functions.
     *
     * @param element is an element to register in the Bloom filter.
     */
    public void add(E element) {
       add(element.toString().getBytes(charset));
    }

    /**
     * Adds an array of bytes to the Bloom filter.
     *
     * @param bytes array of bytes to add to the Bloom filter.
     */
    public void add(byte[] bytes) {
       int[] hashes = createHashes(bytes, k);
       for (int hash : hashes)
           bitset.set(Math.abs(hash % bitSetSize), true);
       numberOfAddedElements ++;
    }

    /**
     * Adds all elements from a Collection to the Bloom filter.
     * @param c Collection of elements.
     */
    public void addAll(Collection<? extends E> c) {
        for (E element : c)
            add(element);
    }
        
    /**
     * Returns true if the element could have been inserted into the Bloom filter.
     * Use getFalsePositiveProbability() to calculate the probability of this
     * being correct.
     *
     * @param element element to check.
     * @return true if the element could have been inserted into the Bloom filter.
     */
    public boolean contains(E element) {
        return contains(element.toString().getBytes(charset));
    }

    /**
     * Returns true if the array of bytes could have been inserted into the Bloom filter.
     * Use getFalsePositiveProbability() to calculate the probability of this
     * being correct.
     *
     * @param bytes array of bytes to check.
     * @return true if the array could have been inserted into the Bloom filter.
     */
    public boolean contains(byte[] bytes) {
        int[] hashes = createHashes(bytes, k);
        for (int hash : hashes) {
            if (!bitset.get(Math.abs(hash % bitSetSize))) {
                return false;
            }
        }
        return true;
    }

    /**
     * Returns true if all the elements of a Collection could have been inserted
     * into the Bloom filter. Use getFalsePositiveProbability() to calculate the
     * probability of this being correct.
     * @param c elements to check.
     * @return true if all the elements in c could have been inserted into the Bloom filter.
     */
    public boolean containsAll(Collection<? extends E> c) {
        for (E element : c)
            if (!contains(element))
                return false;
        return true;
    }

    /**
     * Read a single bit from the Bloom filter.
     * @param bit the bit to read.
     * @return true if the bit is set, false if it is not.
     */
    public boolean getBit(int bit) {
        return bitset.get(bit);
    }

    /**
     * Set a single bit in the Bloom filter.
     * @param bit is the bit to set.
     * @param value If true, the bit is set. If false, the bit is cleared.
     */
    public void setBit(int bit, boolean value) {
        bitset.set(bit, value);
    }

    /**
     * Return the bit set used to store the Bloom filter.
     * @return bit set representing the Bloom filter.
     */
    public BitSet getBitSet() {
        return bitset;
    }

    /**
     * Returns the number of bits in the Bloom filter. Use count() to retrieve
     * the number of inserted elements.
     *
     * @return the size of the bitset used by the Bloom filter.
     */
    public int size() {
        return this.bitSetSize;
    }

    /**
     * Returns the number of elements added to the Bloom filter after it
     * was constructed or after clear() was called.
     *
     * @return number of elements added to the Bloom filter.
     */
    public int count() {
        return this.numberOfAddedElements;
    }

    /**
     * Returns the expected number of elements to be inserted into the filter.
     * This value is the same value as the one passed to the constructor.
     *
     * @return expected number of elements.
     */
    public int getExpectedNumberOfElements() {
        return expectedNumberOfFilterElements;
    }

    /**
     * Get expected number of bits per element when the Bloom filter is full. This value is set by the constructor
     * when the Bloom filter is created. See also getBitsPerElement().
     *
     * @return expected number of bits per element.
     */
    public double getExpectedBitsPerElement() {
        return this.bitsPerElement;
    }

    /**
     * Get actual number of bits per element based on the number of elements that have currently been inserted and the length
     * of the Bloom filter. See also getExpectedBitsPerElement().
     *
     * @return number of bits per element.
     */
    public double getBitsPerElement() {
        return this.bitSetSize / (double)numberOfAddedElements;
    }
}

   实际应用中，重要的往往是m,n,k的选择以及hash函数的设定。在Oracle 11g中就采用了BloomFilter来实现分布式数据库中两表的join操作；在网络中应用则更加广泛，例如分布式web代理Squid；此外还有拼写检查、海量数据处理等领域中都能搞看到它的身影。