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Bloom Filter是由Bloom在1970年提出的一种多哈希函数映射的快速查找算法。通常应用在一些需要快速判断某个元素是否属于集合,但是并不严格要求100%正确的场合。

 

. 实例

  为了说明Bloom Filter存在的重要意义,举一个实例:

  假设要你写一个网络蜘蛛(web crawler)。由于网络间的链接错综复杂,蜘蛛在网络间爬行很可能会形成“环”。为了避免形成“环”,就需要知道蜘蛛已经访问过那些URL。给一个URL,怎样知道蜘蛛是否已经访问过呢?稍微想想,就会有如下几种方案:

  1. 将访问过的URL保存到数据库。

  2. 用HashSet将访问过的URL保存起来。那只需接近O(1)的代价就可以查到一个URL是否被访问过了。

  3. URL经过MD5或SHA-1等单向哈希后再保存到HashSet或数据库。

  4. Bit-Map方法。建立一个BitSet,将每个URL经过一个哈希函数映射到某一位。

  方法1~3都是将访问过的URL完整保存,方法4则只标记URL的一个映射位。

 

  以上方法在数据量较小的情况下都能完美解决问题,但是当数据量变得非常庞大时问题就来了。

  方法1的缺点:数据量变得非常庞大后关系型数据库查询的效率会变得很低。而且每来一个URL就启动一次数据库查询是不是太小题大做了?

  方法2的缺点:太消耗内存。随着URL的增多,占用的内存会越来越多。就算只有1亿个URL,每个URL只算50个字符,就需要5GB内存。

  方法3:由于字符串经过MD5处理后的信息摘要长度只有128Bit,SHA-1处理后也只有160Bit,因此方法3比方法2节省了好几倍的内存。

  方法4消耗内存是相对较少的,但缺点是单一哈希函数发生冲突的概率太高。还记得数据结构课上学过的Hash表冲突的各种解决方法么?若要降低冲突发生的概率到1%,就要将BitSet的长度设置为URL个数的100倍。

 

  实质上上面的算法都忽略了一个重要的隐含条件:允许小概率的出错,不一定要100%准确!也就是说少量url实际上没有没网络蜘蛛访问,而将它们错判为已访问的代价是很小的——大不了少抓几个网页呗。

 

. Bloom Filter的算法

 

  废话说到这里,下面引入本篇的主角——Bloom Filter。其实上面方法4的思想已经很接近Bloom Filter了。方法四的致命缺点是冲突概率高,为了降低冲突的概念,Bloom Filter使用了多个哈希函数,而不是一个。

    Bloom Filter算法如下:

    创建一个m位BitSet,先将所有位初始化为0,然后选择k个不同的哈希函数。第i个哈希函数对字符串str哈希的结果记为h(i,str),且h(i,str)的范围是0到m-1 。

 

(1) 加入字符串过程

 

  下面是每个字符串处理的过程,首先是将字符串str“记录”到BitSet中的过程:

  对于字符串str,分别计算h(1,str),h(2,str)…… h(k,str)。然后将BitSet的第h(1,str)、h(2,str)…… h(k,str)位设为1。

  图1.Bloom Filter加入字符串过程

  很简单吧?这样就将字符串str映射到BitSet中的k个二进制位了。

 

(2) 检查字符串是否存在的过程

 

  下面是检查字符串str是否被BitSet记录过的过程:

  对于字符串str,分别计算h(1,str),h(2,str)…… h(k,str)。然后检查BitSet的第h(1,str)、h(2,str)…… h(k,str)位是否为1,若其中任何一位不为1则可以判定str一定没有被记录过。若全部位都是1,则“认为”字符串str存在。

 

  若一个字符串对应的Bit不全为1,则可以肯定该字符串一定没有被Bloom Filter记录过。(这是显然的,因为字符串被记录过,其对应的二进制位肯定全部被设为1了)

  但是若一个字符串对应的Bit全为1,实际上是不能100%的肯定该字符串被Bloom Filter记录过的。(因为有可能该字符串的所有位都刚好是被其他字符串所对应)这种将该字符串划分错的情况,称为false positive 。

 

(3) 删除字符串过程

   字符串加入了就被不能删除了,因为删除会影响到其他字符串。实在需要删除字符串的可以使用Counting bloomfilter(CBF),这是一种基本Bloom Filter的变体,CBF将基本Bloom Filter每一个Bit改为一个计数器,这样就可以实现删除字符串的功能了。

 

  Bloom Filter跟单哈希函数Bit-Map不同之处在于:Bloom Filter使用了k个哈希函数,每个字符串跟k个bit对应。从而降低了冲突的概率。

 

. Bloom Filter参数选择

 

   (1)哈希函数选择

     哈希函数的选择对性能的影响应该是很大的,一个好的哈希函数要能近似等概率的将字符串映射到各个Bit。选择k个不同的哈希函数比较麻烦,一种简单的方法是选择一个哈希函数,然后送入k个不同的参数。

   (2)Bit数组大小选择

     哈希函数个数k、位数组大小m、加入的字符串数量n的关系可以参考参考文献1。该文献证明了对于给定的m、n,当 k = ln(2)* m/n 时出错的概率是最小的。

     同时该文献还给出特定的k,m,n的出错概率。例如:根据参考文献1,哈希函数个数k取10,位数组大小m设为字符串个数n的20倍时,false positive发生的概率是0.0000889 ,这个概率基本能满足网络爬虫的需求了。 

 

. Bloom Filter实现代码

    下面给出一个简单的Bloom Filter的Java实现代码:

 

复制代码
import java.util.BitSet;
publicclass BloomFilter { /* BitSet初始分配2^24个bit */privatestaticfinalint DEFAULT_SIZE =1<<25; /* 不同哈希函数的种子,一般应取质数 */privatestaticfinalint[] seeds =newint[] { 5, 7, 11, 13, 31, 37, 61 }; private BitSet bits =new BitSet(DEFAULT_SIZE); /* 哈希函数对象 */private SimpleHash[] func =new SimpleHash[seeds.length];
public BloomFilter() { for (int i =0; i < seeds.length; i++) { func[i] =new SimpleHash(DEFAULT_SIZE, seeds[i]); } }
// 将字符串标记到bits中publicvoid add(String value) { for (SimpleHash f : func) { bits.set(f.hash(value), true); } }
//判断字符串是否已经被bits标记publicboolean contains(String value) { if (value ==null) { returnfalse; } boolean ret =true; for (SimpleHash f : func) { ret = ret && bits.get(f.hash(value)); } return ret; }
/* 哈希函数类 */publicstaticclass SimpleHash { privateint cap; privateint seed;
public SimpleHash(int cap, int seed) { this.cap = cap; this.seed = seed; }
//hash函数,采用简单的加权和hashpublicint hash(String value) { int result =0; int len = value.length(); for (int i =0; i < len; i++) { result = seed * result + value.charAt(i); } return (cap -1) & result; } } }
复制代码

 

 

 

参考文献:

 

[1]Pei Cao. Bloom Filters - the math.

http://pages.cs.wisc.edu/~cao/papers/summary-cache/node8.html

[2]Wikipedia. Bloom filter.

http://en.wikipedia.org/wiki/Bloom_filter

 

Google Guava的Bloom filter实现

/*
* Copyright (C) 2011 The Guava Authors
*
* Licensed 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.google.common.hash;

import static com.google.common.base.Preconditions.checkArgument;
import static com.google.common.base.Preconditions.checkNotNull;

import com.google.common.annotations.Beta;
import com.google.common.annotations.VisibleForTesting;
import com.google.common.base.Preconditions;
import com.google.common.hash.BloomFilterStrategies.BitArray;

import java.io.Serializable;

/**
* A Bloom filter for instances of {@code T}. A Bloom filter offers an approximate containment test
* with one-sided error: if it claims that an element is contained in it, this might be in error,
* but if it claims that an element is <i>not</i> contained in it, then this is definitely true.
*
* <p>If you are unfamiliar with Bloom filters, this nice
* <a href="http://llimllib.github.com/bloomfilter-tutorial/">tutorial</a> may help you understand
* how they work.
*
* @param <T> the type of instances that the {@code BloomFilter} accepts
* @author Kevin Bourrillion
* @author Dimitris Andreou
* @since 11.0
*/
@Beta
public final class BloomFilter<T> implements Serializable {
/**
* A strategy to translate T instances, to {@code numHashFunctions} bit indexes.
*/
interface Strategy extends java.io.Serializable {
/**
* Sets {@code numHashFunctions} bits of the given bit array, by hashing a user element.
*/
<T> void put(T object, Funnel<? super T> funnel, int numHashFunctions, BitArray bits);

/**
* Queries {@code numHashFunctions} bits of the given bit array, by hashing a user element;
* returns {@code true} if and only if all selected bits are set.
*/
<T> boolean mightContain(
T object, Funnel<? super T> funnel, int numHashFunctions, BitArray bits);
}

/** The bit set of the BloomFilter (not necessarily power of 2!)*/
private final BitArray bits;

/** Number of hashes per element */
private final int numHashFunctions;

/** The funnel to translate Ts to bytes */
private final Funnel<T> funnel;

/**
* The strategy we employ to map an element T to {@code numHashFunctions} bit indexes.
*/
private final Strategy strategy;

/**
* Creates a BloomFilter.
*/
private BloomFilter(BitArray bits, int numHashFunctions, Funnel<T> funnel,
Strategy strategy) {
Preconditions.checkArgument(numHashFunctions > 0, "numHashFunctions zero or negative");
this.bits = checkNotNull(bits);
this.numHashFunctions = numHashFunctions;
this.funnel = checkNotNull(funnel);
this.strategy = strategy;
}

/**
* Returns {@code true} if the element <i>might</i> have been put in this Bloom filter,
* {@code false} if this is <i>definitely</i> not the case.
*/
public boolean mightContain(T object) {
return strategy.mightContain(object, funnel, numHashFunctions, bits);
}

/**
* Puts an element into this {@code BloomFilter}. Ensures that subsequent invocations of
* {@link #mightContain(Object)} with the same element will always return {@code true}.
*/
public void put(T object) {
strategy.put(object, funnel, numHashFunctions, bits);
}

@VisibleForTesting int getHashCount() {
return numHashFunctions;
}

@VisibleForTesting double computeExpectedFalsePositiveRate(int insertions) {
return Math.pow(
1 - Math.exp(-numHashFunctions * ((double) insertions / (bits.size()))),
numHashFunctions);
}

/**
* Creates a {@code Builder} of a {@link BloomFilter BloomFilter<T>}, with the expected number
* of insertions and expected false positive probability.
*
* <p>Note that overflowing a {@code BloomFilter} with significantly more elements
* than specified, will result in its saturation, and a sharp deterioration of its
* false positive probability.
*
* <p>The constructed {@code BloomFilter<T>} will be serializable if the provided
* {@code Funnel<T>} is.
*
* @param funnel the funnel of T's that the constructed {@code BloomFilter<T>} will use
* @param expectedInsertions the number of expected insertions to the constructed
* {@code BloomFilter<T>}; must be positive
* @param falsePositiveProbability the desired false positive probability (must be positive and
* less than 1.0)
* @return a {@code Builder}
*/
public static <T> BloomFilter<T> create(Funnel<T> funnel, int expectedInsertions /* n */,
double falsePositiveProbability) {
checkNotNull(funnel);
checkArgument(expectedInsertions > 0, "Expected insertions must be positive");
checkArgument(falsePositiveProbability > 0.0 & falsePositiveProbability < 1.0,
"False positive probability in (0.0, 1.0)");
/*
* andreou: I wanted to put a warning in the javadoc about tiny fpp values,
* since the resulting size is proportional to -log(p), but there is not
* much of a point after all, e.g. optimalM(1000, 0.0000000000000001) = 76680
* which is less that 10kb. Who cares!
*/
int numBits = optimalNumOfBits(expectedInsertions, falsePositiveProbability);
int numHashFunctions = optimalNumOfHashFunctions(expectedInsertions, numBits);
return new BloomFilter<T>(new BitArray(numBits), numHashFunctions, funnel,
BloomFilterStrategies.MURMUR128_MITZ_32);
}

/**
* Creates a {@code Builder} of a {@link BloomFilter BloomFilter<T>}, with the expected number
* of insertions, and a default expected false positive probability of 3%.
*
* <p>Note that overflowing a {@code BloomFilter} with significantly more elements
* than specified, will result in its saturation, and a sharp deterioration of its
* false positive probability.
*
* <p>The constructed {@code BloomFilter<T>} will be serializable if the provided
* {@code Funnel<T>} is.
*
* @param funnel the funnel of T's that the constructed {@code BloomFilter<T>} will use
* @param expectedInsertions the number of expected insertions to the constructed
* {@code BloomFilter<T>}; must be positive
* @return a {@code Builder}
*/
public static <T> BloomFilter<T> create(Funnel<T> funnel, int expectedInsertions /* n */) {
return create(funnel, expectedInsertions, 0.03); // FYI, for 3%, we always get 5 hash functions
}

/*
* Cheat sheet:
*
* m: total bits
* n: expected insertions
* b: m/n, bits per insertion

* p: expected false positive probability
*
* 1) Optimal k = b * ln2
* 2) p = (1 - e ^ (-kn/m))^k
* 3) For optimal k: p = 2 ^ (-k) ~= 0.6185^b
* 4) For optimal k: m = -nlnp / ((ln2) ^ 2)
*/

private static final double LN2 = Math.log(2);
private static final double LN2_SQUARED = LN2 * LN2;

/**
* Computes the optimal k (number of hashes per element inserted in Bloom filter), given the
* expected insertions and total number of bits in the Bloom filter.
*
* See http://en.wikipedia.org/wiki/File:Bloom_filter_fp_probability.svg for the formula.
*
* @param n expected insertions (must be positive)
* @param m total number of bits in Bloom filter (must be positive)
*/
@VisibleForTesting static int optimalNumOfHashFunctions(int n, int m) {
return Math.max(1, (int) Math.round(m / n * LN2));
}

/**
* Computes m (total bits of Bloom filter) which is expected to achieve, for the specified
* expected insertions, the required false positive probability.
*
* See http://en.wikipedia.org/wiki/Bloom_filter#Probability_of_false_positives for the formula.
*
* @param n expected insertions (must be positive)
* @param p false positive rate (must be 0 < p < 1)
*/
@VisibleForTesting static int optimalNumOfBits(int n, double p) {
return (int) (-n * Math.log(p) / LN2_SQUARED);
}

private Object writeReplace() {
return new SerialForm<T>(this);
}

private static class SerialForm<T> implements Serializable {
final long[] data;
final int numHashFunctions;
final Funnel<T> funnel;
final Strategy strategy;

SerialForm(BloomFilter<T> bf) {
this.data = bf.bits.data;
this.numHashFunctions = bf.numHashFunctions;
this.funnel = bf.funnel;
this.strategy = bf.strategy;
}
Object readResolve() {
return new BloomFilter<T>(new BitArray(data), numHashFunctions, funnel, strategy);
}
private static final long serialVersionUID = 1;
}
}

  

 

本质是BitSet,适用可容忍一定错误的场景,优势是高效、占用空间小;

3亿个KEY,每个实例映射1W KEY,需要3W个实例,占用空间约600M。

 

posted on 2014-05-22 12:12  夜隼  阅读(2628)  评论(0编辑  收藏  举报