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Hash和Bloom Filter

2011年9月13日 发表评论 阅读评论

这几天的“科研”中涉及到了一个概念,Bloom Filter(有的中文翻译为布隆过滤器,不知道正确否),今天看了下相关的资料,发现这东西和Hash还挺有关系的,在这里一并讲下。

Hash(函数/表)

Hash (中译为哈希,或者散列)函数在计算机领域,尤其是数据快速查找领域,加密领域用的极广。其作用是将一个大的数据集映射到一个小的数据集上面(这些小的数据集叫做哈希值,或者散列值)。Hash table(散列表,也叫哈希表),是根据哈希值(Key value)而直接进行访问的数据结构。也就是说,它通过把哈希值映射到表中一个位置来访问记录,以加快查找的速度。下面是一个典型的hash函数/表示意图:

哈希函数有以下两个特点:

  • 如果两个散列值是不相同的(根据同一函数),那么这两个散列值的原始输入也是不相同的。
  • 散列函数的输入和输出不是唯一对应关系的,如果两个散列值相同,两个输入值很可能是相同的。但也可能不同,这种情况称为“散列碰撞”(或者“散列冲突”)。上图中,John Smith和Sandra Dee就存在hash冲突。

Hash一个应用就是对数据集分类,比如上图,Hash值为0的表示可能在A集合中,Hash值为2的表示B集合中,依次类推,值为15的表示F集合中。但Hash冲突会在这里会导致严重的问题,对于一个未知的新值,其可能不属于上面任何一个集合,但由于冲突,其Hash值和上面的某一个相同,导致误报(因为事先我们不可能做一个含有无限多项输入的完整的Hash表,也就是原来的Hash函数不可能是完美的)。并且,hash冲突也会导致查找效率低下。

Bloom Filter

Bloom Filter是1970年由Bloom提出的。它实际上是一个很长的二进制向量和一系列随机映射函数(Hash函数)。布隆过滤器可以用于检索一个元素是否在一个集合中。它的优点是空间效率和查询时间都远远超过一般的算法,缺点是有一定的误识别率和删除困难。Bloom Filter广泛的应用于各种需要查询的场合中,如Orocle的数据库,Google的BitTable也用了此技术。

如果想判断一个元素是不是在一个集合里,一般想到的是将所有元素保存起来,然后通过比较确定。链表,树等等数据结构都是这种思路. 但是随着集合中元素的增加,我们需要的存储空间越来越大,检索速度也越来越慢(O(n),O(logn))。

这时候就可以利用哈希表这个数据结构(它可以通过一个Hash函数将一个元素映射成一个位阵列(Bit array)中的一个点)。这样一来,我们只要看看这个点是不是1就知道可以集合中有没有它了。这就是Bloom Filter的基本思想。

但这时,哈希冲突会是一个问题:假设Hash函数是良好的,如果我们的位阵列长度为m个点,那么如果我们想将冲突率降低到例如 1%, 这个散列表就只能容纳m/100个元素。显然这就不叫空间效率了(Space-efficient)了。解决方法也简单,就是使用多个Hash,如果它们有一个说元素不在集合中,那肯定就不在。如果它们都说在,虽然也有一定可能性它们都在说谎,不过直觉上判断这种事情的概率是比较低的。这种多个Hash组成的数据结构就叫Bloom Filter。

一个Bloom Filter是基于一个m位的位向量(b1,…bm),这些位向量的初始值为0。另外,还有一系列的hash函数(h1,…hk),这些hash函数的值域属于1~m。下图是一个bloom filter插入x,y,z并判断某个值w是否在该数据集的示意图:

 

上图中,m=18,k=3;插入x是,三个hash函数分别得到蓝线对应的三个值,并将对应的位向量改为1,插入y,z时,类似的,分别将红线,紫线对应的位向量改为1。查找时,当查找x时,三个hash值对应的位向量都为1,因此判断x在此数据集中。y,z也是如此。但是查找w时,w有个hash值对应的位向量为0,因此可以判断不在此集合中。但是,假如w的最后那个hash值比上图中的大1,这是就会认为w在此集合中,而事实上,w可能不在此集合中,因此可能出现误报。显然的,插入数据越多,1的位数越多,误报的概率越大。

Wiki的Bloom Filter词条有关于误报的概率的详细分析:Probability of false positives。从分析可以看出,当k比较大时,误报概率还是比较小的,因此这存储还是很空间有效滴。

Bloom Filter有以下几个特点:

  1. 不存在漏报(False Negative),即某个元素在某个集合中,肯定能报出来。
  2. 可能存在误报(False Positive),即某个元素不在某个集合中,可能也被爆出来。
  3. 确定某个元素是否在某个集合中的代价和总的元素数目无关。

优点:

相比于其它的数据结构,Bloom Filter在空间和时间方面都有巨大的优势。Bloom Filter存储空间和插入/查询时间都是常数。另外, Hash函数相互之间没有关系,方便由硬件并行实现。Bloom Filter不需要存储元素本身,在某些对保密要求非常严格的场合有优势。

缺点:

另外,一般情况下不能从Bloom Filter中删除元素. 我们很容易想到把位列阵变成整数数组,每插入一个元素相应的计数器加1, 这样删除元素时将计数器减掉就可以了。然而要保证安全的删除元素并非如此简单。首先我们必须保证删除的元素的确在Bloom Filter里面. 这一点单凭这个过滤器是无法保证的。另外计数器回绕也会造成问题。

参考资料:

  1. http://en.wikipedia.org/wiki/Hash_function
  2. http://en.wikipedia.org/wiki/Bloom_filter
  3. http://antognini.ch/papers/BloomFilters20080620.pdf


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  1. 2011年9月14日04:47 | #1

    @Yan

    hash函数应该很好取吧,只需要得出的hash值够随机就可以了

    话说bloom filter主要应该是硬件好实现,n个hash一起算

    lsh是什么,没听过。。。

  2. 2011年9月14日15:03 | #2

    它的输入是一个高位向量,输出是一坨bit,因为涉及到单精度数的线性运算,从这个角度来说,实现不是那么简单。

    但它实际的算法很直观,全程只涉及线性运算比如切割超平面之类,如果你玩过水果忍者会有深刻的体会。所以如果你不怕浮点数的话,实现不会很困难。

    里面的数学是一堆科学家在那证误差的理论上限。不看也罢。

  3. 2011年9月14日14:05 | #3

    @sigma

    一般的Hash函数好取,想做到内容相关就不好取了。比如两篇文章讲的都是911,希望他们的Hash值相同,或者两个图像都含有树,希望他们Hash值相近等等。

  4. 2011年9月13日17:30 | #4

    靠,locality.

  5. 2011年9月14日14:45 | #5

    @grapeot

    看了下lsh,一看到表达式那么复杂,还有积分微分。。。

    就没仔细看下去,那你觉得这东西用硬件实现回非常复杂么

  6. 2011年9月13日17:29 | #6

    为毛突然对这个感兴趣了。。。这个的缺点是没有具体说哈希函数怎么取。无法实现基于内容的信息检索。还是lsh(locavity sensitive hashing)靠谱。

  7. 2011年9月15日07:07 | #7

    请教一下,什么是不带引号的科研,什么是带引号的科研

  8. 2011年9月20日13:16 | #8

    @sigma

    <<Currently, I’m a first year Ph.D. student<<

    我是Master Second Year…我应该叫师兄……

  9. 2011年9月20日13:22 | #9

    @赫

    。。。。

    其实我现在也还是master。。。刚second year

  10. 2011年9月20日12:48 | #10

    搜Cloudflare居然遇到了ICT的同门~

    哈希函数无论怎么取都避免不了冲突的吧,一般来说,取8倍于N,k=3时,False Positive概率可以控制到2%左右。

    欢迎围观我的BloomFilter的开源实现

    https://github.com/liheyuan/BloomFilter-For-KeSeek

  11. 2011年9月20日13:01 | #11

    @赫

    哈哈,世界真小~

    膜拜了下师兄的代码,大牛啊~

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