Hešovanie

Zdroje

• Prednaska L10 Erika Demaina z MIT: http://courses.csail.mit.edu/6.851/spring12/lectures/
• Článok z Wikipédia o Bloom filtroch http://en.wikipedia.org/wiki/Bloom_filter
• Učebnica Brass 2008 Advanced data structures (Bloom filters)

Osnova

• vlastnosti hešovacích funkcií: totally random, universal, k-wise independent, simple tabulation;
• chaining, perfect hashing, linear probing,
• Bloom filters, locality sensitive hashing

V sylabe na skúšku je len perfect hashing, rozoberané na Erikovej prednáške

Bloom Filter (Bloom 1970)

• supports insert x, test if x is in the set
  • may give false positives, e.g. claim that x is in the set when it is not
  • false negatives do not occur

Algorithm

• a bit string B[0,...,m-1] of length m, and k hash functions hi : U -> {0, ..., m-1}.
• insert(x): set bits B[h1(x)], ..., B[hk(x)] to 1.
• test if x in the set: compute h1(y), ..., hk(y) and check whether all these bits are 1
  • if yes, claim x is in the set, but possibility of error
  • if no, answer no, surely true

Lemma: If hash functions are totally random and independent, the probability of error is at most (1-exp(-nk/m))^k

• proof later
• totally random hash functions impractical (need to store hash value for each element of the universe), but the assumption simplifies analysis
• if we set k = ln(2) m/n, get error $(1-\exp(-\ln(2)))^{{\ln(2)m/n}}=2^{{-\ln(2)m/n}}=c^{{-m/n}}$, where $c=2^{{\ln(2)}}\approx 1.62$
• to get error rate p for some n, we need $m=n\log _{c}(1/p)=n\lg(1/p)/\lg(c)=n\lg(1/p)/\ln(2)=n\lg(1/p)\lg(e)$
• for 1% error, we need about m=10n bits of space and k=7
• memory size and error rate are independent of the size of the universe U
• compare to a hash table, which needs at least to store data items themselves (e.g. in n lg u bits)
• if we used k=1 (Bloom filter with one hash function), we need m=n/ln(1/(1-p)) bits, which for p=0.01 is about 99.5n, about 10 times more than with 7 hash functions

Use of Bloom filters

• e.g. an approximate index of a larger data structure on a disk - if x not in the filter, do not bother looking at the disk, but small number of false positives not a problem
• Example: Google BigTable maps row label, column label and timestamp to a record, underlies many Google technologies. It uses Bloom filter to check if a combination of row and column label occur in a given fragment of the table
  • For details, see Chang, Fay, et al. "Bigtable: A distributed storage system for structured data." ACM Transactions on Computer Systems (TOCS) 26.2 (2008): 4. [1]
• see also A. Z. Broder and M. Mitzenmacher. Network applications of Bloom filters: A survey. In Internet Math. Volume 1, Number 4 (2003), 485-509. [2]

Proof of lemma

• probability that some B[i] is set to 1 by hj(x) is 1/m
• probability that B[i] is not set to 1 is therefore 1-1/m and since we use k independent hash functions, the probability that B[i] is not set to one by any of them is (1-1/m)^k
• if we insert n elements, each is hashed by each function independently of other elements (hash functions are random) and thus Pr(B[i]=0)=(1-1/m)^{nk}
• Pr(B[i]=1) = 1-Pr(B[i]=0)
• consider a new element y which is not in the set
• error occurs when B[hj(y)]==1 for j=1..k,
• this happens with probability Pr(B[i]=1)^k = (1-(1-1/m)^{nk})^k
• recall that for x>1 we have (1-1/x)^x < 1/e (equality in limit as x->infinity)
• thus probability of error <= (1-exp(-nk/m))^k

• In Mitzenmacher & Upfal (2005), pp. 109–111, 308. - less strict independence assumption

Exercise Let us assume that we have separate Bloom filters for sets A and B with the same set k hash functions. How can we create Bloom filters for union and intersection? Will the result be different from filter created directly for the union or for the intersection?

Theory

• Bloom filter above use about 1.44 n lg(1/p) bits to achieve error rate p. There is lower bound of n lg(1/p) [Carter et al 1978], constant 1.44 can be improved to 1+o(1) using more complex data structures, which are then probably less practical
• Pagh, Anna, Rasmus Pagh, and S. Srinivasa Rao. "An optimal Bloom filter replacement." SODA 2005. [3]
• L. Carter, R. Floyd, J. Gill, G. Markowsky, and M. Wegman. Exact and approximate membership testers. STOC 1978, pages 59–65. [4]
• see also proof of lower bound in Broder and Mitzenmacher 2003 below

Counting Bloom filters

• support insert and delete
• each B[i] is a counter with b bits
• insert increases counters, decrease decreases
• assume no overflows, or reserve largest value 2^b-1 as infinity, cannot be increased or decreased
• Fan, Li, et al. "Summary cache: A scalable wide-area web cache sharing protocol." ACM SIGCOMM Computer Communication Review. Vol. 28. No. 4. ACM, 1998. [5]

References

• Bloom, Burton H. (1970), "Space/Time Trade-offs in Hash Coding with Allowable Errors", Communications of the ACM 13 (7): 422–426
• Bloom filter. In Wikipedia, The Free Encyclopedia. Retrieved January 17, 2015, from http://en.wikipedia.org/w/index.php?title=Bloom_filter&oldid=641921171
• A. Z. Broder and M. Mitzenmacher. Network applications of Bloom filters: A survey. In Internet Math. Volume 1, Number 4 (2003), 485-509. [6]

Locality Sensitive Hashing

• Har-Peled, Sariel, Piotr Indyk, and Rajeev Motwani. "Approximate Nearest Neighbor: Towards Removing the Curse of Dimensionality." Theory of Computing 8.1 (2012): 321-350.
  • Proprocess: A set P of n binary sequences of length k, radius r, gap c
  • Query: binary sequence q of length q
  • Output: If there exist a sequence within Hamming distance r from q, find a sequence within r*c of q with probability at least f.
  • Hash each point in P using L hashing functions, each using O(log n) randomly chosen sequence positions (hidden constant depends on d, r, c), put all results to bins in one table
  • Given q, find it using all L hash function, check every collision, report if distance at most cr. If checked more than 3L collisions, stop.
  • L = O(n^{1/c})
  • Space O(nL), query time O(Ld+L(d/r)log n)
  • Probability of failure less than some constant < 1, can be boosted by repeating independently