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 (1970)

• a bit string of length b, and k hash functions hi : U -> {1, . . . , b}.
• insert(x): set bits h1 (x), . . . , 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 totally random and independent and b = lg(e)kn bits, error is at most 2^{−k}.

• proof later
• totally random hash function impractical (need to store hash value for each element of the universe)
• error independent of the size of the universe U
• lg(e) is approx 1.44. for 1% error, k=7, about 10 bits per element
• compare to a hash table, which needs at lead n lg u bits to store data items themselves

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

Proof of lemma

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

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