Hešovanie

Introduction to hashing

• Universe U = {0,..., u-1}
• Table of size m
• Hash function h : U -> {0,...,m-1}
• Let X be the set of elements currently in the hash table, n = |X|
• We will assume n = Theta(m)

Totally random hash function (a.k.a uniform hashing model)

• select hash value h(x) for each x in U uniformly independently
• not practical - storing this hash function requires u * lg m bits
• used for simplified analysis of hashing in ideal case

Universal family of hash functions

• some set of functions H
• draw a fuction h uniformly randomly from H
• there is a constant c such that for any distinct u,v from U we have Pr(h(u) = h(v)) <= c/m
• probability goes over choices of h

Note

• for totally random hash function, this probability is exactly 1/m

Example of a universal family:

• choose a fixed prime p >= u
• H_p = { h_a | h_a(x) =(ax mod p) mod m), 1<=a<=p-1 }
• p-1 hash functions, parameter a chosen randomly

Proof of universality

• consider x!=y, both from U
• if they collide (ax mod p) mod m = (ay mod p) mod m
• let c = (ax mod p), d = (ay mod p)
• note that c,d in {0..p-1}
• also c!=d because otherwise a(x-y) is divisible by p and both a and |x-y| are numbers from {1,...,p-1} and p is prime
• since c mod m = d mod m, then c-d = qm where 0<|qm|<p
• there are <=2(p-1)/m choices of q where |qm|<p
• we get a(x-y) mod p = qm mod p
• this has exactly one solution a in {1...p-1}
  • if there were two solitions a1 and a2, then (a1-a2)(x-y_ would be divisible by p, which is again not possible
• overall at most 2(p-1)/m choices of hash functions collide x and y
• out of p-1 all hash functions in H_p
• therefore probability of collision <=2/m

Hashing with chaining, using universal family of h.f.

• linked list (or vector) for each hash table slot
• let c_i be the length of chain i
• consider element u from U, let i = h(u)
• any element x from X s.t. u!=x maps to i with probability <= c/m
• so E[c_i] <= 1 + n * c / m = O(1)
  • from linearity of expectation - sum over indicator variables for each x in X if it collides with u
  • +1 in case u is in X
• so expected time of any search, insert and delete is O(1)
  • again, expectation taken over random choice of h from universal H
• however, what is the expected length of the longest chain in the table?
  • O(1+ sqrt(n^2 / m)) = O(sqrt(n))
  • see Brass, p. 382
  • there is a matching lower bound
• for totally random hash function this is "only" O(log n / log log n)
• similar bound proved also for some more realistic families of hash functions
• so the worst-case query is on average close to logarithmic

References

• Textbook Brass 2008 Advanced data structures
• Lecture L10 Erik Demaine z MIT: http://courses.csail.mit.edu/6.851/spring12/lectures/

Perfect hashing

Static case

• Fredman, Komlos, Szemeredi 1984
• avoid any collisions, O(1) search in a static set
• two-level scheme: use a universal hash function to hash to a table of size m = Theta(n)
• each bucket i with set of elements X_i of size c_i hashed to a second-level hash of size m_i = alpha c_i^2 for some constant alpha
• again use a universal hash function for each second-level hash
• expected total number of collisions in a second level for slot i:
  • sum_{u,v in X_i, u!=v} Pr(h(u) = h(v)) <= (c_i)^2 c/m_i = c/alpha = O(1)
  • here c is the constant from definition universal family
• with a sufficently large alpha this expected number of collisions is <=1/2
  • e.g. for c = 2 set alpha = 4
• then with probability >=1/2 no collisions by Markov inequality
  • number of collisions is a random variable Y number with possible values 0,1,2,...
  • if E[Y]<=1/2, Pr(Y>=1)<=1/2 and thus Pr(Y=0)>=1/2
• when building a hash table if we get a collision, randomly sample another hash function
• in O(1) expected trials get a good hash function
• expected space: m + sum_i m_i^2 = Theta(m + sum_i c_i^2)
• sum_i (c_i^2) is the number pairs (x,y) s.t. x,y in X and h(x)=h(y); include x=y
• sum_{x,y} Pr(h(x)=h(y)) = n + sum_{x!=y} Pr(h(x)=h(y)) = n + n^2 c / m = Theta(n)
• so expected space is linear
• can be made worst-case be repeating construction if memory too big
• expected construction time O(n)

Dynamic perfect hashing

• amortized vector-like tricks
• when a 2-nd level hash table get too full, allocate a bigger one
• O(1) deterministic search
• O(1) amortized expectd update
• O(n) expected space

References

• Lecture L10 Erik Demaine z MIT: http://courses.csail.mit.edu/6.851/spring12/lectures/
• Textbook Brass 2008 Advanced data structures

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