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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 x,y from U we have Pr(h(x) = h(y)) <= 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 solutions 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, let for x in U
• any element y from X s.t. x!=y maps to h(x) with probability <= c/m
• so E[C_x] <= 1 + n * c / m = O(1)
  • from linearity of expectation - sum over indicator variables for each y in X if it collides with x
  • +1 in case x 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_{x,y in X_i, x!=y} Pr(h(x) = h(y)) <= (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 of pairs (x,y) s.t. x,y in X and h(x)=h(y); include x=y
• 
• so expected space is linear
• can be made worst-case by 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 gets 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

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

• in fact probability somewhat higher for smaller values
• proof of the bound 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 .
• and thus we get error , where
• to get error rate p for some n, we need
• 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 the bound

• 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 all j=1..k
• assume for now that these events are independent for different j
• error then occurs with probability Pr(B[i]=1)^k = (1-(1-1/m)^{nk})^k
• recall that
• thus probability of error is approximately (1-exp(-nk/m))^k

However, Pr(B[i]=1) are not independent

• consider the following experiment: have 2 binary random variables Y1, Y2, each generated independently, uniformly (i.e. P(Y1=1) = 0.5)
• Let X1 = Ya and X2 = Yb where a and b are chosen independently uniformly from set {1,2}
• X1 and X2 are not independent, e.g. Pr(X2 = 1|X1 = 1) = 2/3, whereas Pr(X2=1) = 1/2. (check the constants!)
• note: X1 and X2 would be independent conditioned on the sum of Y1+Y2.

• 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, delete 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

• todo: expand and according to slides

• typical hashing tables used for exact search: is x in set X?
• nearest neighbor(x): find y in set X minimizing d(x,y) for a given distance measure d
  • or given x and distance threshold r, find all y in X with d(x,y)<=r
• want: polynomial time preprocessing of X, polylogarithmic queries
• later in the course: geometric data structures for similar problems in R^2 or higher dimensions
• but time or space grows exponentially with dimension (alternative is linear search) - "curse of dimensionality"
• examples of high-dimensional objects: images, documents, etc.
• locality sensitive hashing (LHS) is a randomized data structure not guaranteed to find the best answer but better running time

Approximate near neighbour (r,c)-NN

• preprocess a set X of size n for given r,c
• query(x):
  • if there is y in X such that d(y,x)<=r, return z in X such that d(z,x)<=c*r
  • LHS returns such z with some probability at least f

Example for Hamming distance in binary strings of length d

• Hash each string using L hashing functions, each using O(log n) randomly chosen string positions (hidden constant depends on d, r, c), then rehash using some other hash function to tables of size O(n) or a single table of size O(nL)
• 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 + nd), query time O(Ld+L(d/r)log n)
• Probability of failure less than some constant f < 1, can be boosted by repeating independently

Minimum hashing

• similarity of two sets measured as Jaccard index J(A,B) = |A intersect B | / | A union B |
• distance d(A,B) = 1-J(A,B)
• used e.g. for approximate duplicate document detection based on set of words occurring in the document

• choose a totally random hash function h
• minhash_h(A) = \min_{x\in A} h(x)
• Pr(minhash_h(A)=minhash_h(B)) = J(A,B) = 1-d(A,B) if using totally random hash function an no collisions in A union B
  • where can be minimum in union? if in intersection, we get equality, otherwise not
• (r_1, r_2, 1-r_1, 1-r_2)-sensitive family (if we assume no collisions)
• again can we apply amplification

Notes

• Real-valued vectors: map to bits using random projections sgn (w · x + b)
• Data-dependent hashing: Use PCA, optimization, kernel tricks, even neural networks to learn hash functions good for a particular set of points and distance measure (or learn even distance measure from examples). [7]

Sources

• 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. [8]
  • Indyk and Motwani: Approximate nearest neighbors: Towards removing the curse of dimensionality. In Proc. 30th STOC, pp. 604–613. ACM Press, 1998.
• http://www.mit.edu/~andoni/LSH/ Pointers to papers
• Wikipedia
  • http://en.wikipedia.org/wiki/Locality-sensitive_hashing
  • http://en.wikipedia.org/wiki/MinHash
• Several lectures:
  • http://infolab.stanford.edu/~ullman/mining/2009/similarity3.pdf various distances, connection to minhashing
  • https://users.soe.ucsc.edu/~niejiazhong/slides/kumar.pdf nic intro, more duplicate detection, NN at the end
  • http://delab.csd.auth.gr/courses/c_mmdb/mmdb-2011-2012-lsh2.pdf also nice intro through Hamming
• Charikar, Moses S. "Similarity estimation techniques from rounding algorithms." STOC 2002 [9]
  • simHash (cosine measure)