1. CS 361A1 CS 361A (Advanced Data Structures and Algorithms) Lecture 18 (Nov 30, 2005) Fingerprints, Min-Hashing, and Document Similarity Rajeev Motwani

2. CS 361A 2 Game Plan for Week  Fingerprints  Document Similarity  Shingling  Min-Hashing  Min-Wise Independent Permutations

3. CS 361A 3 Fingerprints   – set of large objects (e.g., URLs)  Goal avoid storing large objects explicitly quick-and-dirty equality-testing  Fingerprints? Short tags for objects Distinct fingerprints  distinct objects Distinct objects  probably distinct fingerprints

4. CS 361A 4 Formalization  Fingerprint length k  fingerprint space size N=2 k  Fingerprint function family F = { f :  k }  Random f  R F  f(A)  f(B)  A  Collisions: P[ f(A) = f(B) | A    (ideally 2 O(-k) )  Typical Application Adversarial object-set S with |S| = n << 2 k Goal – |f(S)| = |S| with high probability n 2 pair-wise collisions possible  need 2 k > n 2 (to avoid Birthday Paradox)

5. CS 361A 5 Example – URL Fingerprints  Search Engines Manage large numbers of URL strings Long, variable strings (embedded objects/database-queries)  Desiderata small/fixed-length encodings – hopefully, unique Some scenarios oExact string irrelevant oOnly need ability to distinguish distinct URLs Even otherwise, unique IDs useful for indexing  Numbers? 4 billion webpages  n=2 32 N  n 2  k=64 Fingerprints  8-byte representation

6. CS 361A 6 Fingerprinting vs Hashing  Hashing h:  k Set Membership testing for set S of size n Desire uniform distribution over bin address  k Minimize collisions per bin – reduce lookup time Minimize hash table size  n  N=2 k  Fingerprinting f :  k Object Equality testing over set S of size n Distribution over  k is irrelevant Avoid collisions altogether Tolerate larger k – typically N > n 2

7. CS 361A 7 Fingerprinting Strings  Typical Application – but techniques extend to combinatorial objects (database tuples, trees/graphs)  Obvious techniques Checksum – no worst-case collision probability guarantees MD5 – cryptographically-secure string hashes orelatively slow oavoids leaking information about original string  Rabin’s Scheme Algebraic technique – polynomial arithmetic Efficient – need (1 table lookup + 1 xor + 1 shift) per byte other nice properties…

8. CS 361A 8 Rabin Fingerprints  Consider – m-bit string A=a 1 a 2 … a m  Assume – a 1 =1 and fixed-length strings (wlog)  Encoding Strings Degree-m polynomials over Z 2 A(x) = a 1 x m-1 + a 2 x m-2 + … + a m-1 x 1 + a m  Fingerprints P(x): random, irreducible deg-k polynomial over Z 2 (easy to sample such polynomials) irreducible  unlike x 2 +x+1, can factor x 2 +1=(x+1) 2 f(A) = A(x) mod P(x)

9. CS 361A 9 Analysis  Fix S – n strings of length m  Consider Collision f(A)=f(B)  A(x)=B(x) mod P(x)  Q S =0 mod P(x) Therefore – P(x) is factor of Q S (x)  Collision Probability? degree(Q S ) = n 2 m number of irreducible degree-k factors of Q S (x) is < n 2 m/k Fact: Number of irreducible degree-k polynomials > (2 k -2 k/2 )/k Prob[random P(x) divides Q S (x)] < n 2 m/2 k  Prob [fingerprints not distinct] <

10. CS 361A 10 Beneficial Properties  Hardware-level implementation Z 2 -polynomials same as strings simple shift-register operations  Distributivity – f(A+B) = f(A) + f(B) over Z 2  Let  = concatenation f(A  B) = f(f(A)  ) f(A  B) = A(x)*t m + B(x) mod P(x)  Fingerprint sliding windows over strings – low incremental cost

11. CS 361A 11 Duplicate Document Detection  Problem Given – large collection of arbitrary documents Identify – near-duplicate documents  Web search engines Proliferation of near-duplicate documents oLegitimate – mirrors, local copies, updates, … oMalicious – spam, spider-traps, dynamic URLs, … oMistaken – spider errors 30% of web-pages are near-duplicates [Broder et al 1997] Cost – RAM/disk, search quality, unhappy users Enterprise search – even larger amount of duplication SCAM – plagiarism detection [Shivakumar et al 1998]

12. CS 361A 12 Natural Approaches  Fingerprinting? only works for exact matches here – must identify even near-duplicates  Random Sampling? sample substrings (phrases, sentences, etc) hope: similar documents  similar samples No – even samples of same document will differ  Edit-distance? metric for approximate string-matching expensive – even for one pair of strings impossible – for 10 32 web documents

13. CS 361A 13 Desiderata  Storage only small sketches of each document.  Computation O(n log n) time on n documents  Stream Processing once sketch computed, source is unavailable  Error Guarantees problem scale  small biases have large impact need formal guarantees – heuristics will not do

14. CS 361A 14 Basic Idea [Broder 1997]  Shingling dissect document into q-grams (shingles) represent documents by shingle-sets near-duplicates  shingle-sets intersection is large reduce problem to set intersection  Set Intersection fingerprints of shingles min-hash to estimate intersections sizes

15. CS 361A 15 Shingling  Shingle – q contiguous tokens/words (q-gram)  Consider following “document” a rose is a rose is a rose  Choose q=4  get multi-set of shingles a rose is a rose is a rose is a rose is a rose is a rose is a rose

16. CS 361A 16 Multiset of Fingerprints Doc shingling Multiset of Shingles fingerprint Documents  Sets of 64-bit fingerprints Fingerprints? Use Rabin fingerprints Fingerprint space U = [0, …, N-1] In practice, use 64-bit fingerprints, i.e., N=2 64 Result – uniformity in length of strings

17. CS 361A 17 Similarity of Documents Doc B SBSB SASA Doc A Jaccard measure – similarity of S A, S B  U = [0 … N-1] Claim: A & B are near-duplicates if sim(S A,S B ) is high Claim: A is contained in B if con(S A,S B ) is high

18. CS 361A 18 Remarks  Multiplicities of q-grams – could retain or ignore trade-off efficiency with precision  Shingle Size q ε [3 … 10] Short shingles  increase similarity of unrelated documents oWith q=1, sim(S A,S B ) =1  A is permutation of B oNeed larger q to sensitize to permutation changes Long shingles  small random changes have larger impact  Similarity Measure Similarity is non-transitive, non-metric But – dissimilarity 1- sim(S A,S B ) is a metric [Charikar 02]  [Ukkonen 92] – relate q-gram & edit-distance

19. CS 361A 19 Example  A = “a rose is a rose is a rose”  B = “a rose is a flower which is a rose”  Preserving multiplicity q=1  sim(S A,S B ) = 0.7 oS A = {a, a, a, is, is, rose, rose, rose} oS B = {a, a, a, is, is, rose, rose, flower, which} q=2  sim(S A,S B ) = 0.5 q=3  sim(S A,S B ) = 0.3  Disregarding multiplicity q=1  sim(S A,S B ) = 0.6 q=2  sim(S A,S B ) = 0.5 q=3  sim(S A,S B ) = 0.4285

20. CS 361A 20 Min-Hashing  Consider S A, S B  U Pick – random permutation π of U Define  = π -1 ( min{π(S A )} ) and  = π -1 ( min{π(S B )} ) Meaning? – minimal element under permutation π  Lemma: Let δ = min{ π(S A  S B ) } Claim:  =  π -1 (δ)  S A  S B Clearly

21. CS 361A 21 Min-Hashing  Similarity Sketches Succinct representation of fingerprint sets S A Allows efficient estimation of sim(S A,S B ) Basic idea – use min-hash of fingerprints  sk(A) = k minimal elements under π(S A )  Claim: E[ sim(sk(A), sk(B)) ] = sim(S A,S B ) For each   sk(A)  sk(B)  Observe sketch-similarity is unbiased estimator of similarity reducing variance – use larger k

22. CS 361A 22 Remarks  Implementation shingle/fingerprint/sketch document in streams Issue – cost of pairwise comparison of sketches? ocluster sketch-streams [Broder et al, Guha et al] oOpen? – hashing sketches to identify similarity  [Broder-Mitzenmacher 99] – Min-Hash is only unbiased estimator  [Indyk-Motwani 99] – Locality-Sensitive Hash collisions more likely for similar items Min-Hash is special case

23. CS 361A 23 Multiple Permutations  Better Variance Reduction Instead of larger k, stick with k=1 Multiple, independent permutations  Sketch Construction Pick p random permutations of U – π 1,π 2, …,π p sk(A) = minimal elements under π 1 (S A ), …, π p (S A )  Claim: E[ sim(sk(A),sk(B)) ] = sim(S A,S B ) Earlier lemma  true for p=1 Linearity of expectations Variance reduction – independence of π 1, …,π p

24. CS 361A 24 Min-Wise Indep Permutations  Problem Truly-random π over U = [0 … N-1] is infeasible But – do we really need true randomness?  Solution Poly-size family of permutations F  S N over U Choosing/representing random π  F is easy Min-Wise Independence (MWI) Property: For all sets X  U, for all x  F,

25. CS 361A 25 Minimum-Size MWI Families  [Broder et al 98] Upper/lower bounds of lcm(1,2,…,n) Problem – exponential in N  Approximate MWI Families Relax to Non-constructive – polynomial-size Constructive – size N O(log 1/  ) [Indyk 99]  In practice – 2-universal hashes work well!

26. CS 361A 26 References I  Fingerprinting by random polynomials. M. Rabin. Technical Report TR-15-81, Harvard University (1981). Fingerprinting by random polynomials.  Some applications of Rabin's fingerprinting method. A. Broder. Sequence II (1993). Some applications of Rabin's fingerprinting method.  On the Resemblance and Containment of Documents, A. Broder. SEQUENCES 1997. On the Resemblance and Containment of Documents  Syntactic Clustering of the Web, A. Broder, S. Glassman, M. Manasse, and G. Zweig, WWW 1997. Syntactic Clustering of the Web  Finding near-replicas of documents on the web. N. Shivakumar and H. Garcia-Molina. WebDB 1998. Finding near-replicas of documents on the web.  Identifying and Filtering Near-Duplicate Documents, Andrei Broder. CPM 2000. Identifying and Filtering Near-Duplicate Documents

27. CS 361A 27 References II  Approximate String Matching with q-grams and Maximal Matches. E. Ukkonen. Theoretical Computer Science (1992). Approximate String Matching with q-grams and Maximal Matches  Completeness and Robustness Properties of Min-Wise Independent Permutations. A. Broder and M. Mitzenmacher. Completeness and Robustness Properties of Min-Wise Independent Permutations  Min-Wise Independent Permutations, A. Broder, M. Charikar, A. Frieze and M. Mitzenmacher, JCSS (2000). Min-Wise Independent Permutations  A Small Approximately min-wise Independent Family of Hash Functions. P. Indyk. SODA 1999. A Small Approximately min-wise Independent Family of Hash Functions.  Approximate Nearest Neighbors: Towards Removing the Curse of Dimensionality, P. Indyk and R. Motwani. STOC 1998. Approximate Nearest Neighbors: Towards Removing the Curse of Dimensionality  Similarity Search in High Dimensions via Hashing, A. Gionis, P. Indyk, and R. Motwani. VLDB 1999. Similarity Search in High Dimensions via Hashing  Similarity Estimation Techniques from Rounding Algorithms, M. Charikar, STOC 2002. Similarity Estimation Techniques from Rounding Algorithms