Near-Duplicates Detection Naama Kraus Slides are based on Introduction to Information Retrieval Book by Manning, Raghavan and Schütze Some slides are courtesy of Kira Radinsky Why duplicate detection? About 30-40% of the pages on the Web are (near) duplicates of other pages. E.g., mirror sites Search engines try to avoid indexing duplicate pages Save storage Save processing time Avoid returning duplicate pages in search results Improved user s search experience The goal: detect duplicate pages 1
Exact-duplicates detection A naïve approach detect exact duplicates Map each page to some fingerprint, e.g. 64-bit If two web pages have an equal fingerprint check if content is equal Near-duplicates What about near-duplicates? Pages that are almost identical. Common on the Web. E.g., only date differs. Eliminating near duplicates is desired! The challenge How to efficiently detect near duplicates? Exhaustively comparing all pairs of web pages wouldn t scale. 2
Shingling K-shingles of a document d is defined to be the set of all consecutive sequences of k terms in d k is a positive integer E.g., 4-shingles of My name is Inigo Montoya. You killed my father. Prepare to die : { my name is inigo name is inigo montoya is inigo montoya you inigo montoya you killed montoya you killed my, you killed my father killed my father prepare my father prepare to father prepare to die } Computing Similarity Intuition: two documents are near-duplicates if their shingles sets are nearly the same. Measure similarity using Jaccard coefficient Degree of overlap between two sets Denote by S (d) the set of shingles of document d J(S(d1),S(d2)) = S (d1) S (d2) / S (d1) S (d2) If J exceeds a preset threshold (e.g. 0.9) declare d1,d2 near duplicates. Issue: computation is costly and done pairwise How can we compute Jaccard efficiently? 3
Hashing shingles Map each shingle into a hash value integer Over a large space, say 64 bits H(di) denotes the hash values set derived from S(di) Need to detect pairs whose sets H() have a large overlap How to do this efficiently? In next slides Permuting Let p be a random permutation over the hash values space. π: N N, x y: π x π y Let P(di) denote the set of permuted hash values in H(di) P d i = π H d i Let xi be the smallest integer in P(di) x i = min P d i 4
Illustration Document 1 2 64 2 64 2 64 Start with 64-bit H(shingles) Permute on the number line with p 2 64 Pick the min value Key Theorem Theorem: J(S(di),S(dj)) = P(xi = xj) xi, xj of the same permutation Intuition: if shingle sets of two documents are nearly the same and we randomly permute, then there is a high probability that the minimal values are equal. 5
Proof (1) View sets S1,S2 as columns of a matrix A one row for each element in the universe. a ij = 1 indicates presence of item i in set j Example S 1 S 2 0 1 1 0 1 1 Jaccard(S 1,S 2 ) = 2/5 = 0.4 0 0 1 1 0 1 Proof (2) For columns S i, S j, four types of rows S i S j A 1 1 B 1 0 C 0 1 D 0 0 Let A = # of rows of type A Clearly, J(S1,S2) = A/(A+B+C) 6
Proof (3) Let p be a random permutation of the rows of A Denote by P(Sj) the column that results from applying p to the j-th column Let xi be the index of the first row in which the column P(Si) has a 1 P(S1) P(S2) 0 1 1 0 1 1 0 0 1 1 0 1 Proof (4) We ve seen that J(S1,S2) = A/(A+B+C) Claim: P(xi=xj) = A/(A+B+C) Why? Look down columns Si, Sj until first non-type- D row I.e., look for xi or xj (the smallest or both if they are equal) P(xi) = P(xj) type A row As we picked a random permutation, the probability for a type A row is A/(A+B+C) P(xi=xj) = J(S1,S2) 7
Sketches Thus our Jaccard coefficient test is probabilistic Need to estimate P(xi=xj) Method: Pick k (~200) random row permutations P Sketch di = list of xi values for each permutation List is of length k Jaccard estimation: Fraction of permutations where sketch values agree Sketch di Sketch dj / k Example Sketches S 1 S 2 S 3 R 1 1 0 1 R 2 0 1 1 R 3 1 0 0 R 4 1 0 1 R 5 0 1 0 S 1 S 2 S 3 Perm 1 = (12345) 1 2 1 Perm 2 = (54321) 4 5 4 Perm 3 = (34512) 3 5 4 Similarities 1-2 1-3 2-3 0/3 2/3 0/3 8
Algorithm for Clustering Near-Duplicate Documents 1.Compute the sketch of each document 2.From each sketch, produce a list of <shingle, docid> pairs 3.Group all pairs by shingle value 4.For any shingle that is shared by more than one document, output a triplet <smaller-docid, larger-docid, 1>for each pair of docids sharing that shingle 5.Sort and aggregate the list of triplets, producing final triplets of the form <smaller-docid, larger-docid, # common shingles> 6.Join any pair of documents whose number of common shingles exceeds a chosen threshold using a Union-Find algorithm 7.Each resulting connected component of the UF algorithm is a cluster of near-duplicate documents Implementation nicely fits the map-reduce programming paradigm Implementation Trick Permuting universe even once is prohibitive Row Hashing Pick P hash functions h k Ordering under h k gives random permutation of rows One-pass Implementation For each C i and h k, keep slot for min-hash value Initialize all slot(c i,h k ) to infinity Scan rows in arbitrary order looking for 1 s Suppose row R j has 1 in column C i For each h k, if h k (j) < slot(c i,h k ), then slot(c i,h k ) h k (j) 9
Example C 1 C 2 R 1 1 0 R 2 0 1 R 3 1 1 R 4 1 0 R 5 0 1 C 1 slots h(1) = 1 1 - g(1) = 3 3 - h(2) = 2 1 2 g(2) = 0 3 0 C 2 slots h(3) = 3 1 2 g(3) = 2 2 0 h(x) = x mod 5 g(x) = 2x+1 mod 5 (Denote R 5 as R 0 ) h(4) = 4 1 2 g(4) = 4 2 0 h(5) = 0 1 0 g(5) = 1 2 0 10