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Power-Law Based Estimation of Set Similarity Join Size Hongrae Lee, University of British Columbia Raymond T. Ng, University of British Columbia Kyuseok Shim, Seoul National University

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Highly Similar, but Not The Same, Data 2 Nearly word-for-word copy The duplicate does not cite the original Similar news articles

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Introduction Finding all pairs of similar objects is a very common task – Near duplicate detection – Data integration – Record linkage – Web search 3 Sunita Sarawagi, Alok Kirpal: Efficient set joins on similarity predicates. SIGMOD Conference 2004: 743-754 S. Sarawagi, A. Kirpal, Efficient set joins on similarity predicates. SIGMOD sp|P45680|YFMU_COXBU HYPOTHETICAL 15.8 KD PROTEIN IN FMU-RP... sp|P45680|YFMU_COXBU H 15.8 KD PROTEIN IN FMU-RP...

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Set Similarity Join (SSJoin) SSJoin is proposed as a general framework for finding similar objects Input – two collections of sets, R and S – similarity function sim – similarity threshold τ Output – all pairs (r,s) r ∈ R, s ∈ S, such that sim(r,s) ≥ τ 4 {bolt, destroy, 200, meter, record} {bolt, smashes, 200, meter, world, record, berlin} word n-gram Documents set Jaccard similarity = 0.5

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Estimation of SSJoin Size SSJoin in RDBMS – SSJoin operator as a primitive operator [Chaudhuri, Ganti, Kaushik 06] – Data cleaning as a repetitive operation [Fuxman, Fazli, Miller 05] Efficient and accurate estimation of SSJoin size is crucial in query optimization – Poor size estimations can result in sub-optimal plans 5 SSJ NL Seek S.AR.AT.B SSJ HM Scan S.AR.AT.B different opt-plans depending on SSJ size

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Problem Statement Input – a collection of sets R (self-join) – threshold τ on Jaccard similarity J S Output – the number of pairs (r,s), SSJ(τ), such that J S (r,s) ≥ τ, r, s ∈ R and r≠ s. Jaccard similarity J S – J S (r,s) = |r ∩ s| / |r ∪ s| – e.g., J S ({1,2,3},{2,3,4}) = |{2,3}| / |{1,2,3,4}| = 0.5 6

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Related Work Set similarity join (or selection) – [Sarawagi, Kirpal 04], [Chaudhuri, Ganti, Kaushik 06], [Arasu, Ganti, Kaushik 06], [Bayardo, Ma, Srikant 07], [Xiao, Wang, Lin 08], [Xiao, Wang, Lin, Yu 08], [Hadjieleftheriou, Chandel, Koudas, Srivastava 08], [Xiao, Wang, Lin, Shang 09] Hashed Samples: selectivity estimation of set similarity selection queries – [ Hadjieleftheriou, Yu, Koudas, Srivastava 08] Estimation of the number of frequent patterns – [Chuang, Huang, Chen 08], [Jin, McCallen, Breitbart, Fuhry, Wang 09], [Boley, Grosskreutz 09] 7

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Outline Introduction Signature pattern & Lattice counting Power-law based estimation Correction of the estimation Experimental results 8

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Min-Hash Signature Min-wise hash function – Prob [h(r) = h(s)] = |r ∩ s| / |r ∪ s| Min-Hash signature – Use M min-wise hash functions, h 1,…,h M – J S (r,s) ≈ fraction of signatures for which Min-hash values agree 9 {1,3,53,55,23,534,…} {2,3,50,51,52,53,…} [4,3,5,2] [4,3,3,5] r s sig(r) sig(s) J S (r,s) ≈ 2/4 M=4 [Cohen 97] [Broder, Glassman, Manasse, Zweig 97]

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Min-Hash Representation of Sets We work on Min-Hash signatures of sets – Succinct representation enables faster analysis – Min-Hash signatures preserve Jaccard similarity between original sets – Might be readily available 10 DB r1{7,10,19,52,67} r2{10,19,43,52} r3{10,13,43,52,67,85} r4{10,38,43,49,80,94} r5{3,25,29,47,50,66,73,75} Sig(DB) sig (r1)[4,3,5,2] sig (r2)[4,3,3,5] sig (r3)[4,3,2,2] sig (r4)[3,3,3,2] sig (r5)[1,1,1,2] M (signature size) = 4

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Signature Pattern Define signature pattern to represent frequently co- occurring signature values Signature pattern – A Min-Hash signature possibly with ‘X’ ‘X’: don’t care position – A signature (set) matches a pattern if it (its signature) agrees on all non-X positions with the pattern e.g., [4,3,5,2] matches patterns [4,3,X,X] or [X,3,5,2] (and many more), but does not match [4,3,2,X] (position matters) – length: # non-X positions – freq (support count): # matching signatures in the DB 11

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An Example Signature Pattern 12 Sig(DB) sig (r1)[4,3,5,2] sig (r2)[4,3,3,5] sig (r3)[4,3,2,2] sig (r4)[3,3,3,2] sig (r5)[1,1,1,2] [4,3,X,X] Signature pattern Pattern Length 2 Pattern Freq 3 (r1,r2,r3)

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# Similar Pairs by Pattern Frequency Pattern freq f, length i pairs have at least i matching positions in their signatures (J S ≥ i /M) – pattern length J S (estimated) – pattern frequency # pairs 13 Sig(DB) sig (r1)[4,3,5,2] sig (r2)[4,3,3,5] sig (r3)[4,3,2,2] sig (r4)[3,3,3,2] sig (r5)[1,1,1,2] [4,3,X,X] Signature pattern Pattern Length 2 Pattern Freq 3 (r1,r2,r3) signature pairs match at least 2 positions J S (r1,r2), J S (r2,r3), J S (r3,r1) ≥ 2/4 (est.) 3232 ( ) f2f2 ( )

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SSJoin Size By Pattern Frequency Given threshold τ, we find all patterns with length ≥ τ*M For each pattern, pairs satisfy τ 14 LengthMatching setFreqMatching pair set# pairs 2r1, r2, r33{(r1,r2),(r1,r3),(r2,r3)}3 2r1, r32{(r1,r3)}1 2r2, r42{(r2,r4)}1 2r1, r3, r43{(r1,r3),(r1,r4),(r3,r4)}3 3r1, r32{(r1,r3)}1 Signature pattern sig1=[4, 3, X, X] sig2=[4, X, X, 2] sig3=[X, 3, 3, X] sig4=[X, 3, X, 2] sig5=[4, 3, X, 2] Naïve approach for SSJoin Size: sum # pairs from all patterns ∑=9 There are overlaps in pattern frequency and thus # pairs We need the cardinality of union of matching pair sets when τ = 0.5 freq 2 ( )

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Computing the Union Size We can compute the union size with Inclusion-Exclusion (IE) formula – Combinatorial # operations! 15 Signature patternMatching pair set sig1=[4, 3, X, X]S1={(r1,r2),(r1,r3),(r2,r3)} sig2=[4, X, X, 2]S2={(r1,r3)} sig3=[X, 3, X, 2]S3={(r1,r3),(r1,r4),(r3,r4)} |S1 ⋃ S2 ⋃ S3| = |S1| + |S2|+ |S3| − (|S1 ⋂ S2|+ |S2 ⋂ S3| +|S3 ⋂ S1|) + |S1 ⋂ S2 ⋂ S3|

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Efficient Evaluation of IE-Formula [4,3,X,X][4,X,X,2][X,3,X,2] [4,3,X,2] SSJ(0.5)=|S1 ⋃ S2 ⋃ S3| =|S1| + |S2|+ |S3| − (|S1 ⋂ S2|+ |S2 ⋂ S3| +|S3 ⋂ S1|) + |S1 ⋂ S2 ⋂ S3| =1*(|S1| + |S2|+ |S3|) + (−3 + 1) *|S4| 16 (r1,r2) (r1,r3) (r2,r3) (r1,r3) (r1,r4) (r3,r4) (r1,r3) S2 S1 S3 S4 Pattern LatticeMatching Pair Lattice Patterns and matching pairs exhibit lattice structure layer nodes according to the pattern length (= level) edges: inclusion relationship patterns length < τ*M are not shown

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Lattice Counting Compute SSJoin size from ‘pattern distribution’ (# patterns per each length and frequency) Basically simplified IE-formula computation using lattices Does not store actual matching sets or pair sets, only counts! 17 Signature patternLengthMatching setFreqMatching pair set# pairs sig1=[4, 3, X, X]2r1, r2, r33{(r1,r2),(r1,r3),(r2,r3)}3 sig2=[4, X, X, 2]2r1, r32{(r1,r3)}1 sig3=[X, 3, 3, X]2r2, r42{(r2,r4)}1 sig4=[X, 3, X, 2]2r1, r3, r43{(r1,r3),(r1,r4),(r3,r4)}3 sig5=[4, 3, X, 2]3r1, r32{(r1,r3)}1 Pattern Distribution LengthFrequency# pattern 222 32 321 See the paper for details Please see the paper for details

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Outline Introduction Signature pattern & Lattice counting Power-law based estimation Correction of the estimation Experimental results 18

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Pattern Distribution LengthFrequency# pairs 222 32 321 pattern frequency # of patterns (pattern count) level 2 (pattern length=2) level 3 (pattern length=3) there 2 patterns that match 3 sets and whose length is 2 i.e., sig1=[4,3,X,X] (r1,r2,r3) sig4=[X,3,X,2] (r1,r3,r4) 19 If we have exact pattern dist., we can exactly estimate SSJoin size

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Exact Pattern Distribution Computing exact pattern distribution is infeasible – We need pattern distribution for freq >= 2 (min freq for generating a pair) Minimum support threshold = 2 – Most frequent pattern mining algorithms are not designed to handle such a low support threshold – Even if they could, it would take too long to be used for query optimization purposes 20

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Power-Law Distribution of Pattern Count 21 minimum support threshold mined pattern distribution missing pattern distribution A Power-law distribution is observed in # patterns-frequency relationship (or pattern count-support count) [Chuang, Huang, Chen 08] Power law: count = β*frequency -α

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SSJoin Size Estimation 1.Find frequent patterns with ξ > 2 2.Estimate the parameters of the Power-law distribution at each level with the acquired patterns 3.Compute the full pattern distribution based on the estimated parameters 4.Compute SSJoin size with Lattice Counting formula 22

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Outline Introduction Signature pattern & Lattice counting Power-law based estimation Correction of the estimation Experimental results 23

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Systematic Overestimation By Min- Hash Big overestimation is observed e.g., relative error J S =0.4: 10332% J S =0.5: 2614% J S =0.6 : 573% 24 # pair – similarity plot of exhaustive pair-wise comparison

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Effect of Skewed Distribution 01234 # matching position T(i) 10,0001,000100 10 1 1001 100 – 2*10 + 100 + 1 = 181 # pairs with J S =i/M 25 Assume 10% of pairs have +1 or -1 more matching positions in their Min-Hash signatures

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Probabilistic Modeling s={1,2,3,4,5,6,8} r={1,2,4,5,7} J S (r,s) =4/8=0.5 0.5 sig(r) sig(s) 0.5 Pr (J=j | I=i) ≡ Prob (j matching position when J S =i/M) E [ # matching position] = 2 Prob (3 matching position) ? 0.5 3 (1-0.5) 4-3 4 3 () 26

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Considering All # Pairs # matching position 01234 T(i): # pairs with J S =i/M O(j): # pairs with j matching pos in sig T(0) O(0) T(1)T(2) T(3)T(4) O(1) O(2) O(3)O(4) O(2) = T(0)*P(2|0) +T(1)*P(2|1) +T(2)*P(2|2) +T(3)*P(2|3) +T(4)*P(2|4) O(0) O(1) O(2) O(3) O(4) T(0) T(1) T(2) T(3) T(4) P(0|0) P(0|1) P(0|2) P(0|3) P(0|4) P(1|0) P(1|1) P(1|2) P(1|3) P(1|4) P(2|0) P(2|1) P(2|2) P(2|3) P(2|4) P(3|0) P(3|1) P(3|2) P(3|3) P(3|4) P(4|0) P(4|1) P(4|2) P(4|3) P(4|4) = AT=O Observed size by Min-Hash True Size Transition Probability 27

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NNLS Optimization AT=O T=A -1 O Subject to X ≥ 0 A is non-singular We actually have an estimated vector O’, not the exact O O is highly skewed and lower entries make higher entries negligible We solve Non-negative least square (NNLS) constrained optmization problem Scale the matrix by a weight matrix W, W i,i =1/O(i) and W i,j =0,i ≠j ∥ WAX – WO ∥ 28 T may have negative values

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SSJoin Size Estimation Algorithm 29 Min-Hash Signatures of DB Partial Pattern Distribution (# patterns for each length) SSJoin Size Error Correction Est. Full Pattern Distribution Freq. pattern mining algorithm No need for actual patterns Only count # patterns Power-law parameter estimation Lattice Counting NNLS optimization

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Outline Introduction Signature pattern & Lattice counting Power-law based estimation Correction of the estimation Experimental results 30

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Experimental Setup Dataset – DBLP, 800K – IBM Quest synthetic data, 50K Compared algorithms – LC(ξ) : the proposed solution with a minimum support threshold of ξ – Independent Sum (IS) : without lattice counting – LCNC(ξ) : LC without the error correction step – HS(ρ) : Hashed samples[Hadjieleftheriou, Yu, Koudas, Srivastava 08] adapted to SSJoin Opt_Merge [Sarawagi, Kirpal 04] ρ: sampling ratio Evaluation metric – Accuracy: actual count, relative error – Runtime: pre-processing time, estimation time 31

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Accuracy 32 LC delivers accurate estimations for high similarity thresholds HS: random samples will miss many highly similar pairs DBLP Synthetic Data HS: accurate enough for very low similarity thresholds

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Runtime 33 DBLP 40K Estimation timePre-processing time LC is faster (with better accuracy) LC’s pre-processing time is smaller

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Effect of Error Correction Step 34 Huge overestimation without considering the overlaps Error correction step effectively reduces the overestimation Computational overhead of the error correction step is negligible AccuracyRuntime

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Scalability 35 Estimation timePre-processing time Much slower increase in runtime and pre-processing time than HS, random sampling

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Summary Proposed a SSJoin size estimation algorithm based Min-hash signatures and frequent pattern mining technique with the error correction Evaluated the proposed algorithm with synthetic and real-world databases Future work – Apply recent developments in estimating the number of frequent patterns: random sampling 36

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Thank you 37

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Lattice Structure in Patterns Patterns and corresponding matching pair sets have lattice structure – Partial order by inclusion relationship, lub and glb by intersection and union – E.g, if a set matches [4,3,X,2] it matches all of its children – If a set matches both [4,3,X,X] and [4,X,X,2], it also matches [4,3,X,2] We can compute the union size by Inclusion-Exclusion (IE) formula Lattice structures greatly simplifies the IE-formula computation 38 [4,3,X,X][4,X,X,2][X,3,X,2] [4,3,X,2] (r1,r2) (r1,r3) (r2,r3) (r1,r3) (r1,r4) (r3,r4) (r1,r3) S2 S1 S3 S4 Pattern LatticeMatching Pair Lattice

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Lattice Counting Lattice Counting (LC) – Efficient computation of IE-formula exploiting the underlying lattices – Level sum F i : # pairs with i matching values in their signatures – Coefficient C i collapses repeated computation of the same results into a single operation [Lee, Ng, Shim 07] Only needs # patterns of freq f and length i – e.g., if τ = 0.5 and M=4, SSJ(0.5) = LC(2) and LC needs # patterns of length 2,3 and 4 for each frequency – Does not need actual patterns 39 LC(t) = ∑ t≤i≤M C i *F i, t= τ*M C i : coefficient for level i F i : level i sum

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Parameter Estimation Might Fail for Longer Patterns There are in general a smaller number of higher-level (longer) patterns We may not have enough points for parameter estimation LC(t) requires all pattern dist. for level t ~ M – LC(t) = ∑ t ≤ i ≤ M C i *F i Our solutions – Approximate Lattice Counting – Interpolation 40 Enough points for parameter estimation, i = 3 Not enough points for parameter estimation, i = 9

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Approximate Lattice Counting [_,_,X,X][_,X,_,X][_,X,X,_][X,_,_,X][X,_,X,_][X,X,_,_] [_,_,_,X][_,_,X,_][_,X,_,_][X,_,_,_] [_,_,_,_] [_,_,X,X][_,X,_,X][_,X,X,_][X,_,_,X][X,_,X,_][X,X,_,_] [_,_,_,X][_,_,X,_][_,X,_,_][X,_,_,_] LC(t) = ∑ t ≤ i ≤ M C i *F i t level M … t t + k … LC k (t) = ∑ t ≤ i ≤ t+k C k,i *F i 41 Partial independence assumption: ignore high level nodes only considering nodes up to level t+k t = τ*M, k: approximation constant Full lattice Partial lattice

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Estimation with Limited Pattern Distribution An observation – SSJoin size is highly skewed and Pair count – Jaccard similarity exhibits a Power-law relationship Used for interpolation when very low support thresholds or NNLS optimization failure 42 Jaccard similarity Pair count (SSJoin size) Jaccard similarity Pair count (SSJoin size) DBLP

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Power Hypothesis 43 DBLP Synthetic Data

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