Outline. Approximation: Theory and Algorithms. Reference Sets [GJK + 02] Illustration: Triangle Inequality. Pruning with Reference Sets

Transcription

1 Otine Approximation: Theory and Agorithms Prning with Nikoas Agsten Free University of Bozen-Bozano Facty of Compter Science DIS Unit 10 May 15, Upper and Lower Bond Exampe: 2 Concsion Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24 Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24 [GJK + 02] Istration: Triange Ineqaity c c Reference sets take advantage of the triange ineqaity for metrics. An extended version of the triange ineqaity is: δ(a, b) δ(b, c) δ(a, c) δ(a, b) + δ(b, c), where δ() is a metric distance fnction compted between eements a, b, and c. a b a b δ(a, c) < δ(a, b) + δ(b, c) δ(a, b) δ(b, c) < δ(a, c) a b c a c b δ(a, c) = δ(a, b) + δ(b, c) δ(a, b) δ(b, c) = δ(a, c) Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24 Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24

2 Notation Reference Vector S 1 and S 2 are sets of trees (nordered forests) K S 1 S 2 is caed the reference set S 1 and S 2 are trees k K, 1 K, is a tree of the reference set v i is a vector of size K that stores the distance from S 1 to each tree k K the -th eement of v i stores the edit distance between and k : v i = δ t (, k ) v j is the respective vector for S 2 Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24 Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24 Metric Space Upper and Lower Bond Upper and Lower Bond Metric space of the reference set: the eements of the reference set define the basis of a metric space the vector v k represents tree T k as a point in this space v k is the coordinate of the point on the -th axis Exampe: Two trees and in the metric space defined by the reference set K = {k 1, k 2 }. v k2 v j2 v i2 v j1 v i1 v k1 From the triange ineqaity it foows that for a 1 K Upper bond: Lower bond: v i v δ t (, ) v i + v δ t (, ) δ t (, ) min v i + v = t (, ) max v i v = t (, ) Approximate join: match a trees with δ t (T 1, T 2 ) τ Upper bond: If t (, ) τ, then and match. Lower bond: If t (, ) > τ, then and do not match. Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24 Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24

3 Reference Set Tradeoff Upper and Lower Bond We Separated Csters Upper and Lower Bond Sma reference set: efficient reference vector comptation we have to compte S distances for each additiona tree in the reference set to constrct the vectors for sma reference sets the constrction of the vectors is cheaper Large reference set: effective fiters arge reference sets make t and t more effective ths, once the reference vectors are compted, ess distance comptations are needed Where is the optimm? We cster the set S = S 1 S 2. The csters are we separated for threshod τ if trees within a cster have sma distance (within τ 2 ) trees from different cster have arge distance (more than 3τ 2 ) Exampe: Two we separated csters C 1 and C 2. > 3τ 2 C 1 C 2 Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24 Upper and Lower Bond Trees within the Same Cster Upper Bond Appies Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24 Upper and Lower Bond Trees in Different Csters Lower Bond Appies Upper bond: δ t (, ) v i + v = δ t (, k ) + δ t (k, ) Assmption: S 1 and S 2 are in the same cster C. The csters are we separated. The cster C contains a reference tree k K. In this case v i and v δ t(, ) τ. Rest: If,, and k are in the same cster from v i and v we concde that T 1 and T 2 match ths we need not to compte δ t (, ) τ 2 Lower bond: δ t (, ) v i v = δ t (, k ) δ t (k, ) Assmption: S 1 is in cster C 1, S 2 is in cster C 2. The csters are we separated. The cster C 1 contains a reference tree k K. In this case v i and v > 3τ 2 δ t(, ) > τ. Rest: If and k are in the same cster, bt is in a different cster from v i and v we concde that T 1 and T 2 do not match ths we need not to compte δ t (, ) v i k δ t (, ) v v i > 3τ 2 C 1 C 2 k δ t (, ) v Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24 Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24

4 Upper and Lower Bond Bonds for We Separated Csters Upper and Lower Bond Optimm We Separated Csters Scenario: Cost: Assme that S = S 1 = S 2 is we separated into csters. Consider a cster of size n that contains a tree k K of the reference set. We need to compte S 1 distances to the tree k K Benefit: From the pper bond we concde that a trees within a cster match ( t τ) we save n (n 1)/2 edit distance comptations From the ower bond we concde that trees of different csters do not match ( t > τ) we save n ( S n ) edit distance comptations Optimm: csters are we separated the reference set contains one tree per cster Gha et a. [GJK + 02] find csters by samping and estimate a reference set size. Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24 Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24 Exampe Exampe: S 1 = {T 1, T 2, T 3 }, S 2 = {T 4, T 5, T 6 } Reference set K = {T 1 }, 1 K = 1 Distances: T 1 T 2 T 3 T 4 T 5 T 6 T T T T T T 6 0 Reference Vectors: v 1,1 = (0), v 2,1 = (4), v 3,1 = (1), v 4,1 = (1), v 5,1 = (4), v 6,1 = (1) The csters C 1 = {T 1, T 3, T 4, T 6 } and C 2 = {T 2, T 5 } are we separated. We combine the previos approaches: ower bond with traversa strings pper bond with constrained edit distance reference sets Instead of one vector v i we compte two vectors for each tree : ower bond: vi contains the traversa string distance pper bond: vi contains the constrained edit distance Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24 Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24

5 Metric Space : Triange Ineqaity Metric space of the reference set: the eements of the reference set define the basis of a metric space each tree T k is represented by a rectange in this metric space vk and v k are two opposite corners of this rectange Exampe: Two trees and in the metric space defined by the reference set K = {k 1, k 2 }. / k2 v j2 v i2 The triange eqations changes as foows: (a) For a 1 K δ t (, ) vi + v (b) For a 1 K v v i if v > vi δ t (, ) vi v if vi > v 0 otherwise j2 i2 j1 v j1 i1 v i1 / k1 Note: If v > vi or vi > v then [vi, v i ] and [v, v ] are disjoint intervas. In a other cases we can not give a ower bond (other than 0). Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24 : Upper and Lower Bonds Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24 Istration: Cases for the Lower Bond Upper bond: Lower bond: t (, ) = t (, ) = max min v i + v v v i if v > vi vi v if vi > v 0 otherwise Case v > vi : ower bond is v v i i v i ower bond v v k Case vi > v : ower bond is v i v v ower bond i vi v k A other cases ([vi, v i ] and [v, v ] overap): no ower bond v i v vi v k Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24 Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24

6 Smmary Concsion What s Next? Concsion : Upper and Lower Bond Combination of reference sets with other bonds Binary Branch Distance ower bond proof compexity Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24 Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24 Sdipto Gha, H. V. Jagadish, Nick Kodas, Divesh Srivastava, and Ting Y. Approximate XML joins. In Proceedings of the ACM SIGMOD Internationa Conference on Management of Data, pages , Madison, Wisconsin, ACM Press. Nikoas Agsten (DIS) Approximation: Theory and Agorithms Unit 10 May 15, / 24