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Cheng, Chen, Chen, Xie Evaluating Probability Threshold k- Nearest-Neighbor Queries over Uncertain Data Reynold Cheng (University of Hong Kong) Lei Chen (Hong Kong University of Science &Tech) Jinchuan Chen (Hong Kong Polytechnic University) Xike Xie (University of Hong Kong) International Conference on Extending Database Technology 2009

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Cheng, Chen, Chen, Xie Agenda 1. Introduction 2. Problem Definition 3. Basic Solution 4. Efficient Solution 5. Results

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Cheng, Chen, Chen, Xie Data Uncertainty Inherent in various applications Location-based services (e.g., using GPS, RFID) [TDRP98, SSDBM99] Natural habitat monitoring with sensor networks [VLDB04a] Biomedical and biometric databases[ICDE06, ICDE07]

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Cheng, Chen, Chen, Xie Attribute Uncertainty Model [TDRP98,ISSD99,VLDB04b] pdf y (pdf) Uncertainty region We represent an uncertainty pdf as a histogram

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Cheng, Chen, Chen, Xie k-NN Queries k-NN Query over Precise Data - application in LBS [VLDB03] - natural habitat monitoring system [VLDB04a] - network traffic analysis [ICDCS07] - pattern matching in CAM [VLDB04c] k-NN over Uncertain Objects - [VLDB08a] ranks the probability each object is the NN of the query point. - [ICDE07a] use expected distance and does not discuss the probability.

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Cheng, Chen, Chen, Xie Agenda 1. Introduction 2. Problem Definition 3. Basic Solution 4. Efficient Solution 5. Results

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Cheng, Chen, Chen, Xie Probability Threshold k-Nearest-Neighbor Query (T-k-PNN) INPUT 1. A query point q, parameter k, threshold T 2. A set of n objects with uncertainty regions and pdfs OUTPUT A number of k-subset p(S) is the qualification probability of the k-subset S

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Cheng, Chen, Chen, Xie Example of a k-PNN query (k=3) {O 1, O 2, O 3 } {O 1, O 2, O 4 } O2O2 O3O3 O1O1 O4O4 O5O5 O6O6 O7O7 O8O8 q

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Cheng, Chen, Chen, Xie Example of a k-PNN query (k=3) O2O2 O3O3 O1O1 O4O4 O5O5 O6O6 q {O 1, O 2, O 3 } {O 1, O 2, O 4 } … {O 6, O 7, O 8 } k-bound {O 1, O 2, O 3 } {O 1, O 2, O 4 } … {O 4, O 5, O 6 } O7O7 O8O8

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Cheng, Chen, Chen, Xie k-bound Filtering (k=3) O2O2 O3O3 O1O1 O4O4 O5O5 O6O6 q k-bound O7O7 O8O8 f1f1 f2f2 f3f3 f k (k-bound): is the k-th minimum maximum distance Since min(r 7 )> f 3, O 7 can not be 3-NN of q. Because there are always 3 objects with distances smaller than f 3. We apply k-bound filtering on an index (e.g. R-tree) to prune unqualified objects.

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Cheng, Chen, Chen, Xie Agenda 1. Introduction 2. Problem Definition 3. Basic Solution 4. Efficient Solution 5. Results

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Cheng, Chen, Chen, Xie Basic solution for a T-k-PNN query (k=3,T=0.1) 3-subsetQP {O 1, O 2, O 3 }0.2 {O 1, O 2, O 4 } 0.1 {O 1, O 3, O 4 } 0.1 {O 2, O 3, O 4 } 0.1 0.05 {O 2, O 3, O 5 } 0.05 {O 1, O 3, O 5 } …… 0.05 {O 1, O 2, O 5 } 0.05 {O 1, O 2, O 5 } 0.1 {O 2, O 3, O 4 } 0.1 {O 1, O 3, O 4 } 0.1 {O 1, O 2, O 4 } 0.2{O 1, O 2, O 3 } QP3-subset O2O2 O3O3 O1O1 O4O4 O5O5 O6O6 q k-bound T=0.1 Exact QP is expensive to compute! Too many k-subsets! Step1: k-bound filteringStep2: QP CalculationStep3: Accept S, if qp(S)≥T SymbolMeaning riri |o i − q| d i (r)pdf of r i (distance pdf) D i (r)cdf of r i (distance cdf)

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Cheng, Chen, Chen, Xie Qualification Probability

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Cheng, Chen, Chen, Xie Basic Solution [TKDE04] O2O2 q n1n1n1n1 f O1O1 O3O3 O4O4 d i (r): distance pdf of O i from qd i (r): distance pdf of O i from q D i (r): distance cdf of O i from qD i (r): distance cdf of O i from q r i : distance of O i from qr i : distance of O i from q n i : smallest distance of O i from q (min(r i ))n i : smallest distance of O i from q (min(r i )) f: shortest max distance of all objects from qf: shortest max distance of all objects from q O5O5 O6O6

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Cheng, Chen, Chen, Xie Agenda 1. Introduction 2. Problem Definition 3. Basic Solution 4. Efficient Solution 5. Results

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Cheng, Chen, Chen, Xie Efficient Solution Framework (GVR) Lower bound Upper bound 3. 4. Refinement k-subset Generation k-subset Verification And Refinement k-subsets rejected k-subsets accepted k-subsets Candidate Objects 1. k-bound Filtering 2. Probabilistic Candidate Selection k-subsets Generation Verification Refinement

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Cheng, Chen, Chen, Xie Probabilistic Candidates Selection O2O2 O3O3 O1O1 O4O4 O5O5 O6O6 q k-bound 0.1 0.2 0.5 Cutoff Probability of O i : Pr(r i ≤f k ) S 1 ={O 4, O 5,O 6 } cp(S 1 )=0.5*0.2*0.1 = 0.01 S 2 ={O 4, O 5 } cp(S 2 )=0.5*0.2 = 0.1 Given T=0.2, if cp(S 2 ) < T, then qp(S 1 )

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Cheng, Chen, Chen, Xie Probabilistic Candidates Selection 0.5{O 4 } 0.2{O 5 } 0.1{O 6 } 1{O 3 } 1{O 2 } 1{O 1 } CP1-subset 0.2{O 2, O 3, O 5 } 0.2{O 1, O 3, O 5 } 0.1{O 1, O 4, O 5 } 0.5{O 2, O 3, O 4 } 0.1{O 2, O 4, O 5 } 0.1{O 3, O 4, O 5 } 0.5{O 1, O 3, O 4 } 0.2{O 1, O 2, O 5 } 0.5{O 1, O 2, O 4 } 1{O 1, O 2, O 3 } CP3-subset 1{O 2,O 3 } 0.5{O 2,O 4 } 0.2{O 2,O 5 } 0.5{O 3,O 4 } 0.2{O 3,O 5 } 0.2{O 1,O 5 } 0.1{O 4,O 5 } 0.5{O 1,O 4 } 1{O 1,O 3 } 1{O 1,O 2 } CP2-subset T=0.2, k=3

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Cheng, Chen, Chen, Xie Storage Efficient Compression 1{O 2,O 3 } 0.5{O 2,O 4 } 0.2{O 2,O 5 } 0.5{O 3,O 4 } 0.2{O 3,O 5 } 0.2{O 1,O 5 } 0.5{O 1,O 4 } 1{O 1,O 3 } 1{O 1,O 2 } CP2-subset Subsets are sorted in descending order of their CPs. {O 3,O 5 } {O 2,O 5 } {O 1,O 5 } Size-2 Set Original subsets Compressed subsets Store the common prefix of the subsets And the last element of the subset that has the minimum product of cutoff probability greater than T

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Cheng, Chen, Chen, Xie Storage Efficient Compression 0.5{O 4 } 0.2{O 5 } 0.1{O 6 } 1{O 3 } 1{O 2 } 1{O 1 } CP1-subset 0.2{O 2, O 3, O 5 } 0.2{O 1, O 3, O 5 } 0.1{O 1, O 4, O 5 } 0.5{O 2, O 3, O 4 } 0.1{O 2, O 4, O 5 } 0.1{O 3, O 4, O 5 } 0.5{O 1, O 3, O 4 } 0.2{O 1, O 2, O 5 } 0.5{O 1, O 2, O 4 } 1{O 1, O 2, O 3 } CP3-subset 1{O 2,O 3 } 0.5{O 2,O 4 } 0.2{O 2,O 5 } 0.5{O 3,O 4 } 0.2{O 3,O 5 } 0.2{O 1,O 5 } 0.1{O 4,O 5 } 0.5{O 1,O 4 } 1{O 1,O 3 } 1{O 1,O 2 } CP2-subset {O 4 } {O 5 } {O 3 } {O 2 } {O 1 } Size-1 Set {O 3,O 5 } {O 2,O 5 } {O 1,O 5 } Size-2 Set Size-3 Set {O 1,O 2,O 5 } {O 1,O 3,O 5 } {O 2,O 3,O 5 } T=0,2, k=3

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Cheng, Chen, Chen, Xie O3O3 Seeds Pruning O1O1 O2O2 q O4O4 k=3 f1f1 f2f2 f3f3 min(r 4 ) > f 2 > f 1 Seeds: o 1, o 2, o 3 If o 4 belongs to a 3-nn set S, o 1 and o 2 must also belong to S. r 4 > r 2 r 4 > r 1 min(r 4 ) For example, we can prune the set {o 1,o 3,o 4 }, according to the above rule. max(r 1 ) =f 1 max(r 2 ) =f 2 max(r 3 ) =f 3 No CP calculation is needed. Can prune more candidate k-sets

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Cheng, Chen, Chen, Xie Verifiers: Upper and Lower Bounds (T=0.2) Candidates k-subsets (After PCS) 0 1 S1 S1 1 0 0.19 0.6 0.1 0.5 ? 0.4 0.54 0.14 0.15 0.18 0.03 Verifier Incremental Refinement Classifier 1 1 0 S2 S2 1 S3 S3 0 1

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Cheng, Chen, Chen, Xie Verification and Refinement PartitionsStair-Case Model Divide the range [min(r 1 ), f k ] into a series of partitions. Extended from the probabilistic verifiers in [ICDE08b] Build a data structure, i.e. stair-case model, to store the distance cdf of each object. Derive the lower and upper bounds of a k-set’s QP based on the stair-case model. Reject (Accept) a k-set once its QP must be lower (larger) than the threshold.

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Cheng, Chen, Chen, Xie Lower and Upper Bounds Given that

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Cheng, Chen, Chen, Xie Upper- Lower- Bound Verifiers

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Cheng, Chen, Chen, Xie Complexity of Verifiers ItemsCost Upper Bound for one ObjO(kM|C|) Lower Bound for one ObjO(kM|C|) Total complexity of verificationO(kM|C||Q|) |Q|=no. of k-subsets generated from PCS |C|=no. of candidates with non-zero prob. M= no. of subregions

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Cheng, Chen, Chen, Xie Agenda 1. Introduction 2. Problem Definition 3. Basic Solution 4. Efficient Solution 5. Results

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Cheng, Chen, Chen, Xie Experiment Setup Uncertain Object DB Long Beach (53k) ( http://www.census.gov/geo/www/tiger/ ) Uncertainty pdf Uniform (default) Gaussian (represented by histograms) Threshold (T)0.1 k6

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Cheng, Chen, Chen, Xie 1. k-bound Filtering

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Cheng, Chen, Chen, Xie 2. Performance of GVR

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Cheng, Chen, Chen, Xie 3. k-subset Generation

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Cheng, Chen, Chen, Xie 3. k-subset Generation

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Cheng, Chen, Chen, Xie 4. Verification and Refinement

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Cheng, Chen, Chen, Xie 5. Time Analysis

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Cheng, Chen, Chen, Xie 6. Gaussian Distribution

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Cheng, Chen, Chen, Xie Conclusion We proposed an efficient evaluation framework for T-k- PNN query We proposed various techniques: - k-bound to filter away those unqualified objects - PCS to reduce the number of k-subsets - verification/refinement methods to avoid exact calculation Future Work - extend the techniques to other queries

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Cheng, Chen, Chen, Xie Reference [TDRP98] P. A. Sistla, O. Wolfson, S. Chamberlain, and S. Dao,“Querying the uncertain position of moving objects,” in Temporal Databases: Research and Practice, 1998. [SSDBM99] D.Pfoser and C. Jensen, “Capturing the uncertainty of moving-objects representations,” in Proc. SSDBM, 1999. [VLDB04a] A. Deshpande, C. Guestrin, S. Madden, J. Hellerstein, and W. Hong, “Model-driven data acquisition in sensor networks,” in Proc. VLDB, 2004. [ICDE06] C. Böhm, A. Pryakhin, and M. Schubert, “The gauss-tree: Efficient object identification in databases of probabilistic feature vectors,” in Proc. ICDE, 2006. [ICDE07a] V. Ljosa and A. K. Singh, “APLA: Indexing arbitrary probability distributions,” in Proc. ICDE, 2007. [SIGMOD03] R. Cheng, D. Kalashnikov, and S. Prabhakar, “Evaluating probabilistic queries over imprecise data,” in Proc. ACM SIGMOD, 2003. [ICDE07b] J. Chen and R. Cheng, “Efficient evaluation of imprecise location-dependent queries,” in Proc. ICDE, 2007. [VLDB06a] M. Mokbel, C. Chow, and W. G. Aref, “The new casper: Query processing for location services without compromising privacy,” in VLDB, 2006. [TKDE92] D. Barbara, H. Garcia-Molina, and D. Porter, “The management of probabilistic data,” TKDE, vol. 4, no. 5, 1992. [VLDB04b] N. Dalvi and D. Suciu, “Efficient query evaluation on probabilistic databases,” in VLDB, 2004. [VLDB06b] P. Agrawal, O. Benjelloun, A. D. Sarma, C. Hayworth, S. Nabar, T. Sugihara, and J. Widom, “Trio: A system for data, uncertainty, and lineage,” in VLDB, 2006. [VLDB03] G. Iwerks, H. Samet, and K. Smith, “Continuous k-nearest neighbor queries for continuously moving points with updates,” in Proc. VLDB, 2003. [ICDCS07] S. Ganguly, M. Garofalakis, R. Rastogi, and K. Sabnani, “Streaming algorithms for robust, real-time detection of ddos attacks,” in ICDCS, 2007. [AKDDM96] U. Fayyad, G. Piatesky-Shapiro, P. Smyth, and R. Uthurusamy, Advances in Knowledge Discovery and Data Mining. AAAI Press/MIT Press, 1996. [VLDB04c] N. Koudas, B. Ooi, K. Tan, and R. Zhang, “Approximate NN queries on streams with guaranteed error/performance bounds,” in Proc. VLDB, 2004. [VLDB08a] G. Beskales, M. Soliman, and I. Ilyas, “Efficient search for the top-k probable nearest neighbors in uncertain databases,” in VLDB, 2008. [VLDB06c] O. Mar, A. Sarma, A. Halevy, and J. Widom, “ULDBs: databases with uncertainty and lineage,” in VLDB, 2006.

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Cheng, Chen, Chen, Xie Reference [VLDB07a] L. Antova, C. Koch, and D. Olteanu, “Query language support for incomplete information in the maybms system,” in Prof. VLDB, 2007. [SIGMOD08a] S. Singh et al, “Orion 2.0: Native support for uncertain data,” in Prof. ACM SIGMOD, 2008. [ICDE08a] Singh et al, “Database support for pdf attributes,” in Proc. ICDE, 2008. [TKDE04] R. Cheng, D. V. Kalashnikov, and S. Prabhakar, “Querying imprecise data in moving object environments,” IEEE TKDE, vol. 16, no. 9, Sept. 2004. [DASFAA07] H. Kriegel, P. Kunath, and M. Renz, “Probabilistic nearest-neighbor query on uncertain objects,” in DASFAA, 2007. [MUD08] Y. Qi, S. Singh, R. Shah, and S. Prabhakar, “Indexing probabilistic nearest-neighbor threshold queries,” in Proc. Workshop on Management of Uncertain Data, 2008. [TKDE08] X. Lian and L. Chen, “Probabilistic group nearest neighbor queries in uncertain databases,” IEEE Trans. On Knowledge and Data Engineering, vol. 20, no. 6, 2008. [ICDE08b] R. Cheng, J. Chen, M. Mokbel, and C. Chow, “Probabilistic verifiers: Evaluating constrained nearest-neighbor queries over uncertain data,” in Proc. ICDE, 2008. [VLDB05] Y. Tao, R. Cheng, X. Xiao, W. K. Ngai, B. Kao, and S. Prabhakar, “Indexing multi-dimensional uncertain data with arbitrary probability density functions,” in Proc. VLDB, 2005. [VLDB07b] J. Pei, B. Jiang, X. Lin, and Y. Yuan, “Probabilistic skylines on uncertain data,” in Proc. VLDB, 2007. [SIGMOD08b] X. Lian and L. Chen, “Monochromatic and bichromatic reverse skyline search over uncertain databases,” in Proc. SIGMOD, 2008. [ICDE07c] M. Soliman, I. Ilyas, and K. Chang, “Top-k query processing in uncertain databases,” in Proc. ICDE, 2007. [SIGMOD08c] M. Hua, J. Pei, W. Zhang, and X. Lin, “Ranking queries on uncertain data: A probabilistic threshold approach,” in Proc. SIGMOD, 2008. [VLDB08b] V. Rastogi, D. Suciu, and E. Welbourne, “Access control over uncertain data,” in Proc. VLDB, 2008. [VLDB08c] C. Koch and D. Olteanu, “Conditioning probabilistic databases,” in Proc. VLDB, 2008. [VLDB08d] R. Cheng, J. Chen, and X. Xie, “Cleaning uncertain data with quality guarantees,” in Proc. VLDB, 2008. [SIGMOD84] A. Guttman, “R-trees: A dynamic index structure for spatial searching,” Proc. of the ACM SIGMOD Int’l. Conf., 1984.

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Cheng, Chen, Chen, Xie Q & A Thanks!

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