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Approximate NN queries on Streams with Guaranteed Error/performance Bounds

Approximate NN queries on Streams with Guaranteed Error/performance Bounds. Nick Koudas @ AT&T labs-research Beng Chin Ooi , Kian-Lee Tan , Rui Zhang @ National University of Singapore. Problem. Problem: kNN search. Environment: data stream (one scan; memory constraint).

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Approximate NN queries on Streams with Guaranteed Error/performance Bounds

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  1. Approximate NN queries on Streams with Guaranteed Error/performance Bounds Nick Koudas @ AT&T labs-research Beng Chin Ooi , Kian-Lee Tan , Rui Zhang @ National University of Singapore

  2. Problem • Problem: kNN search. • Environment: data stream (one scan; memory constraint). • Approximate Solution: e-approximate kNN (ekNN). • Motivation: Applications in which absolute error is preferable or more straightforward. IP: 137.132.48.120 137.132.48.121 …

  3. Two Optimization Problems: • memory optimization for a given error bound: given an error bound e, use as little memory as possible to answer ekNN queries. • error minimization for a given memory size: given a fixed amount of memory, achieve the best accuracy for ekNN queries. • Requirements: • One scan algorithm. • Satisfies the constraints. • Efficient updates and query processing.

  4. A Framework • Divide space into equal square-shaped cells. • Maintain at most K points in each cell. • For any k≤K, absolute error of kNN distance is bounded by dM, the maximum distance within a cell. For Euclidean distance: dM = where d is dimensionality; u is the number of cells each dim is divided to.

  5. Maintenance of the Points--aDaptive Indexing on Streams by space-filling Curves (DISC) • Cells are not explicitly maintained, only points. • Cells linearized according to Z-curve. • Z-value of the cell is the key of a point. • Points maintained in a B*-tree. • An efficient merge-cell algorithm possible.

  6. Algorithm: Build index • m: the order of Z-curve, 2m cells each dim. • If e given, , we get . me is integer, so • If memory constraint given, set a large enough m. • Build index • Initialize m • Read a record P, calculate Z-value, search the B*-tree and find out Nc: number of existing points in the cell P belongs to. • If Nc <K • Insert P to the B*-tree. • Else • Discard one and insert P. • If memory runs out //this only happens for the error minimization problem • Merge cells and let m=m-1 • Go back to Step 2 (Read next record)

  7. Algorithm: Merge Cells • General Merge-Cell • Apply to any structure. • For each new cell, find all the points of the old cells in it, and merge them. • Bulk Merge-Cell • Only apply to DISC. • Scan all the leaf pages once.

  8. Algorithm: KNN search • W: a window query centered at the center of the cell Q is in; and with gradually increasing side length s. • Find the kNN to Q within W. • If the kNN distance is no larger than the distance between the nearest side of W to Q and Q, search terminates; • Else increase s by 1/u .

  9. Experiments

  10. Questions ?

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