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iDistance -- Indexing the Distance iDistance -- Indexing the Distance An Efficient Approach to KNN Indexing C. Yu, B. C. Ooi, K.-L. Tan, H.V. Jagadish. Indexing the distance: an efficient method to KNN processing, VLDB 2001.

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Similarity queries: Similarity range and KNN queries Similarity range query: Given a query point, find all data points within a given distance r to the query point. KNN query: Given a query point, find the K nearest neighbours, in distance to the point. r Kth NN Query Requirement

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SS-tree : R-tree based index structure; use bounding spheres in internal nodes Metric-tree : R-tree based, but use metric distance and bounding spheres VA-file : use compression via bit strings for sequential filtering of unwanted data points Psphere-tree : Two level index structure; use clusters and duplicates data based on sample queries; It is for approximate KNN A-tree: R-tree based, but use relative bounding boxes Problems: hard to integrate into existing DBMSs Other Methods

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Basic Definition Euclidean distance: Relationship between data points: Theorem 1: Let q be the query object, and Oi be the reference point for partition i, and p an arbitrary point in partition i. If dist(p, q) <= querydist(q) holds, then it follows that dist(Oi, q) – querydist(q) <= dist(Oi, p) <=dist(Oi,q) + querydist(q).

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Basic Concept of iDistance Indexing points based on similarity y = i * c + dist (Si, p) S1S1 S2S2 S3S3 SkSk S k+1 Reference/anchor points S1S1 S2S2 S3S3... d S1+dS1+d c

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Data points are partitioned into clusters/ partitions. For each partition, there is a Reference Point that every data point in the partition makes reference to. Data points are indexed based on similarity (metric distance) to such a point using a CLASSICAL B+-tree Iterative range queries are used in KNN searching. iDistance

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Searching region is enlarged till getting K NN. A range in B+-tree KNN Searching... S1S1 S2S2

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dist (S 1, q) S2S2 S1S1 Increasing search radius : r Dis_min(S 1 ) Dis_max(S 1 ) q S1S1 S2S2 0 dist (S 2, q) Dis_max(S 2 ) Dis_min(S 2 ) Dis_min(S 1 ) Dis_max(S 1 ) Dis_max(S 2 ) r dist (S 1, q) dist (S 2, q) KNN Searching

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Q2

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dist (S, q) Inefficient situation: When K= 3, query sphere with radius r will retrieve the 3 NNs. Among them only the o 1 NN can be guaranteed. Hence the search continues with enlarged r till r > dist(q, o 3) S q r o2o2 o1o1 o3o3 Over Search?

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Stopping Criterion Theorem 2: The KNN search algorithm terminates when K NNs are found and the answers are correct. Case 1: dist(furthest(KNN’), q) < r Case 2: dist(furthest(KNN’), q) > r r Kth ? In case 2

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(centroid of hyperplane, closest distance) (external point, closest distance) Space-based Partitioning: Equal-partitioning

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(centroid of hyper-plane, furthest distance) Space-based Partitioning: Equal-partitioning from furthest points (external point, furthest distance)

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Using external point to reduce searching area Effect of Reference Points on Query Space

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Using (centroid, furthest distance) can greatly reduce search area The area bounded by these arches is the affected searching area. Effect on Query Space

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0.671.0 0.31 0.20 0.70 0 1.0 Using cluster centroids as reference points Data-based Partitioning I

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0.671.0 0.31 0.20 0.70 0 1.0 Using edge points as reference points Data-based Partitioning II

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100K uniform data set Using (external point, furthest distance) Effect of search radius on query accuracy Dimension = 8Dimension = 16 Dimension = 30 Performance Study: Effect of Search Radius

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10-NN queries on 100K uniform data sets Using (external point, furthest distance) Effect of search radius on query cost I/O Cost vs Search Radius

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10-NN queries on 100K 30-d uniform data set Different Reference Points Effect of Reference Points

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KNN queries on 100K 30-d clustered data set Effect of query radius on query accuracy for different partition number Effect of Clustered # of Partitions on Accuracy

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10-NN queries on 100K 30-d clustered data set Effect of # of partitions on I/O and CPU Costs Effect of # of Partitions on I/O and CPU Cost

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KNN queries on 100K, 500K 30-d clustered data sets Effect of query radius on query accuracy for different size of data sets Effect of Data Sizes

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10-KNN query on 100K,500K 30-d clustered data sets Effect of query radius on query cost for different size of data set Effect of Clustered Data Sets

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10-KNN query on 100K 30-d clustered data set Effect of Reference Points: Cluster Edge vs Cluster Centroid Effect of Reference Points on Clustered Data Sets

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10-KNN query on 100K,500K 30-d clustered data sets Query cost for variant query accuracy on different size of data set iDistance ideal for Approximate KNN?

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10-KNN query on 100K 30-d clustered data sets C. Yu, B. C. Ooi, K. L. Tan. Progressive KNN search Using B+- trees. Performance Study -- Performance Study -- Compare iMinMax and iDistance

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iDistance vs A-tree

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Summary of iDistance iDistance is simple, but efficient It is a Metric based Index The index can be integrated to existing systems easily.

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