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**High-dimensional Similarity Join**

Presented by Yang Xia Wongsodihardjo, Hariyanto Wang Hao

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**Agenda -kdb tree join Introduction Motivation R*-tree based join**

Epsilon grid order join Summary

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Introduction Extracting knowledge from large multi-dimensional databases. Many data mining algorithms require to process all pair of points which have a distance not exceeding a user-given parameter . The operation of generating all such pairs is in essence a similarity join. Data mining algorithms can be directly performed on top of a similarity join.

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Motivation Conventional joining algorithms cannot be directly applied to high-D similarity join, such as nested-loop join, sort-merge join, and hash-based join. Make use of the index built on the high-D data.

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**Efficient Processing of Spatial Joins Using R-trees**

by T. Brinkhoff, H. P. Kriegel, and B. Seeger SIGMOD 1993 Presented by Hariyanto Wongsodihardjo 6 September 2001

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**Efficient Processing of Spatial Joins Using R-trees**

Presenting a study of spatial join processing using R-trees, particularly R*-trees, which is one of the most efficient members of the R-tree family Presenting several techniques for improving spatial join execution time with respect to CPU and I/O time

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**R-tree Basic Algorithms**

Let S be a query rectangle of a window query. The query is performed by starting in the root and computing all entries which rectangles intersects S For these entries, the corresponding child nodes are read into main memory and the query is performed like in the root node The efficiency of queries depends on the goodness how R-trees assign rectangles to nodes.

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**A First Approach of a Spatial Join for R-trees**

CPU Time Tuning The consumption of CPU time is proportional to the number of floating point comparisons required for computing the join condition (i.e. the test whether two rectangles intersect). Several constraints should be considered The storage utilization and the query performance of the original R*-tree should not be affected Expensive preprocessing steps for the nodes of the R*-tree should be avoided The algorithm should be robust and easy to implement

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**CPU-Time and I/O-Time Tuning**

CPU-Time Tuning Restricting the search space Spatial Sorting and plane sweep I/O-Time Tuning Local plane-sweep order with pinning Local z-order

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**Restricting the search space**

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**Restricting the search space**

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**Restricting the search space**

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**Spatial sorting and plane sweep**

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**Spatial sorting and plane sweep**

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**Spatial sorting and plane sweep**

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**Spatial sorting and plane sweep**

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**Local plane-sweep order**

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**Local plane-sweep order**

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**Local plane-sweep order with pinning (SJ4)**

Sequence for local plane-sweep order on example 2 is II, I,IV, III and the read schedule is <r1, s2, s1, r2, s2, r4, r3> Pinning algorithm is based on the degree of the rectangles of both entries. The degree of an rectangle E is given by the number of intersections between rectangle E and the rectangles which belong to entries of the other tree not processed until now. Thus for ex. 2 the read schedule is <r1, s2, r4, r3, s1, r2>. The page whose rectangle has a max degree is pinned and the join is performed for the pinned page.

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Local z-order (SJ5)

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Local z-order (SJ5) Compute intersection between each rectangle of R with all rectangles of S Sort resulting rectangles on the spatial location of their centers Use z-ordering to sort resulting rectangles Then pin pages as before. The sequence for Figure 7 is I, II, III, V, IV and the read schedule is <s1, r2, r1, s2, r4, r3, s3>.

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**I/O Performance Comparison**

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**I/O Performance Comparison**

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**Conclusion R* tree join algorithm is straightforward**

R* tree join algorithm improves CPU-time by applying spatial sorting and restricting the search space R* tree join algorithm improves I/O-time by applying local sweep order with pinning or local z-order

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**High-dimensional similarity joins ( tree)**

Presented By Yang Xia References:K. Shim, R. Srikant, and R. Agrawarl, High-dimensional similarity joins, Proc. 13th IEEE Internat. Conf. on Data Engineering, 1997, pp

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Introduction tree is a main-memory data structure optimized for performing similarity joins. It uses the similarity distance limit as a parameter in building the tree. Problem Definition -Self-join -Non-self-join -Distance metric:

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**Problems with Current Indices**

Number of Neighboring Leaf Nodes Storage Utilization Traversal Cost Build Time Skewed Data

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tree Definition The co-ordinates of the points in each dimension lie between 0 and +1. Start with a single leaf node. Whenever the number of points in a leaf node exceeds a threshold, the leaf node is split. If the leaf node was at level i, the i dimension is used for splitting. The node is split into parts.

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Example of tree

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**Similarity Join using the tree**

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Memory Management Main-memory can hold all points within a 2 distance on the first dimension.

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Memory Management Main-memory cannot hold all points within a 2 distance on the first dimension.

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Design Rationale Biased Splitting: The dimension used in previous split is selected again for splitting as long as the length of the dimension in the bounding rectangle of each resulting leaf node is at least . Sized Splitting: When we split a node, we split the node in sized chunks.

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**Design Rationale Number of Neighboring Leaf Nodes. Space Requirements.**

Traversal Cost. Build time. Skewed data.

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An example

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Experiments Synthetic Data Parameters

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Experiments(1)

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Experiments(2)

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Experiments(3)

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Conclusions tree reduces the number of neighbor leaf nodes that are considered for the join test. tree reduces the traversal cost of finding appropriate branches in the internal nodes. The storage cost for internal nodes is independent of the number of dimensions.

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**Presented By Wang Hao 6 September 2001**

Epsilon Grid Order: An Algorithm for the Similarity Join on Massive High-Dimensional Data Christian Bhm, Bernhard Braunmller, Florian Krebs, and Hans-Peter Kriegel SIGMOD 2001 Presented By Wang Hao 6 September 2001

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**Motivation Indexing Based Join Join without Index**

R-tree family, MuX (Multipage Index) tree, etc.. Optimization conflict between CPU and IO [BK01]. Optimize CPU: fine-gained partitioning with page capacities of a few points. Optimized IO: large block size requires less IO. Join without Index Seeded tree, spatial hash join, -kdb tree, etc.. Not scalable to large data sets. -kdb tree: cache size can be from 36% to 60% of database size.

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**Design Objectives Join without Index. Optimize both CPU and IO.**

Scalable to large data set of size well beyond 1GB.

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**Basic Ideas Define a sort order of data: epsilon grid order.**

Laying an equi-distant grid cell with cell length , over the data space and comparing the cells lexicographically. Use external sort to sort the data. Schedule the IO carefully during join phase.

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Epsilon Grid Order For two vectors p, q is true iff there exists a dimension di, such that Epsilon grid order is a strict order: irreflexive, asymmetric, and transitive.

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**Epsilon Grid Order (Cont.)**

A point with cannot be a join mate or p, of any point p’ which is not A point with cannot be a join mate or p, of any point p’ which is not

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**I/O Scheduling Using the Grid Order**

Unbuffered IO operations. Example: IO Units in a 2-D data space

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I/O Scheduling (Cont.) Illustration: Pairs of IO units that must be considered for join. In the picture, each entry in the matrix stands for one pair of IO Units. IO thrashing effects

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Scheduling Mode

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Scheduling Algorithm

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**Joining Two IO Units Active dimensions**

Minlen: minimum of length of sequences for join.

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**Optimization Potentials**

Use larger sequences to optimize IO. Optimize minlen for minimal CPU processing time. Comparing with -kdb tree and MuX tree, no directory is constructed. The only space overhead is the recursion stack: O(log n) Other possible optimizations Modification of sort order. Optimization in the recursion in join_sequence.

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**Experiments Settings: Buffer memory: 10% of database size.**

Use Euclidean distance. Distance parameter : determined using algorithm in [SEKX98] such that they are suitable for clustering. Compare with Nested-loop join, Z-ordering R-tree based join, and MuX tree based join.

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**Experiments on Uniformly Distributed 8-D Data.**

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**Experiments on Real 16-D Data from CAD Database.**

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**Conclusions and Future work**

Define a strict order: epsilon grid order. A sophisticated scheduling algorithm. Several optimization techniques. Experiments show it outperforms competitive algorithms for data sets with size up to 1.2 GB. Future work Parallel version of the join algorithm. Extend the cost model to query optimizer.

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Overall Summary We have covered three joining algorithms: R* tree-based join, e-kdb tree join, and epsilon grid order join. Specific algorithms have been proposed to perform similarity join for each of the following cases: Both data set have index, Only one data set has index, None of them have index. High-D similarity joins can be applied in data mining algorithms such as clustering.

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**Resource Links Readings on High-dimensional Similarity Join**

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