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Clustered Pivot Tables forI/O-optimized Similarity SearchJuraj Moško, Jakub Lokoč, Tomáš Skopal

Department of Software Engineering

Faculty of Mathematics and Physics

Charles University in Prague

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Presentation outlineSimilarity search in metric spaces

Pivot tables

Clustered pivot tables◦Static variant◦Dynamic variant

Experiments

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Similarity searchSuitable for unstructured data, query often not

in DB

Similarity is often modeled by a metric distance

Expensive distance functions - EMD, SQFD, DTW, …

Metric indexing◦ Based on lower-bounding◦ If abs(d(p, q) – d(p, o)) > r

filter out object o

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Pivot tables Simple yet efficient main memory metric index Having k static pivots Pi and database S of n objects

Oj, pivot table stores all the distances d(Pi, Oj) in the matrix of size k x n

Pivot tables = two structures - distance matrix + data file

Cheap filtering of non-relevant objects (lower-bounding)

Non-filtered objects are refined by the original expensive distance function

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Clustered pivot tablesWhat if the pivot table does not fit into

main memory?

Solution 1 – just slice datafile◦+ simple to construct◦ - sequential scan => high I/O cost

Solution 2 – reorganize and slice datafile◦+ similar objects in one page (page = cluster)

=> higher probability that all objects are filtered=> lower I/O cost

◦ - metric clustering is expensive

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Metric clustering? M-tree!Dynamic, persistent, balanced

structureLeaf node represents cluster of similar

objectsMany construction strategies

considering quality of M-tree hierarchy with complexity < O(n2)◦Single/Multi/Hybrid-way leaf

selection◦Slim-down algorithm◦Reinsertions

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Static CPTData file = objects serialized from M-

tree leaves◦Classic pivot table reorganizing input

Fixed page size in a paged data file

Preserve M-tree?◦ Future re-indexing◦ Query processing

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Dynamic CPTData file = set of M-tree leaves

◦ Distance matrix connected to the M-tree leaves

Internal fragmentation◦ M-tree leaves contain different number of

data objects, utilization is not 100%Dynamic operations do not

degenerate created clusters

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CPT - QueryingFiltering based on lower-bounding

If all data objects from one page are filtered out, page from data file is not loaded into memory => I/O optimization

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CPT - Querying problemsProblem 1 – LAESA kNN algorithm

sorts DB objects according to their lower bound to the query object – not optimal for I/O cost◦ Solution - CPT does not sort objects =>

objects are processed sequentially

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CPT – Querying problemsProblem 2 – in CPT the dynamic radius

decreases slower during the kNN processing◦ Solution - First bunch of objects is not

clustered

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CPT – Querying problemsProblem 2 – in CPT the dynamic radius

decreases slower during the kNN processing◦ Solution - First bunch of objects is not

clustered

Qx

Qx

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Experiments (1)2 real datasets

◦subset of CoPhIR, subset of Corel2 synthetic datasets

◦Cloud, PolygonSetWe considered more M-tree variants

◦Single/Multi way leaf selection◦Reinsertions

Measured I/O costCPT vs. PT vs. M-tree

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Experiments (2)

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Experiments (3)

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ConclusionWe have designed I/O-optimized

method for persistent pivot tables

Future work◦Thorough experiments on SSD disks◦Use other metric clustering

techniques

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Thank you