Christian B

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Christian Böhm Hans-Peter Kriegel,Ludwig
Maximilians Universität MünchenA Cost Model
and Index Architecture for the Similarity Join
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Feature Based Similarity
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Simple Similarity Queries

• Specify query object and
• Find similar objects range query
• Find the k most similar objects nearest
  neighbor q.

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Join Applications Catalogue Matching

• Catalogue matching
• E.g. Astronomic catalogues

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Join Applications Clustering

• Clustering (e.g. DBSCAN)
• Similarity self-join

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R-tree Spatial Join (RSJ)
procedure r_tree_sim_join (R, S, e) if IsDirpg
(R) Ù IsDirpg (S) then foreach r Î R.children
do foreach s Î S.children do if
mindist (r,s) e then CacheLoad(r)
CacheLoad(s) r_tree_sim_join (r,s,e)
else ( assume R,S both DataPg ) foreach
p Î R.points do foreach q Î S.points do
if p - q e then report (p,q)
R
S
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Cost Modeling

• Single similarity queries Access prob. of pages
  modeled using the concept of Minkowski Sum

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Cost Modeling

• Binomial formula

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Cost Modeling

• Mating probability of index pages
• Probability that distance between two pages e
• Two-fold application of Minkowski sum

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Page Capacity Optimization

• Cost model can determine index selectivity which
  depends on various parameters
• Page capacity (number of stored points) is an
  important parameter
• Known from similarity search Page capacity
  optimization yields considerable improvement

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Analysis of the Index Overhead

• Assuming 100 selectivity (index doesnt work)How
  much more expensive is index usage ?
• CPU
• Distance betw. boxes more expensive to compute
  than distance betw. points a 5
• Smaller capacity ? more box distance computations

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Analysis of the Index Overhead

• Disk I/O
• High constant cost per page access (move disk
  head)
• Page access is by factor b 10000 / d more
  expensive than continuous reading of a point
• Smaller capacity ? more disk head movement

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Analysis of the Index Overhead

• What selectivity is needed that index pays off ?

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Optimization

• I/O cost functionis optimized by
• CPU cost functionis optimized by

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Optimization

• I/O cost
• Large capacity optimum (several 10,000 points,
  typically)
• CPU cost
• Small capacity optimum (lt 100 points, typically)
• No compromise achievable

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Multipage Index (MuX)
separate optimization

• CPU-performance like CPU optimized index
• I/O- performance like I/O optimized index

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Experimental Evaluation
Uniform 4D
Uniform 8D
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Experimental Evaluation
CAD Data 16D
Color Images 64D
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Conclusions

• Summary
• High potential for performance gains of the
  similarity join by page capacity optimization
• Necessary to separately optimize I/O and CPU
• Future research potential
• Similarity join for metric index structures
• Approximate similarity join
• Parallel similarity join algorithms

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Consequences

• Assume for I/O optimization selectivity 100
• Page accesses in a nested block loop like style

fill cache with pages of R (1 page free)
foreach S-page s do
if s joins some of the cached R-pg then
load (s)
foreach joining R-page r in cache do
if mindist(r,s) lt e then join (r,s)
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R-tree Spatial Join (RSJ)
procedure r_tree_sim_join (R, S, e) if IsDirpg
(R) Ù IsDirpg (S) then foreach r Î R.children
do foreach s Î S.children do if
mindist (r,s) e then CacheLoad(r)
CacheLoad(s) r_tree_sim_join (r,s,e)
else ( assume R,S both DataPg ) foreach
p Î R.points do foreach q Î S.points do
if p - q e then report (p,q)
R
S