NNH: Improving Performance of Nearest-Neighbor Searches Using Histograms

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NNH Improving Performance of Nearest-Neighbor
Searches Using Histograms

• Liang Jin (UC Irvine)
• Nick Koudas (ATT Labs Research)
• Chen Li (UC Irvine)
• Supported by NSF CAREER No. IIS-0238586
• EDBT 2004

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NN (nearest-neighbor) search

• KNN find the k nearest neighbors of an object.

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Example image search
Query image

• Images represented as features (color histogram,
  texture moments, etc.)
• Similarity search using these features
• Find 10 most similar images for the query image
• Other applications
• Web-page search Find 100 most similar pages for
  a given page
• GIS find 5 closest cities of Irvine
• Data cleaning

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NN Algorithms

• Distance measurement
• For objects are points, distance well defined
• Usually Euclidean
• Other distances possible
• For arbitrary-shaped objects, assume we have a
  distance function between them
• Most algorithms assume a high-dimensional tree
  structure for the datasets (e.g., R-tree).

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Search process (1-NN for example)

• Most algorithms traverse the structure (e.g.,
  R-tree) top down, and follow a branch-and-bound
  approach
• Keep a priority queue of nodes (MBR) to be
  visited
• Sorted based on the minimum distance between q
  and each node
• Improvement
• Use MINDIST and MINMAXDIST
• Reduce the queue size
• Avoid unnecessary disk IOs to access MBRs

Priority queue
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Problem

• Queue size may be large
• 60,000 objects, 32d (image) vectors, 50 NNs
• Max queue size 15K entries
• Avg queue size half (7.5K entries)
• If queue cant fit in memory, more disk IOs!
• Problem worse for k-NN joins
• E.g., 1500 x 1500 join
• Max queue size 1.7M entries gt 1GB memory!
• 750 seconds to run
• Couldnt scale up to 2000 objects!
• Disk thrashing

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Our Solution Nearest-Neighbor Histogram (NNH)

• Main idea
• Utilizing NNH in a search (KNN, join)
• Construction and incremental maintenance
• Experiments
• Related work

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NNH Nearest-Neighbor Histograms
pm
p2
p1
m of pivots
Distances of its nearest neighbors r1, r2, ,
They are not part of the database
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Structure

• Nearest Neighbor Vectors

each ri is the distance of ps i-th NN T length
of each vector

• Nearest Neighbor Histogram
• Collection of m pivots with their NN vectors

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Outline

• Main idea
• Utilizing NNH in a search (KNN, join)
• Construction and incremental maintenance
• Experiments
• Related work

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Estimate NN distance for query object

• NNH does not give exact NN information for an
  object
• But we can estimate an upper bound for the k-NN
  distance ?qest of q

Triangle inequality
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Estimate NN for query object(cont)

• Apply the triangle inequality to all pivots
• Upper bound estimate of NN distance of q

• Complexity O(m)

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Utilizing estimates in NN search

• More pruning prune an mbr if

mbr
MINDIST
q
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Utilizing estimates in NN join

• K-NN join for each object o1 in D1, find its
  k-nearest neighbors in D2.
• Traverse two trees top down keep a queue of pairs

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Utilizing estimates in NN join (contt)

• Construct NNH for D2.
• For each object o1 in D1, keep its estimated NN
  radius ?o1est using NNH of D2.
• Similar to k-NN query, ignore mbr for o1 if

MINDIST
o1
mbr
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More powerful prune MBR pairs
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Prune MBR pairs (cont)
mbr1
mbr2
MINDIST
Prune this MBR pair if
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Outline

• Main idea
• Utilizing NNH in a search (KNN, join)
• Construction and incremental maintenance
• Experiments
• Related work

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NNH Construction

• If we have selected the m pivots
• Just run KNN queries for them to construct NNH
• Time is O(m)
• Offline
• Important selecting pivots
• Size-Constraint Construction
• Error-Constraint Construction (see paper)

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Size-constraint NNH construction

• of pivots m determines
• Storage size
• Initial construction cost
• Incremental-maintenance cost
• Choose m best pivots

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Size-constraint NNH construction

• Given m ( of pivots), assume
• query objects are from the database D
• H(pi,k) doesnt vary too much
• Goal Find pivots p1, p2, , pm to minimize
  object distances to the pivots
• Clustering problem
• Many algorithms available
• Use K-means for its simplicity and efficiency

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Incremental Maintenance

• How to update the NNH when inserting or deleting
  objects?
• Need to shift each vector
• Associate a valid length Ei to each NN vector.

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Outline

• Main idea
• Utilizing NNH in a search (KNN, join)
• Construction and incremental maintenance
• Experiments
• Related work

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Experiments

• Datasets
• Corel image database
• Contains 60,000 images
• Each image represented by a 32-dimensional float
  vector
• Time-series data from ATT
• Similar trends. Report results for Corel data set
• Test bed
• PC 1.5G Athlon, 512MB Mem, 80G HD, Windows 2000.
  
• GNU C in CYGWIN

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Goal

• Is the pruning using NNH estimates powerful?
• KNN queries
• NN-join queries
• Is it cheap to have such a structure?
• Storage
• Initial construction
• Incremental maintenance

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Improvement in k-NN search

• Ran k-means algorithm to generate 400 pivots for
  60K objects, and constructed NNH
• Performed K-NN queries on 100 randomly selected
  query objects.
• Queue size to measure memory usage.
• Max queue size
• Average queue size

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Reduced Memory Requirement
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Reduced running time
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Effects of different of pivots
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Join Reduced Memory Requirement
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Join Reduced running time
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JoinRunning time for different data sizes
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Cost/Benefit of NNH
For 60,000 float vectors (32-d).
Pivot (m) 10 50 100 150 200 250 300 350 400
Construction time (sec) 0.7 3.59 6.6 9.4 11.5 13.7 15.7 17.8 20.4
Storage space (kB) 2 10 20 30 40 50 60 70 80
Incr mantnce. time (ms) 0 0 0 0 0 0 0 0 0
Improved q-size(kNN)() 40 30 28 24 24 24 23 20 18
Improved q-size(join)() 45 34 28 26 26 25 24 24 22
0 means almost zero.
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Conclusion

• NNH efficient, effective approach to improving
  NN-search performance.
• Can be easily embedded into current
  implementation of NN algorithms.
• Can be efficiently constructed and maintained.
• Offers substantial performance advantages.

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Related work

• Summary histograms
• E.g., Jagadish et al VLDB98, Mattias et al
  VLDB00
• Objective approximate frequency values
• NN Search algorithms
• Many algorithms developed
• Many of them can benefit from NNH
• Algorithms based on pivots/foci/anchors
• E.g., Omni Filho et al, ICDE01, Vantage objects
  Vleugels et al VIIS99, M-trees Ciaccia et al
  VLDB97
• Choose pivots far from each other (to represent
  the intrinsic dimensionality)
• NNH pivots depend on how clustered the objects
  are
• Experiments show the differences

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Work conducted in the Flamingo Project on Data
Cleansing at UC Irvine