Approximate Spatial Query Processing Using Raster Signatures

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Approximate Spatial Query Processing Using
Raster Signatures
Federal University of Rio de Janeiro

• Leonardo Guerreiro Azevedo, Rodrigo Salvador
  Monteiro, Geraldo Zimbrão Jano Moreira de Souza
• Coppe Graduate School of Engineering
• Institute of Mathematics Computer Science
  Department

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Common Spatial Queries

• Area of polygon
• Area of polygon within window
• Spatial Joins
• polygon ? polygon, polygon ? polyline polyline
  ? polyline
• Distance
• Buffer
• Perimeter
• Topological queries

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Common Spatial Queries

• Approximate Area of polygon
• Approximate Area of polygon within window
• Approximate Spatial Joins
• polygon ? polygon, polygon ? polyline polyline
  ? polyline
• Approximate Distance
• Approximate Buffer
• Approximate Perimeter
• Approximate Topological queries

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Approximate Answers to Spatial Queries

• What is an approximate answer?
• If the exact result is a number, the
  approximate result will be a number and a
  confidence interval
• If not, the graphical display of approximate
  answers is something like a fuzzy map

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Motivation

• The increase of storage capacity
• The decrease of hardware costs
• Disk access time is still high
• Complex queries
• Data stored in devices that are not on-line.

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Motivation

• Approximate answer may be enough
• exact answers are itself approximations
• Approximate answers can be computed quickly
• Spatial query processing
• Scale
• Quality
• Round-off errors

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Scenarios and Applications

• Decision Support System
• Increasing business competitiveness
• More use of accumulated data
• Data mining
• During drill down query sequence in ad-hoc data
  mining
• Earlier queries in a sequence can be used to find
  out the interesting queries.
• Data warehouse
• Performance and scalability when accessing very
  large volumes of data during the analysis
  process.

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Scenarios and Applications

• Query optimization
• To define the most efficient access plan for a
  given query
• Distributed data recording and warehousing
  environments
• Data may be remote, and even may be unavailable
• Old data can be disposed in order to make room
  for new ones. Therefore it becomes impossible to
  answer to queries on deleted information.

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Scenarios and Applications

• Mobile computing
• An approximate answer may be an alternative
• When the data is not available
• To save storage space

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A framework for approximate query processing
Data environment set-up for providing approximate
answers
New data
Approx. Query Engine
Database
Queries
Responses
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Four Color Raster Signature (4CRS)

• Raster approximation (VLDB98)
• Object representation upon a grid of cells.
• Each cell stores relevant information using few
  bits.
• Grid resolution can be changed
• Precision ? storage requirements
• 4 types of cells

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4CRS Approximation Construction of Signatures
Polygon
4CRS
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Polygon approximate area

• The algorithm is based on the sum of the expected
  area of each cell grid
• Empty cells 0
• Full cells 100
• Weak and Strong cells ? supposing uniform
  distribution
• Weak cells (0, 0.5 interval ? mean 0.25
• Strong cells (0.5, 1) interval ? mean 0.75
• Count the number of each cell type in the
  polygons 4CRS, and multiply these values by the
  presumed cell area.

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Confidence interval

• A measure of answer accuracy
• The polygon area inside weak or strong cell is
  assumed to be uniformly distributed.
• Weak cells
• Strong cells
• Using Central Limit Theorem ? confidence interval
• 95
• 99

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Confidence interval (example)

• Query results
• weak cells 100
• strong cells 120
• full cells 400
• Confidence interval 95
• Weak cells
• Strong cells
• Full cells 400 (full cells have the exact area!)
• Total
• Error between -1.15 and 1.15

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Cell Area Distribution
Weak
Strong
Comparable to an uniform distribution Variance
0.021369 (U 0.020833) Mean 0.246453 (U 0.25)
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Example

• empty cells 55
• weak cells 27
• strong cells 26
• full cells 79
• Approximate area( S weak 0.25 S strong
  0.75 S full ) cellArea
• Exact area 106.40
• Appr. area 105.25
• Error 1.07

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Approximate area of polygon ? window intersection

• This algorithm is similar to the approximate
  polygon area algorithm
• There are two kinds of cell overlap
• The cell may be completely contained by the
  window
• The cell may be partially contained by the window
• proportional to its overlapping area

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Experimental tests

• Computer PC Pentium IV 1,8 GHz, 512 MB RAM
• Page size 2,048 Bytes
• Target to evaluate the use of 4CRS for
  approximate query processing against exact query
  processing related to the following aspects
• Response time
• Storage requirements
• Accuracy
• The algorithms tested were
• Polygon approximate area
• Approximate area of polygon x window intersection
• 100 random windows for each data set (different
  sizes and positions)

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Experimental tests

• Use of R-trees in order to reduce the search
  space.

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Experimental tests

• The polygon real data sets used in the
  experiments consist of township boundaries,
  census block-group, topography, geologic map and
  hydrographic map from Iowa (USA), and Brazilian
  municipalities.

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Approximate polygon area
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Approximate polygon area
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Approximate polygon ? window area
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Approximate polygon ? window area
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Conclusion

• The experimental results demonstrated the
  efficiency of the 4CRS use for approximate query
  processing.
• Storage requirements
• 4CRS has an average of 3.75 of the real data set
  size
• Accuracy
• Approximate area average error of 2.62
• Window query approximate area average error of
  1
• Response time
• Approximate area average 28.41
• Window query approximate area average 7.22
• Disk access
• Approximate area average 1.90
• Window query approximate area average 7.04

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Future works

• Algorithms for the other operations
• Approximate area of polygon x polygon
  intersection algorithm is being evaluated
• Use of approximations for mobile computing