DenseRegion Based Compact Data Cube

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Dense-Region Based Compact Data Cube

• Presented by Kan Kin Fai

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Outline

• Background
• Introduction to Compact Data Cube
• Pros and cons of the Compact Data Cube method
• Dense-Region Based Compact Data Cube

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Background

• Why is a data cube?
• Some pre-computed aggregates on the underlying
  data warehouse.
• System constraints on materializing data cube(s)
• Disk space, maintenance cost, etc.
• Common approach materialize parts of a data
  cube.
• Alternative use approximation technique
• Reason OLAP applications accept approximate
  answers in many scenarios.

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Introduction to Compact Data Cube

• Compact Data Cube was proposed by Vitter and Wang
  in Approximate Computation of Multidimensional
  Aggregates of Sparse Data Using Wavelets (SIGMOD
  99).
• Main Ideas
• Offline phase perform Haar wavelet transform on
  the underlying data (i.e. the base cuboid) and
  store the k most significant coefficients.
• Online phase process any given query based on
  the k most significant coefficients.

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Introduction to Compact Data Cube

• Basics of Haar wavelet transform
• Building Compact Data Cube
• Thresholding and Ranking
• Answering On-Line Queries

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Introduction to Compact Data Cube

• Basics of Haar wavelet transform
• e.g. S 2, 2, 0, 2, 3, 5, 4, 4

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Introduction to Compact Data Cube

• Basics of Haar wavelet transform
• For compression reasons, the detail coefficients
  are normalized.
• The coefficients at the lower resolutions are
  weighted more heavily.
• Approximates the original signal by keeping only
  the most significant coefficients.
• Requires only O(N) CPU time and O(N/B) I/Os to
  compute for a signal of N values.
• Multidimensional wavelet transform a series of
  one-dimensional wavelet transforms.

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Introduction to Compact Data Cube

• Building the Compact Data Cube
• Problem 1 the size of the multidimensional array
  representing the underlying data is too large
  (assume the data are very sparse).
• Solution Divide the wavelet transform process
  into multiple passes.

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Introduction to Compact Data Cube

• Building the Compact Data Cube
• Problem 2 The density of the intermediate
  results would increase from pass to pass.
• Solution truncate the intermediate
  multidimensional array by cutting off entries
  with small magnitude.
• I/O complexity

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Introduction to Compact Data Cube

• Thresholding and Ranking
• Choice 1 keep the C largest (in absolute value)
  wavelet coefficients.
• Choice 2 keep the C wavelet coefficients with
  the largest weights among the C largest
  coefficients (C lt C).
• The weight of a coefficient equals to the number
  of its dimensions with value zero.

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Introduction to Compact Data Cube

• Answering On-Line Queries
• Space ((d1)k), CPU time

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Pros and cons of the compact data cube method

• Pros
• Requires little disk spaces (a small number of
  disk blocks).
• Responds to on-line query fast.
• Answers OLAP queries more accurately than other
  approximation techniques like histogram and
  random sampling.
• Can progressively refine the approximate answer
  with no added overhead.

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Pros and cons of the compact data cube method

• Cons
• Approximates a vast amount of useless empty cells
  in base cuboid together with useful non-empty
  cells in base cuboid.
• Needs to cut off entries with small magnitude at
  the end of each pass in order to maintain a
  constant amount of I/O operations from pass to
  pass.

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Dense-Region Based Compact Data Cube

• Aim
• To enhance the ability of the compact data cube
  method to handle datasets having
  dense-regions-in-sparse-cube property.
• Main Idea
• To exclude empty cells in base cuboid from
  approximation.
• Two-phase approach
• Compute dense regions in base cuboid.
• Approximate each dense region independently.

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Dense-Region Based Compact Data Cube

• Question 1 how can we find the dense regions
  efficiently?
• Efficient DEnse region Mining (EDEM) algorithm
  proposed by Cheung et al. in DROLAP -- A
  Dense-Region-Based Approach to On-line Analytical
  Processing (DEXA99)

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Dense-Region Based Compact Data Cube

• Basic ideas of EDEM
• Build a k-d tree to store the valid cells.
• Grow dense region covers along boundaries.
• Search dense regions among the covers.
• Complexity of EDEM linear to the number of
  dimensions and sub-quadratic to the number of
  data points.

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Dense-Region Based Compact Data Cube

• Question 2 how should we allocate disk space in
  approximating the dense regions?
• Choice 1 allocate disk space equally to each
  dense regions.
• Choice 2 allocate disk space according to the
  sizes of dense regions.
• Choice 3 order the wavelet coefficients of all
  the dense regions and keep the most significant
  ones (in absolute value).

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Dense-Region Based Compact Data Cube

• Question 3 how should we treat the data points
  outside the dense regions?
• Keep all or keep only significant ones.
• Question 4 how do we answer on-line queries
  using the dense-region based approach?
• Check if a dense region covered by the given
  query.
• Check if the stored coefficients contribute to
  the range sum and compute the amount of
  contribution if needed.

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Dense-Region Based Compact Data Cube

• One favorable side effect
• we may parallelize the construction of compact
  data cube.
• More questions
• How can we handle updates to the underlying data?
• How can we approximate iceberg cube? Can we apply
  the idea of compact data cube to iceberg cube?
• Can compact data cube be used to answer other
  types of OLAP queries besides range-sum?