Data Cube Materialization: Choices and Strategies for Cube Computation

Introduction

Data cube computation is an essential task in data warehouse implementation. The precomputation (materialization) of all or part of a data cube can greatly reduce response time and enhance OLAP performance. However, it is challenging because it may require huge computational time and storage space.

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Choices for Data Cube Materialization

There are three main choices for data cube materialization:

1. Full Materialization

• All cuboids in the data cube lattice are precomputed and stored.
• Provides the fastest query response time since all aggregations are already computed.
• Disadvantage: Requires enormous storage space and high computation cost, especially when the number of dimensions is large.

2. No Materialization

• No cuboids are precomputed; all queries are computed on-the-fly from the base cuboid (raw data).
• Disadvantage: Very slow query response time, especially for complex OLAP queries on large datasets.

3. Partial Materialization

• Only a selected subset of cuboids is precomputed and stored.
• Provides a balance between storage cost and query response time.
• This is the most practical and widely used approach.

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Strategies for Cube Computation

i) Full Cube Computation

• Computes every possible cuboid for all combinations of dimensions.
• For an n-dimensional data cube, the total number of cuboids is 2^n.
• Suitable only when n is small and storage is not a constraint.

ii) Iceberg Cube

• Instead of computing the complete cube, only those cuboids (cells) that satisfy a minimum support threshold (called the iceberg condition) are computed.
• Cells with aggregate values below the threshold are discarded.
• Example condition: HAVING count(*) >= min_support
• This avoids computing and storing many low-value, sparse cells, saving both time and space.
• The name comes from the idea that only the "tip of the iceberg" (the significant aggregates) is materialized.

iii) Shell Cube (Partial Materialization)

• A strategy for partial materialization where only cuboids involving a small number of dimensions (such as 3 to 5) are precomputed.
• These precomputed cuboids form a shell cube.
• For example, in an n-dimensional data cube, we could compute all cuboids with 3 dimensions or less, resulting in a Shell Cube of size 3.
• Remaining higher-dimensional cuboids are computed on-the-fly using the precomputed shell cuboids.
• Advantage: Significantly reduces storage while still speeding up many common queries.

iv) Closed Cube

• Only closed cells are materialized. A cell is closed if there is no supercell with the same measure value (e.g., same count).
• Reduces redundancy in stored data by avoiding cells whose values can be directly derived from a more specific cell.

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Summary Table

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Conclusion

Among all strategies, partial materialization (especially Iceberg Cube and Shell Cube) provides the best trade-off between storage cost and query performance, making them the most practical choices for real-world data warehouse implementations.

Data Discretization and Data Preprocessing Tasks

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Part 1: Data Discretization (Definition)

Data discretization is a technique of data preprocessing that transforms continuous (numeric) data into discrete intervals or categories. Instead of working with raw continuous values, the data is divided into a finite number of intervals or bins, and each interval is assigned a label or concept.

For example, a continuous attribute like age (values: 1 to 100) can be discretized into categories such as:

• Young: 1-30
• Middle-aged: 31-60
• Senior: 61-100

Discretization reduces the number of distinct values for a continuous attribute, making data easier to handle in mining tasks like classification and association analysis.

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Part 2: Tasks for Data Preprocessing

Raw data collected from the environment can be dirty - meaning it is incomplete, noisy, or inconsistent. Poor quality data leads to poor quality mining results. Therefore, data preprocessing is a critical step in data mining.

The main tasks (categories) of data preprocessing are:

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1. Data Cleaning

• The process of removing noise and inconsistent data from the dataset.
• Handles missing values, smooths noisy data, identifies or removes outliers, and resolves inconsistencies.
• Example: Filling in missing age values with the mean age of the dataset.

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2. Data Integration

• The process of combining data from multiple sources (databases, files, data cubes) into a coherent data store.
• Handles issues like redundancy, duplicate records, and conflicting attribute names.
• Example: Merging customer data from two different databases into one unified dataset.

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3. Data Transformation

• Data is transformed into forms appropriate for mining.
• Includes operations such as:
  • Normalization: Scaling values to a small range (e.g., 0.0 to 1.0)
  • Aggregation: Summarizing data
  • Generalization: Replacing low-level data with higher-level concepts
• Example: Normalizing salary values so they fall between 0 and 1.

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4. Data Reduction

• The process of reducing the volume of data while maintaining analytical integrity.
• Produces a reduced representation of the dataset that is much smaller in size yet produces the same (or nearly the same) analytical results.
• Techniques include: dimensionality reduction, numerosity reduction, and data compression.
• Example: Removing irrelevant attributes that do not contribute to the mining task.

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5. Data Discretization

• Transforms continuous attributes into discrete intervals or categories as defined above.
• Reduces the number of distinct values for continuous attributes.
• Particularly useful for algorithms that require categorical data.
• Example: Converting income (continuous) into Low, Medium, High categories.

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Summary Table

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Note: Data preprocessing is one of the most critical steps in data mining because the quality of data directly affects the quality of the mining results.