Current, Up-to-date Day to day operations Application Oriented Flat Relational Tables of medium size
• On-Line Analytical Processing ● ● ● ●
Historical, Non Volatile Decision Support Subject Oriented Multidimensional processing
Multidimensional Model World Sales ($b) Paris Islamabad Bandung Kuala Lumpur
Cell
Bread Wine Milk Dimensions
2006
2005
2004
2003
Flour
Measure
Members
Conceptual Level • Two Kinds of Tables – Several Dimensional tables – One Fact Table
• Example
World Sales
– Dimensional Tables • Product, Location, Time
Location
– Fact Table • World Sales
Product Time
Conceptual Level • Additional Knowledge – Hierarchies on Dimensions
• Example – Product < Category – Day < Month < Year – City < Country < Continent
Star Schema • Specific Features – Sizes: Dimensional Tables 4 ISB KL
7 8
6
2003
2004
4
7
3
8 6
Wine Milk
8
6
1
5
Flour
6
9
Wine
8 6
Milk Flour
5 2005 2006
9
6
2006
1
2005
5
3
Bread
5
Bread
2003 2004
Paris ISB BDG KL 4 2
Projection Time, Product
4
7
3
8 6
Wine Milk
8
6
1
5
Flour
Wine Milk
38 47 76 98
Flour
85 46 55 63
75 84 72 49 2005 2006
6
24 48 52 25
2004
9
3
Bread
2003
Bread
2006
1
2005
5
2003 2004
Paris ISB BDG KL 4 2
Roll-Up Location: City < Continent EU Asia 56 28 32 40
Bread
47 28 64 42 38 56 50 65
Milk Flour
7
3
8 6
Wine Milk
8
6
1
5
Flour
2005 2006
Wine
4
2004
65 43 26 72
6
2003
9
3
2006
1
2005
5
Bread
2003 2004
Paris ISB BDG KL 4 2
Efficiency Issues • Joins with the Fact Table – Join indexes – Bit map indexes
• Computation of Aggregations – Pre-compute and store intermediate aggregations – Cube by operator: pre-computation of projections
Further Issues • « Sorting » a cube according to the measure values – Without changing its content – Problem: no standard algorithm
• Summarizing a data cube – Change the content – Identify sub-cubes whose measure values are « almost » the same
Sorting a Cube
3
4
Wine Milk
6
5
1
3
9
7
4 6
Flour
8
4
1
2
2005
2006
Increasing Bottom-Up Left-Right
Milk Bread Wine Flour
4
6
7
9
3
4
6
7
1
3
5 6
1 2
4
8 2003
6
2004
7
2005 2006
Bread
2003 2004
• Sort the Data Cube According to Criteria on the Measure Values
– Achieved based on Switch – Polynomial algorithm under restrictions
Optimal Representations • Sorting is not always possible • With null values, no polynomial algorithm to compute the « best » representation – According to the number of misplaced cells
• Use AI techniques to reach one solution – Hill climbing: at each step the best solution is chosen – Genetic algorithm
Data Cube Summarization • Compute areas of the data cube in which the sales have (almost) the same value If CITY ∈ [C1, C2] And PRODUCT ∈ [P1, P2] Then SALES is 6
• Apply Apriori over the set of dimensions to optimize the scans of the Data Cube
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bDepartment of Mathematics, Washington University, St. Louis, Missouri 63130, USA;. cMathematical Institute, JustusâLiebigâUniversity Giessen, 35392 ...
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