1. Chapter 4: Dimensions, Hierarchies, Operations, Modeling
Prof. Bayer, DWH, Ch.4, SS 2000

2. Chapter 4.1 Hierarchical Dimensions
Def: Hierarchical Dimensions are composite keys with
an order on the key attributes. Prefixes are allowed as keys.
Ex: dimension Time = ( Year, Month, Day)
legal keys are:
(Year) or
(Year, Month) or
(Year, Month, Day)
Def: Basic facts are values in cells with full foreign keys
Prof. Bayer, DWH, Ch.4, SS 2000

3. Aggregations, Summaries
Def: Aggregations are facts in cells with partial keys. These facts are derived by aggregation functions. In a cube with derived facts the aggregation function must be specified.
Ex: Sales on a monthly basis
Sales (Year, Month) = S Sales (Year, Month, Days)
Aggregation Functions: count, sum, avg, min, max, ...
Prof. Bayer, DWH, Ch.4, SS 2000

4. Note on Aggregations
Aggregations may be stored explicitely in the cube, but then they should be secured by integrity constraints
Aggregations may be virtual and must be computed on demand when needed
i.e., classical tradeoff between storage space, performance, flexibility
Prof. Bayer, DWH, Ch.4, SS 2000

5. Relational Modeling Expand and complete partial key by ALL
(Year, Month, ALL)
(ALL, Month, ALL)
(ALL, ALL, ALL)
to obtain simple and complete relational keys via special symbol ALL
Question: SQL to compute complete cube with all aggregations from base-cube?
Prof. Bayer, DWH, Ch.4, SS 2000

6. Hierarchy Example
Prof. Bayer, DWH, Ch.4, SS 2000

7. Chapter 4.2: OLAP Operations
Def: Roll-up computes higher aggregations from lower aggregations or base facts according to hierarchies
Ex: for base facts (Year, Month, Day) there are 3 roll-up functions:
Roll-up (Year, Month, ALL)
Roll-up (Year, ALL, ALL)
Roll-up (ALL, ALL, ALL)
which are supported in general (canonical roll-ups)
Prof. Bayer, DWH, Ch.4, SS 2000

8. therefore 23 -1 aggregations or in general 2m -1 aggregations
Additional Roll-ups:
(ALL, Month, ALL) etc.
therefore aggregations or in general
2m -1 aggregations
for m hierarchy levels
Note: see later chapters for the support of arbitrary aggregations
Note: for m dimensions with h1, h2, ...hm hierarchy levels there are
different aggregations for a given aggregation function.
Prof. Bayer, DWH, Ch.4, SS 2000

9. Dim1: (4, 5) = cardinality of the dimension levels Dim2: (6, 7, 2)
Size of base cube
2-dim example
Dim1: (4, 5) = cardinality of the dimension levels
Dim2: (6, 7, 2)
(4 5) ( 6 7 2) = Size of base cube 42 20 84
Prof. Bayer, DWH, Ch.4, SS 2000

10. Size of hierarchically aggregated Cube4 - 6 7 2
336 5
840 84
168
120 42 24 20 1
Number of cells per aggregation function 1645
Prof. Bayer, DWH, Ch.4, SS 2000

11. Size of completely aggregated cube4 5 6 7 2 |
1 2 7 14 24
24 x 6 =144
168
5 x 168 = x x 1008 = x 1008 = = 5040 : :
Prof. Bayer, DWH, Ch.4, SS 2000

12. Computation with binary Tree4 5 1 20 4 1 6 1 6 24
120 20 4 1 1 1 7 7 1 7 7 20
168 24 28 4
140
840
120 2 1 2 1 1 1 2 1 2 1 2 1 2 2 1 2
120
140 40 20
336
168 48 24 56 28 8 4
1680
840
240
280
Prof. Bayer, DWH, Ch.4, SS 2000

13. Size of the Cube
Lemma: Given a data cube with m dimensions with h1, ..., hm hierarchy levels resp. Let the hierarchy levels of dimension i have
Then the base cube has
and the cube with all aggregations has
Prof. Bayer, DWH, Ch.4, SS 2000

14. Size of the Cube (2)
The aggregated cube is larger than the base cube by the
factor
Prof. Bayer, DWH, Ch.4, SS 2000

15. Size of the hierarchically aggregated Cube
For a hierarchy i with hi levels and
there are
hierarchical aggregation possibilities , i.e.
Lemma: A hierarchically completely aggregated data cube has
Prof. Bayer, DWH, Ch.4, SS 2000

16. size of the hierarchically aggregated cube plus base cube
Ex: (4 5) ( )
size of the hierarchically aggregated cube plus base cube
= ( ) * ( )
= 25 * 133 = 3325
Ex: (4 5) ( ) ( 8 3)
size of base cube: 40,320
hierarchically aggregated cube plus base:
= ( ) * ( ) * ( )
= 3325 * 33 = 109,725
Prof. Bayer, DWH, Ch.4, SS 2000

17. hierarchically aggregated cube plus base:
Ex: (4 5) ( ) ( 8 3) (5 9)
size of base cube: 1 814,400
hierarchically aggregated cube plus base:
= 109,725 * ( ) = 5 595,975
Prof. Bayer, DWH, Ch.4, SS 2000

18. Additional comments on aggregations
1. In addition to the size of the complete cube there is a factor of 5 for the various aggregation functions, e.g. sum, avg, min, max, count, ...
2. So far we did not consider general restrictions, e.g. „all Saturdays in March“ or „vacation months July and August“, which cross bounds of hierarchy levels
Interactive query formulation results in an unlimited number of aggregations
Optimization: restrictions corresponding to hierarchy levels shoud be pushed down, since they lead to query boxes
Prof. Bayer, DWH, Ch.4, SS 2000

19. Roll-up (Year, Month, ALL) Roll-up (Year, ALL, ALL)
Note: See later chapters for multidimensional indexes and MHC techniques and optimization of ROLAP-algebra to support hierarchical canonical aggregations like
Roll-up (Year, Month, ALL)
Roll-up (Year, ALL, ALL)
Roll-up (ALL, ALL, ALL)
but not
Roll-up ( ALL, Month, ALL)
Prof. Bayer, DWH, Ch.4, SS 2000

20. Non-hierarchical aggregation, e.g. March for all years
Optimization Problem
Non-hierarchical aggregation, e.g.
March for all years
decompose into union of several restrictions, e.g.
S Sales (Year, Month, Day)
where Month = March and
(Year = 1996 or Year = 1997 or Year = 1998)
see later for translation into ROLAP expression and transformations for optimization
Prof. Bayer, DWH, Ch.4, SS 2000

21. Aggregation for month e.g. by covering QB of weeks and postfiltering
Multiple Hierarchies
e.g. the time hierarchy
Aggregation for month e.g. by covering QB of weeks and postfiltering
Prof. Bayer, DWH, Ch.4, SS 2000

22. Navigation Operations
Drill Down: first show single result for aggregated value, e.g. sales per day, then show:
hourly values for days with very high or very low sales
in order to plan working hours for sales people better
Other Examples:
daily sales during Christmas season
vacation bookings for skiing on fasching
Prof. Bayer, DWH, Ch.4, SS 2000

23. Roll-up: Compute Aggregations
Prof. Bayer, DWH, Ch.4, SS 2000

24. Slicing
Selection of a smaller data cube or even reduction of a multidimensional datacube to fewer dimensions by a point restriction in some dimension (becomes pivot element)
Prof. Bayer, DWH, Ch.4, SS 2000

25. Dicing (würfeln)
rotate result, to show another view, e.g. exchanging rows and columns
Slice management
precomputing and caching of several slices for later or special use, e.g. for a special sales person
Prof. Bayer, DWH, Ch.4, SS 2000

26. Chapter 4.3 Modeling
Purpose: analysis of business processes, characteristic facts (Kennzahlen) for managers to support decisions (DSS)
Steps of Decision Process:
1. Which business processes to model and analyze?
2. What are the measures, where do they come from?
3. Which degree of details, e.g. minutes like in SAP? Which precision is required for OLAP?
4. Common properties of measures to determine dimensions? Brand, Time, geogr. Region, Productgroup? Dependencies between levels of hierarchies?
Prof. Bayer, DWH, Ch.4, SS 2000

27. 5. Attributes of dimensions, e.g. screen size of TV cc and PS for cars
focal length for camera
Problem: how common are properties and dimensions? Non common properties cannot be modeled by levels of dimensions, are called features at GfK (up to 50), are numbered with meaning dependent on specific dimension element, e.g.
TV: screen size color audio system
Car: transmission cc PS #cyl ...
Prof. Bayer, DWH, Ch.4, SS 2000

28. 6. Constant or changing attributes of dimensions? E.g.
New models of car makers
new powersource: electrical, hydrogen, solar
attributes are rather stable, but still should be planned ahead! (mergers like Daimler-Crysler)
7. Sparsity: one hypercube or several, i.e. multicube model? Influences storage requirements, query formulation and performance, cannot be hidden easily from user, maybe by views?
Prof. Bayer, DWH, Ch.4, SS 2000

29. 8. Caching and management of aggregates?
Number of aggregates
Maintenance costs
Avg.
Response time
100% 0%
Total costs
Time
Optimal
Number of
aggregates
Prof. Bayer, DWH, Ch.4, SS 2000

30. Chapter 4.4 Comparison of OLAP Architectures
MOLAP: Multidimensional OLAP
ROLAP: Relational OLAP
3. HOLAP: Hybrid OLAP
Prof. Bayer, DWH, Ch.4, SS 2000

31. MOLAP Architecture
Prof. Bayer, DWH, Ch.4, SS 2000

32. MDDBMS in ANSI-X3-Sparc
Prof. Bayer, DWH, Ch.4, SS 2000

33. Logical components of a MDDBMS
Prof. Bayer, DWH, Ch.4, SS 2000

34. ROLAP Architecture
Prof. Bayer, DWH, Ch.4, SS 2000

35. HOLAP Architecture
Prof. Bayer, DWH, Ch.4, SS 2000

36. flexible precomputations, partial aggregates parallelism
Reasons for MOLAP
performance
write access
Data Marts
functional power
Reasons for ROLAP
scalability
flexible precomputations, partial aggregates
parallelism
DB-mamagement and ACID
Prof. Bayer, DWH, Ch.4, SS 2000