1. Chapt. 7 Multidimensional Hierarchical Clustering
Fig. 3.1 Hierarchies in the `Juice and More´ schema
Year (3)
Month (12)
TIME
Region (8)
Nation (7)
Trade
Type (2)
Business
Type (7)
CUSTOMER
Type (5)
Brand (8)
Category (19)
Container (10)
PRODUCT
Sales
Organization (5)
Distribution
Channel (3)
DISTRIBUTION
All Products
All Distributions
All Customer
All Time
Prof. Bayer, DWH, Ch.7, SS2000

2. ... 36 PRODKEY CUSTKEY DISTKEY TIMEKEY SALES DISTCOST PRODUCT
2180 rows
TYPE
BRAND
CATEGORY
CONTAINER
...
CUSTOMER
7064 rows
REGION
NATION
TRADE-TYPE
BUSINESS-TYPE
DISTRIBUTION 12
rows
SALESORG
CHANNEL
TIME 36
YEAR
MONTH
FACT
26M rows
(b)
Prof. Bayer, DWH, Ch.7, SS2000

3. Size of completely aggregated Cube
(6*9*20*11)*(9*8*3*8)*(6*4)*(4*13) =
(5*8*19*10)*(8*7*2*7)*(5*3)*(3*12)
4*6*6*9*11*
= = larger than base cube
5*5*7*7*
Base Cube has cells * 4 B ~ 9 GB
Number of available facts: 26 million
Prof. Bayer, DWH, Ch.7, SS2000

4. Sparsity: 26*106 -------------- = 0,0116 2,245* 109
= 0,0116
2,245* 109
= % sparsity
Prof. Bayer, DWH, Ch.7, SS2000

5. Hierarchically aggregated Cube (1+5+40+760+7600) = 8406
( ) = 8406
( ) = 961
(1+5+15) =
(1+3+24) =
P =
Size of base cube
Number of aggregate cells
==> Juice and More database has 96 times more hierarchically aggregated cells than occupied base cells!
Prof. Bayer, DWH, Ch.7, SS2000

6. In addition: grouping, computation of aggregates, sorting of results.
Star-Joins
Restrictions on several dimension tables, which are then joined with fact table
In addition: grouping, computation of aggregates, sorting of results.
Example:
Select
<MEASURE AGGREGATION>
From
Fact F, Customer C, DISTRIBUTION D,
Product P, Time T
Where F.
ProdKey
= P.
AND
CustKey
= C.
F.TIMEKEY = T.TIMEKEY AND
F.DISTKEY = D.DISTKEY AND
<CUSTOMER RESTRICTION> AND
<DISTRIBUTION RESTRICTION> AND
<PRODUCT RESTRICTION> AND
<TIME RESTRICTION>
Prof. Bayer, DWH, Ch.7, SS2000

7. <MEASURE AGGREGATION>
Select
<MEASURE AGGREGATION>
From
Fact F
Where F.
ProdKey
BETWEEN Pkey1 AND Pkey2 AND
DistKey
BETWEEN Dkey1 AND Dkey2 AND
CustKey
BETWEEN Ckey1 AND Ckey2 AND
TimeKey
BETWEEN Tkey1 AND Tkey2
Prof. Bayer, DWH, Ch.7, SS2000

8. How to compute star-joins efficiently?
Key Question:
How to compute star-joins efficiently?
Secondary indexes on foreign keys of fact table (standard B-trees), see chapter 5 for details
- intersect result lists
retrieve tuples from fact table randomly
Bitmaps
Prof. Bayer, DWH, Ch.7, SS2000

9. Bitmap Index Intersection
bitmap for organization
34 % of
= „TM“
tuples
bitmap for region
32 % of
= „
Asia “
tuples
result of bitmap
intersection
10 % of
tuples
80 % of
accessed disk pages
Page 1
Page 2
Page 3
Page 4
Page 5
pages
(shaded)
Bitmap Index Intersection
Prof. Bayer, DWH, Ch.7, SS2000

10. Problem: for small result sets of a few %, almost all pages of the facts table must be fetched from disk, if the hits in the result set are not clustered on disk.
Ex: with 8 KB pages 20 to 400 tuples per page, i.e. at 0.25% to 5% hits in the result almost all pages must be fetched.
At least tuple clustering, preferably page clustering, are desirable, but how??
Goal: Code hierarchies in such a way, that for star-joins with the Fact table we have to join only with a query box on the Fact table
Prof. Bayer, DWH, Ch.7, SS2000

11. Example Hierarchy in Member Set Representation
Basic Idea for Multidimensional Clustering
All
1L}
0.5L;
Juice
Apple
1L; OJ
0.7L;
0.33L;
{OJ 1 = m
All
Products
AppleJuice
Orange Juice
Apple Juice
1L} OJ
0.7L;
0.33L;
{OJ 1 = m
1L}
Juice
Apple
0.5L { 1 2 = m
Product
Category 1
0,33L 1
0,7L 2 1L
0,5L 1 1L
0.33L}
{OJ 2 1 = m
0.7L}
{OJ 2 = m
1L} OJ { 2 3 = m
0.5L}
{A-Juice 2 4 = m
1L}
{A-Juice 2 5 = m
Level
Label
Member Ordinal (e.g.,1)
Member
Label (e.g., 0.7L)
Legend:
Example Hierarchy in Member Set Representation
Prof. Bayer, DWH, Ch.7, SS2000

12. Dimension D consists of Value Set V = [[ v1, v2, ... vn ]]
Hierarchy H of height h consisting of h+1 hierarchy levels H = [[L0 , L1 ,..., Lh ]]
Level Li is a set of sets = [[m1i, ..., mji ]] with mki elof V
mki get names, e.g. „Orange Juice“ as label(m11), in general label(mki)
Constraint: every mli+1 must be a subset of some mki
Prof. Bayer, DWH, Ch.7, SS2000

13. Hierarchic Relationships
The children of mki are all those sets mli+1 of the lower level i+1 with the property:
mli+1 elof??? mki , formally:
children(mki ) := [[mli+1 subsetof??? Li+1 : mli+1 subof??? mki ]]
parent(mki ) := [[mli-1 subsetof??? Li-1 : mli-1 superof??? mki ]]
Principle: the children of m are numbered by the bijective function ordm starting at 1 or 0
Prof. Bayer, DWH, Ch.7, SS2000

14. Enumeration and Surrogate Functions Let A be an enumeration type
A = [[ a0, a1, ... ak ]]
f : A --> (0, 1 ,..., k ) defined as
f (ai ) = i
then i is called the surrogate of ai
Prof. Bayer, DWH, Ch.7, SS2000

15. Hierarchies and composite Surrogates
Basic Idea: concatenate the surogates of successive hierarchy levels (compound surrogates cs)
Note: the root ALL of the hierarchy is not encoded
Def: compound surrogate cs for hierarchy H
ordm : children (m) --> [[0, 1, ..., |children(m)| -1]]
cs (H, mi) := ord father (mi) (mi) if i=1
:=cs (H, father ( mi)) comp??? ord father (mi) (mi) otherwise
Prof. Bayer, DWH, Ch.7, SS2000

16. Example: REGION f(REGION) South Europe Middle Europe 1 Northern Europe
Middle Europe 1
Northern Europe 2
Western Europe 3
North America 4
Latin America 5
Asia 6
Australia 7
(a)
Prof. Bayer, DWH, Ch.7, SS2000

17. Surrogates for Region and the entire Costumer Hierarchy
CUSTOMER
South
Europe
North America
Asia
Retail
Wholesale
Kana ´s
Sushi
Bar
Joe
‘s Sports Bar
... 4 6 2 1
USA
Canada
Australia 7
Surrogates for Region and the entire Costumer Hierarchy
Prof. Bayer, DWH, Ch.7, SS2000

18. North America --> USA --> Retail --> Bar
Example: the path
North America --> USA --> Retail --> Bar
has the compound surrogate 4?1?1?2
Next Idea: for every hierarchy level determine the higest branching degree (plus a safety margin for future extensions) and code by fixed number of bits.
surrogates (H,i) := max [[ cardinality (children (H,m)) : m in??? level (H, i-1) ]]
Prof. Bayer, DWH, Ch.7, SS2000

19. handgeschriebene Seite 6.6 ??? Problem mit doppelten Indices?
Prof. Bayer, DWH, Ch.7, SS2000

20. Properties of MHC Encoding very compact coding of fixed length
lexicographic order of composite keys remains, i.e. isomorphic to integer ordering
point restrictions on arbitrary hierarchy levels lead to interval restrictions on the compound surrogates
Prof. Bayer, DWH, Ch.7, SS2000

21. Example: path to USA is: North America --> USA 4 = 1002 1 = 0012
4 = = 0012
leads to range on cs: to
and to the decimal range: to 543 or [528 : 543]
==> star join with restriction North America.USA leads to an interval restriction on the fact table
==> point restrictions on arbitrary hierarchy levels of several dimensions lead to Query Boxes on the fact table.
Prof. Bayer, DWH, Ch.7, SS2000

22. Complex Hierarchies
time with months and weeks, both restrictions lead to intervals on the level of days
Example of Fig. 4-4
proposal for multiple hierarchies: choose the most useful (depending on the query profile) or consider multiple hierarchies as several independent hierarchies. Caution, this increases the number of dimensions !!!
Time variant hierarchies: extend by time interval of validity , see Example Fig. 4-5,
Prof. Bayer, DWH, Ch.7, SS2000

23. Complex Hierarchy Graphs
REGION
YEAR
NATION
CUSTOMER TYPE
MONTH
WEEK
TRADE TYPE
CUSTOMER SIZE
DAY
CUSTOMER
(b)
(a)
Complex Hierarchy Graphs
Prof. Bayer, DWH, Ch.7, SS2000

24. Change of a hierarchy over the time
CUSTOMER
South Europe
North
America
...
Canada
USA
Retail
Wholesale
Bar
Restaurant
Year
<= 1997
Year
> 1997
Joe
‘s Sports Bar
Change of a hierarchy over the time
Prof. Bayer, DWH, Ch.7, SS2000

25. Orange
Juice
Asia
Prof. Bayer, DWH, Ch.7, SS2000

26. Processing a query box in sort order with the Tetris algorithm
Apple
Juice
Asia
Processing a query box in sort order with the Tetris algorithm
Prof. Bayer, DWH, Ch.7, SS2000