Chapt. 7 Multidimensional Hierarchical Clustering

Prof. Bayer, DWH, Ch.7, SS2002 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)

DistributionChannel (3)

DISTRIBUTION

All Products All DistributionsAll Customer All Time


(b)

PRODKEY

CUSTKEY

DISTKEY

TIMEKEY

SALES

DISTCOST

PRODKEY

PRODUCT2180 rows

TYPE

BRAND

CATEGORY

CONTAINER

...

CUSTKEY

CUSTOMER7064 rows

REGION

NATION

TRADE-TYPE

BUSINESS-TYPE

...

DISTKEY

DISTRIBUTION12 rows

SALESORG

CHANNEL

...

TIMEKEY

TIME36 rows

YEAR

MONTH

FACT26M rows

...


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*13 185.328-------------------- = ----------- = 7.96 larger than base cube 5*5*7*7*19 23.275

Base Cube has 2.245.024.000 cells * 4 B ~ 9 GB

Number of available facts: 26 million


Sparsity:

26*106

-------------- = 0,01162,245* 109

100 - 1.16 = 98.84 % sparsity


Hierarchically aggregated Cube

(1+5+40+760+7600) = 8406

(1+8+56+112+784) = 961

(1+5+15) = 21

(1+3+24) = 28

= 4.749.961.608

Size of base cube 2.145.024.000

Number of aggregate cells 2.504.937.608

==> Juice and More database has 96 times more hierarchically aggregated cells than occupied base cells!


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 TWhere F. ProdKey = P. ProdKey AND

F. CustKey = C. CustKey ANDF.TIMEKEY = T.TIMEKEY ANDF.DISTKEY = D.DISTKEY AND<CUSTOMER RESTRICTION> AND<DISTRIBUTION RESTRICTION> AND<PRODUCT RESTRICTION> AND<TIME RESTRICTION>


Select <MEASURE AGGREGATION>

From Fact F

Where F. ProdKey BETWEEN Pkey1 AND Pkey2 AND

F. DistKey BETWEEN Dkey1 AND Dkey2 AND

F. CustKey BETWEEN Ckey1 AND Ckey2 AND

F. TimeKey BETWEEN Tkey1 AND Tkey2


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


Bitmap Index Intersection

bitmap for organization = „TM“

bitmap for region = „

Asia “

1.....1.11 1.1...1.1. 1.1...1.1. ...1.1.... ..1.1...1.

11.1...... 1.11.....1 .1.1..1... 1.1.1..... .1..1.1...

1......... 1.1....... ......1... .......... ....1.....

Page 1 Page 2 Page 3 Page 4 Page 5

result of bitmap intersection

accessed disk pages (shaded)

34 % of tuples

32 % of tuples

10 % of tuples

80 % of

pages


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


Basic Idea for Multidimensional Clustering

1L} 0.5L; Juice Apple 1L; OJ 0.7L; OJ 0.33L; {OJ

0

1 m

1L} OJ 0.7L; OJ 0.33L; {OJ1

1 m

0.5L} {A-Juice2 4 m

1L} Juice Apple 0.5L Juice Apple {1

2 m

0.33L} {OJ2 1 m 0.7L} {OJ

22 m 1L} OJ {

23 m 1L} {A-Juice

2 5m

Orange Juice Apple Juice

0,33L 0,7L 1L0,5L

Product Category

All Products

All

0 1

10 0 2

Level Label Member Ordinal (e.g.,1) Member Label (e.g., 0.7L)Legend:

Example Hierarchy in Member Set Representation

AppleJuice

1 1L


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, ..., mj

i ]] with mki V

mki get names, e.g. „Orange Juice“ as label(m1

1), in general label(mk

i)

Constraint: every mli+1 must be a subset of some mk

i


Hierarchic Relationships

The children of mki are all those sets ml

i+1 of the lower level i+1 with the property:

mli+1 mk

i , formally:

children(mki ) := [[ml

i+1 Li+1 : mli+1 mk

i ]]

parent(mki ) := [[ml

i-1 Li-1 : mli-1 mk

i ]]

Principle: the children of m are numbered by the bijective function ordm starting at 1 or 0


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


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)) ord father (mi) (mi) otherwise


Example:

REGION f(REGION)

South Europe 0

Middle Europe 1

Northern Europe 2

Western Europe 3

North America 4

Latin America 5

Asia 6

Australia 7

(a)


0

CUSTOMER

South Europe North America Asia

RetailWholesale Kana´s Sushi Bar

Joe‘s Sports Bar

... ...

Bar

4 6

2

1

10

RetailUSACanada 10

... ...

... ...

... ...

Australia7

Wholesale0

Surrogates for Region and the entire Costumer Hierarchy


Example: the path

North America --> USA --> Retail --> Bar

has the compound surrogate 4112

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 level (H, i-1) ]]


let li := log2 surrogates (H,i)

then li bits are needed for the surrogates of level i

let be a path = m0 m1 m2 ... mh

to a leaf mh of hierarchy H:


cs (H,) = cs (H,mh)

hlll

mfathermord 32

1 21

hlll

mfathermord 43

2 22

h

mfathermord h

:=

... +

+

+


Example:

cs (H, Bar) = 100 001 1 010 = 538

l1=3 l2=3 l3=1 l4=3

number of bits needed at certain level


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


Example: path to USA is:

North America --> USA

4 = 1002 1 = 0012

leads to range on cs:100 001 0 0002 to 100 001 1 1112

and to the decimal range:528 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.


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,


(a) (b)

YEAR

MONTH WEEK

DAY

REGION

NATION

TRADE TYPE

CUSTOMER TYPE

CUSTOMER SIZE

CUSTOMER

Fig. 4-4 Complex Hierarchy Graphs


CUSTOMER

South Europe North America ...

USACanada

Retail Wholesale

Bar Restaurant

Joe ‘s Sports BarYear <= 1997 Year > 1997

Fig. 4-5 Change of a hierarchy over the time


Orange

Juice

Asia


Apple

Juice

AsiaProcessing a query box in sort order with the Tetris algorithm

Chapt. 7 Multidimensional Hierarchical Clustering

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