A Multidimensional Modeling Approach for OLAP
within the Framework of the Relational Model based on
Quotient Relations
O. Mangisengi, A M. Tjoa
Institute of Software Technology
Technical University of Vienna, Austria
{oscar,tjoa}@ifs.tuwien.ac.at
Abstract
The paper introduces the concept of quotient
relations to model and query OLAP data. The use of
quotient relation inherits the advantages of the original
relational model as it was introduced by Codd. Drilldown and roll-up operations can be performed by the
powerful partitioning and de-partitioning operators on
quotient relations. The proposed approach fulfils the
requirements for a formal data model, namely the
existence of an implementation independent formalism,
the separation of structure and content and the existence
of a declarative query language.
1.
Introduction
The On-line Analytical Processing (OLAP) is
emerging as the most important approach in Data
Warehousing. OLAP allows to model data in a
multidimensional way as a cube and to query and analyze
data from many different perspectives. Independent from
the different implementation aspects, OLAP data are
presented to the user in a multidimensional data model
[CCS93].
There are several ways how to formally define
multidimensional models and their query languages.
However until now there do not exist a commonly
accepted formal multidimensional data model. Such a
model is necessary as a basis for an accepted standardized
logical data model for OLAP data. This would allow
practitioners and researchers to specify their data
warehouses in a unified way.
The aim of this paper is to propose an approach for
the conceptual modeling of multidimensional data which
is entirely based on a relational data model. It seems to the
authors that such a model would very much correspond
with the original intuition of Codd when he introduced the
concept of OLAP in his pioneering white paper [CCS93].
Among the different ways to define a data cube the
star schema approach could be regarded as the most
dominant one. A data cube is defined as a collection of at
least one fact table and a set of dimension tables.
In this paper we will introduce a meta model with the
capability to describe these tables and the OLAP queries in
an elegant manner. The general requirements for such a
formal data model that can serve as a foundation for
multidimensional database systems, are similar to those that
made the relational model successful, namely the existence
of an implementation independent formalism, the
separation of structure and contents, and the existence of
declarative query language [BSHD98].
Recently a number of approaches are proposed in the
literature for the formal foundation of multidimensional
modeling. [BSHD98] compares and describes the most
important modeling approaches in this area, namely the
approaches [AGS97] of Agrawal, Gupta, and Sarawagi,
[CT97a,CT97b] of Cabbibo and Torlone, [LW96] of Li and
Wang, and [GL97] of Gyssens and Lakshmanan.
The aim of the paper is to introduce the well known
concept of quotient relations which was introduced in
[FK77] for OLAP modeling of multidimensional data
within the framework of the relational model.
The remainder of this paper is structured as follows. In
section 2 we briefly present the quotient relation approach.
The partitioning operator is very well suited for the
construction of n-dimensional cubes. With the use of the
partitioning and de-partitioning operator drill-downs and
roll-ups can be performed in an easy manner.The modeling
of OLAP-data by means of quotient relation is given in
section 3. Section 4 presents an illustrative example for the
use of quotient relations for multidimensional queries.
Section 5 presents our conclusion.
2.
Quotient Relations
2.1. Basic Concepts
In this section we will briefly introduce the basic
concepts of quotient relation as it was introduced in
[FK77].
Let D1 , D2 , , Dn be (not necessarily distinct) sets. A
relation R on D1 , D2 , , Dn is a subset of the
Cartesian
product
D1  D2  Dn ;
i.e.
R  D1  D2  Dn .
-
Corresponding to each D j , j 1,2,, n is an Aj called an
attribute of R. The attributes of R are distinct.
Let I denote the collection of attributes of the n-ary
relation (The relation R is often represented as R(I)).
Let R be an n-ary relation and A a subset of I. The
relation  A  R  R is said to be an equivalence on R
if for tuples
t, t   R, t  A t  iff tA  t A. Thus if
-
tuples t and t  have the same attribute values with
respect to the attributes of A they are equivalent
under  A .  A is reflexive, symmetric, and
transitive, hence an equivalence relation.
As such, we say R is partitioned by A into disjoint
blocks of equivalent tuples, each of which is denoted
by t   t  | t   R  t   A t
The collection of all blocks, written as
R A   t  | t   R  t  R
Example: For the given relation R(A,B,C,D) in Table 2.1
the relation R A is presented in Table 2.2.
Table 2.2 Relation R A
-
B
2
2
3
1
2
3
C
X
X
Z
X
X
X
D
X
Y
W
Y
Z
Y
3. The modeling of OLAP-data by means of
quotient relation
Let R(I) be an n-ary relation. For different choices of
A, different quotient relations are obtained. Together
they constitute the family of quotient relations QR
QR  R A | A  I  . The partitioning
attribute set for every R  QR (i.e. A  I ) is
denoted as quot R .
It can be easily shown that for a given n-ary relation
R, the family QR is a finite lattice with universal
over
-
A
1
1
1
2
2
3
R,

bounds the quotient relation R consisting of a single

3.1 Quotient Algebra in a Multidimensional Model
Based on the theory of the quotient algebra, a relation
can be partitioned by attributes due to its attributes values.
The relation partitioned by these attributes can be viewed
as a multidimensional representation of this relation.
Furthermore, we will show how the operations of the
quotient algebra could be used for handling
multidimensional cubes.
As a first example, we introduce a relation
R(Year,Item,City,Sales). Let relation R contains the
following data:

block (i.e. R  R ), and the quotient relation R in
which each tuple is itself a block. The partial order in
the lattice is given by the degree of refinement. Then

-
R participates in T only if tC tD (where
 , , ,  ), provided that the underlying domains
of
are   compatible. The resultant quotient relation T is still
partitioned by A.

Cartesian Product
The Cartesian product, T : R  S  is defined such that for
i 1, 2, , k and j 1, 2, , p , a block [tij] of T is
defined as [tij] = [ri]  [sj], where  denotes the Cartesian
product of the conventional algebra; r1 , r2 , , rk  denote
the blocks of R ; and [s1],[s2],…,[sp], denotes the blocks of
S  . Note that T is partitioned by the union A  B.
Set operators like union, set-difference, and
intersection are obviously also defined in the quotient
algebra. It can easily be shown that the quotient algebra is
relationally complete. A detailed description of the quotient
algebra approach will be given in [KT98a].
Table 2.1 Relation R(A,B,C,D)
R
A
B
C
D
1
2
X
X
1
2
X
Y
1
3
Z
W
2
1
X
Y
2
2
X
Z
3
3
X
Y
R/A
S   QS partitioned by sets A and B respectively (i.e.
quot R   A and quot S   B ). Further let C and D be
other attribute sets of R.

Projection
The projection T : R C  is defined only if quot R   C ,
in which case T is partitioned by A. Blocks are treated as
unit by the operators, and in case of projection each block
is projected on C and possible tuples eliminated.

Restriction
The restriction T : RC  D is defined so that a block [t]

for any R  QR we have R  R  R .
Let R(I) be an n-ary relation with A and B denoting
any two sets of attributes of R, and R a member of
QR . The partitioning operator (/) induces a partition
on a quotient relation, while the de-partitioning
operator (*) eliminates a particular partition.
2.2. The Quotient Algebra
Let R and S be n-ary and m-ary relations,
respectively. Consider the quotient relations R  QR and
Table 3.1 Relation R(Year,Item,City,Sales)
Year
Item
City
Sales
Y
I
Ci
Sa
1994
1
New York City
40
1994
2
New York City
30
1994
1
Los Angeles
30
1994
2
Los Angeles
20
1994
1
San Francisco
10
1994
2
San Francisco
20
1995
3
New York City
20
1995
4
New York City
60
1995
2
Los Angeles
10
1995
4
Los Angeles
50
1995
2
San Francisco
30
1995
4
San Francisco
40
Example 3.1: Partitioning of relation R by the Year:
Table 3.4 Quotient relation X2
derived by partitioning of Year and City
City
Item
Sales
Year
Ci
I
Sa
Y
NYC
1
40
2
30
LA
1
30
1994
2
20
SF
1
10
2
20
NYC
3
20
1
60
LA
2
10
1995
4
50
SF
2
30
4
40

X 1 : R Y ear
This operation denotes that a quotient relation R is
partitioned by attribute Year. The result of this
partitioning is shown by the following table 3.2. The
attributes names of those attributes which are used for
partitioning are highlighted by bold letters
Year
Y
1994
1994
1994
1994
1994
1994
1995
1995
1995
1995
1995
1995
Table 3.2 Quotient relation X1
derived by partitioning of Year
City
Item
Ci
I
New York City(NYC)
1
New York
2
Los Angeles (LA)
1
Los Angeles
2
San Francisco(SF)
1
San Francisco
2
New York City
3
New York City
1
Los Angeles
2
Los Angeles
4
San Francisco
2
San Francisco
4
Sales
Sa
40
30
30
20
10
20
20
60
10
50
30
40
For simplicity reasons we will represent the quotient
relation by omitting the explicit repeating of attribute
values in the tuples of the quotient relation.
The next table shows a representation of relation
X1 in this manner.
Table 3.3 Simplified representation of table 3.2
City
Item
Sales
Year
Ci
I
Sa
Y
NYC
1
40
NYC
2
30
LA
1
30
1994
LA
2
20
SF
1
10
SF
2
20
NYC
3
20
NYC
1
60
LA
2
10
1995
LA
4
50
SF
2
30
SF
4
40
Example 3.2: Partitioning of relation X1 of the previous
example by City.
3.2. Fact & Dimension
Multidimensional model are mostly designed
consisting of a fact relation and some dimension relations.
Therefore, we will introduce the following definition of a
join operator in the quotient algebra between a fact relation
and its dimension relations. Let A be the set
A1, A2 ,, An , X1, X 2 ,, X k and B be the set
B , B ,, B , X  , X  ,, X  ,
where
X1, X 2 , , X k



and X 1 , X 2 ,  , X k are the attributes in common in the
attribute sets A and B.
The natural join between two quotient relations
Fact(A) and Dim(B), denoted as Fact  A  DimB  , is
defined as follows:
Fact  A  DimB  :
Fact A  DimBX  X A  B quot Fact  quot Dim
1
2
n
1
2
k
Example 3.3: Let Fact(A) with A={Year,Item,City,Sales}
be the flat fact relation given in Table 3.1.
Let Dim(B) with B={City,State} be the following
dimension quotient relation (partitioned by City) given in
Table 3.5.
Table 3.5 Quotient Relation B{City,State}
which is partitioned by City
State
City
ST
Ci
SF
CA
LA
CA
Dallas
TX
X 2 : X1 City
The join of Fact(A) with Dim(B), denoted
Fact  A  DimB  , has as its resulting quotient relation
Table 3.6. It is obvious that this result is partitioned by
City.
Table 3.6 Result of Fact  A  DimB 
Year
Item
Sales
State
City
Y
I
Sa
ST
Ci
1994
1
10
CA
1994
SF
2
20
CA
1995
2
30
CA
1995
4
40
CA
1994
1
30
CA
1994
LA
2
20
CA
1995
2
10
CA
1995
4
50
CA
Example 4.4: Let Fact2(A) be a fact quotient relation
partitioned
by
the
attribute
Year
(i.e.
quot(Fact2(A))={Year}) as given in table 3.3.
Let Dim(B) be a dimension quotient relation with
B={City,State} which is already partitioned by the
attribute State (i.e. quot(Dim(B))={State}) given in Table
3.7.
Table 3.7 Relation Dim(B){City,State}
partitioned by State
City
State
Ci
ST
SF
CA
LA
Dallas
TX
NYC
NY
The join Fact  A  DimB  has the following result (Table
3.8).
Table 3.8 Result of Fact  A  DimB 
partitioned by State
City
Item
Sales
Year
State
Ci
I
Sa
Y
ST
1994
LA
1
30
LA
2
20
1995
LA
2
10
LA
4
50
CA
1994
SF
1
10
SF
2
20
1995
SF
2
30
SF
4
40
1994
NYC
1
40
NYC
2
30
NY
1995
NYC
3
20
NYC
1
60
3.3. Rolling-up & Drilling-down
In this section, we apply the partitioning and departitioning operator for the drilling-down and rolling-up
operations in the multidimensional model. The quotient
algebra supports the partitioning operator (/) used to
induce a partition on a quotient relation. On the other
hand, it also supports the de-partitioning operator (*) to
eliminate a particular partition. As stated above the
following quotient algebra operation provides the operation
of the partitioning operator.
 
X :  R A  B


The operation X denotes that a quotient relation R is
partitioned by attribute A, and B. To eliminate the
partitioning by B, the de-partitioning operator, *, can be

   
used as follows: Y : X * B    R A  B   B  R A.
 

In the multidimensional model, the two operations,
rolling-up and drilling-down, can be applied using the
partitioning and de-partitioning operator.
3.3.1. Rolling-Up
Rolling-up is an operation used to aggregate data to a
higher level in the dimension hierarchy. In the quotient
algebra, the partition operator can be used to denote rollingup operations.
Let R(A) be a quotient relation with quot(R(A)) as its
partitioning attributes. The rolling-up operation in the
quotient relation R(A) (with quot(R(A))) by aggregating the
attribute X (with X  quot R A ) is denoted by:


R  UPRA, X  : R A X , (i.e.  R quot R A X )


with quot R  UPR A, X  : quot R A  X .
To illustrate the rolling-up operation in the quotient
algebra, we introduce the following quotient relation
Q(ST,Ci,STO,Sa) with the aggregation hierarchy
StoreCityState. We use a sample quotient relation
Store1 by partitioning of Q using STO given by table 3.9.

Table 3.9 Relation Store1  Q STO
State
ST
CA
CA
CA
CA
CA
CA
NY
City
Ci
SF
SF
SF
LA
LA
LA
NY
Store
STO
4
5
6
2
3
7
1
Sales
Sa
100
300
500
200
400
600
500
Let us assume as above that the lowest level of
aggregation in the quotient relation Store1 is the Store
attribute. To roll-up the sales data from Store to City, the
quotient algebra operation is given as follows:
City1 : Store1 Ci ty .
The result of rolling-up the sales data from Store to City is
given in the table 3.10.
Remark: It could be useful to introduce derived attributes
in quotient relations (e.g. sum, average, maximum,
minimum, etc.) which are functionally dependent from the
partitioning attribute. Table 3.10 shows the introduction of
sums for the partitioning attribute City.
City1 : City1 State, City , Store, Sales , SumSales , City 
Table 3.10 Rolling-up Store to City and the
introduction of derived attributes Sum(Sales,City)
State
Store
Sales
Sum
City
ST
STO
(Sales,
Ci
City)
4
100
CA
SF
5
300
900
6
500
2
200
CA
LA
3
400
1200
7
600
NY
NY
1
500
500
If a higher summarization of data is required, then a
quotient relation Q has to be partitioned by this higher
level attribute. In our case, a rolling-up to State requires a
further partitioning by ST and the quotient algebra
operation can be written as follows: State1 : City1 State .
The result of this roll up operation is given in table 3.11.
Table 3.11 Rolling-up from City to State and the
introduction of the derived attributes Sum(Sales,State)
Store
Sales
Sum(
Sum(
State
City
STO
Sales,C Sales,S
ST
Ci
ity)
tate)
4
100
SF
5
300
900
CA
6
500
2100
2
200
LA
3
400
1200
7
600
NY
NY
1
500
500
500
3.3.2. Drilling-down
Drilling-down in a multidimensional model is used
to obtain the data of a finer granularity level within the
aggregation hierarchy of a given dimension. In the
quotient algebra, the de-partitioning operator can be used
for this purpose. The de-partitioning operator can be
regarded as the inversion of the partitioning operator.
Let R(S) be a quotient relation partitioned by the
attribute-set quot(R(S)). The de-partitioning operator (*)
reduces the degree of partitioning of a quotient relation.
R(S) can be de-partitioned by an attribute set X  S .

RS   X : R / (S \ X)
with quot(R(S)*X) := quot(R(S)) \ quot(X). It is obvious

that RS  S  R   R and RS  X  X  RS  .
The drilling-down operation on a quotient relation

R A into a relation where the details of B  A are of
interest can be performed by the following operation:
 
D  DOWN :  R A  * B , with B  A , where



R A is a quotient relation, and A denotes attribute set
used for the original partition, and B is the partitioning
attribute to be eliminated. In this operation, data is
disaggregated from a higher level to a lower level.
We obtain a drill-down of the sales data from State to
City, in the following way:
City1 : State1 * ST , (i.e. City1 : City1 ST   ST ).
In this example, the partitioning of the attribute ST is
eliminated, and the table obtained by drilling down table
3.11 is table 3.10 again. If we want to obtain a more
detailed view of the data, e.g. at store level, the required
drill-down from City to Store can be realized by:
Store1 : City1 * Ci  Store1 Ci   Ci or
 





Store1 :     Q STO  Ci  ST   ST   Ci : Q STO








The result of this drilling-down operation is again table 3.9.
4. An extended example for the use of the quotient
algebra in a Data Warehouse
Table 4.1 Relation Sa(T,Pr,STO,SL)
Time
Product
Store
Sales
T
Pr
STO
SL
1
3
1
30
2
2
1
20
13
3
4
10
14
2
4
50
397
3
1
40
398
3
1
30
401
3
4
20
402
3
4
60
1
5
5
25
14
3
5
45
Store
STO
1
4
5
7
Product
Pr
2
3
4
5
Relation S(STO,Ci,ST)
City
Ci
New York City
Los Angeles
San Francisco
Austin
State
ST
NY
CA
CA
TX
Relation P(Pr,I,Ct)
Item
Category
I
Ct
Lots of Nuts
Food
Sweet tooth
Food
Fizzy Light
Soft drinks
Fizzy Classic
Soft drinks
Relation D(T,M,Y)
Time
Month
Year
T
M
Y
1
10
1994
2
10
1994
13
11
1994
14
11
1994
397
10
1995
398
10
1995
401
11
1995
402
11
1995
To show the use of the quotient algebra in a Data
Warehouse, we use a small data warehouse schema
example. In this example, the Data Warehouse has a sales
fact relation (Sa) and three dimension relations. The
dimension relations are given by the relation Store (S),
Product (P), and Day (D). A sample illustration of the
relation Sa, S, P, and D is given in the tables 4.1.
To demonstrate the relevance of the use of quotient
algebra in a data warehouse environment, a representative
multidimensional query will be described by means of
quotient algebra operations.
Example: Comparison of the sales of a product `sweet
tooth` for stores in state CA for the year 1994 and 1995.
The steps used to process the multidimensional query
using quotient relations are:
1. Restriction of stores which are located in California
and partitioning by the attribute ST.


R1 :  S ST ST ' CA' STO, ST 


The result of the quotient algebra operation R1 is
given in the table 4.2.
4.
Join between the fact relation Sa with R1, R2, and R3.
a. Join with R1 . R4 : Sa  R1 . The table 4.5 shows
the result of the quotient algebra operation R4.
Table 4.5 Result of the quotient algebra operation R4
Time
Product
Store
Sales
State
T
Pr
STO
SL
ST
13
3
4
10
14
2
4
50
401
3
4
20
CA
402
3
4
60
1
5
5
25
14
3
5
45
As a next step the (now) irrelevant attribute store can be
omitted by the following projection.

R4 : R4 T , Pr , SL, ST 

The quotient algebra operation R 4 is given in table 4.6.
Table 4.6 Result of the quotient algebra operation R4
Time
T
13
14
401
402
1
14
Table 4.2 Result of quotient algebra operation R1
Store
City
State
STO
Ci
ST
4
LA
CA
5
SF
2. Restriction of product “Sweet tooth” and projection
on {Product,Item}

R2 : PI ' Sweet tooth' P r , I 
b.
The following table shows the result of the quotient
algebra operation R2.
Table 4.3 Result of the quotient algebra operation R2
Product
Item
Pr
I
3
Sweet tooth
3.
Restriction to relevant Time elements in the Year
1994 and 1995. The quotient algebra operation is
given as follows:

R3 : D Y Y '1994 '  Y '1995 ' T , Y 
The following table 4.4 shows the relation D
partitioned by Year in 1994 and 1995
Table 4.4 Result of quotient algebra operation R3
Time
Year
T
Y
1
2
1994
13
14
397
398
1995
401
402
Product
Pr
3
2
3
3
5
3
Sales
SL
10
50
20
60
25
45

State
ST
CA
Join with R2 and projection on the relevant
attributes Time, Item, Sales, and State.
R : R   R Time, Item, Sales , State  .
5
4
2
Table 4.7 shows the result of the quotient algebra
operation R5.
Table 4.7 Result of the quotient algebra operation R5
Time
Sales
Item
State
T
SL
I
ST
13
10
401
Sweet tooth
20
CA
402
60
14
45
c.
Join with R3 . R6 : R5  R3
Table 4.8 Result of the quotient algebra operation R6
Time
Sales
Item
State
Year
Y
SL
I
ST
Y
13
Sweet tooth
10
CA
1994
14
45
401
Sweet tooth
20
CA
1995
402
60
The (now) irrelevant attribute Time can be ignored by the

following projection. R6 : R6 ST , Y , I , SL
Table 4.9 Result of the quotient algebra
operation R6
5.
State
ST
CA
Year
Y
1994
Item
I
Sweet tooth
CA
1995
Sweet tooth
Introduction
of
the
Sum(Sales(Year)) in R6 .
derived
Sales
SL
10
45
20
60
attribute
R6   R6 State , Year , Item, Sales , SumSales , Year 
Table 4.10 Introduction of the aggregation function
sum(Sales(Year))
Sales
Sum
State
Year
Item
SL
(Sales, Year)
ST
Y
I
1994
Sweet tooth
10
55
CA
45
CA
1995
Sweet tooth
20
80
60
Roll-up of R6 by ST and introduction of derived
attribute Sum(Sales,Year). The result is given in
Table 4.11.
State , Year , Item, Sales ,

R7 : R6 ST 

SumSales , Year , SumSales , State 
6.
Table 4.11 Roll-up and introduction of the aggregation
function sum(Sales(State))
Sales Sum(
Sum(
State
Year
Item
SL
Sales,
Sales,
ST
Y
I
Year)
State)
1994 Sweet tooth
10
55
CA
45
135
1995 Sweet tooth
20
80
60
5. Conclusion and Future Work
Our work is to study the use of quotient relations as
an instrument of modeling and querying OLAP data. In
this approach OLAP cubes are represented by quotient
relations for fact and dimensions. It has been shown that
the partitioning and de-partitioning operators can be easily
used for drilling-down and rolling-up. The advantages of a
modeling approach using quotient relations are given by
the implementation independent formalism of the quotient
algebra. This advantage is comparable with the
implementation independent formalism of Codd’s original
relational model. Furthermore the approach fulfils the
criterion of separation of structure and contents. The
intentional description of quotient relations (i.e. its
schemas consisting of the set of attributes of the relations
together with the partitioning sets) are clearly separated
from its extensions (i.e. the block-tuples of the quotient
relation).
The criterion of the existence of a declarative query
language is given by the quotient algebra and has been
shown in illustrative examples in the paper. Further
investigation will be made on the object-oriented
implementation of a quotient relation based OLAP
representation and on the use of quotient relations for
mandatory access control security concept for OLAP as it
is represented for the traditional relational model in
[KKST97]. An implementation of an object-oriented
realization is under work.
Acknowledgement
The authors are very indebted to Andreas Kurz for
many creative remarks. This work is supported by the
Austrian Federal Bank, Project No. 6681. For the objectoriented implementation we thank Informix for their
“Innovative Software Grant”.
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