Sl02 2x2 (1)

Design Problem

Distributed Database Systems
Fall 2012
Design problem of distributed systems: Making decisions about
the placement of data and programs across the sites of a computer
Distributed Database Design network as well as possibly designing the network itself.
SL02 In DDBMS, the distribution of applications involves
Distribution of the DDBMS software
Distribution of applications that run on the database
Distribution of applications will not be considered in the following;
Design problem instead the distribution of data is studied.
Design strategies (top-down, bottom-up)
Fragmentation (horizontal, vertical)
Allocation and replication of fragments, optimality, heuristics

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Framework of Distribution Design Strategies/1
Dimension for the analysis of distributed systems
Level of sharing: no sharing, data sharing, data + program sharing
Behavior of access patterns: static, dynamic
Level of knowledge on access pattern behavior: no information,
partial information, complete information
Top-down approach
Designing systems from scratch
Homogeneous systems
Bottom-up approach
The databases already exist at a number of sites
The databases should be connected to solve common tasks

Distributed database design should be considered within this
general framework.
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Design Strategies/2 Design Strategies/3
Top-down design strategy
Distribution design is the central part of the design in DDBMSs
(the other tasks are similar to traditional databases).
Objective: Design the LCSs by distributing the data over the sites.
Two main aspects have to be designed carefully:
Fragmentation: Relation may be divided into a number of
sub-relations, which are distributed
Allocation and replication: Each fragment is stored at site(s) with
”optimal” distribution
Distribution design issues
Why fragment at all?
How to fragment?
How much to fragment?
How to test correctness?
How to allocate?

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Design Strategies/4 Fragmentation/1
Bottom-up design strategy What is a reasonable unit of distribution? Relation or fragment of
relation?
Relations as unit of distribution:
If the relation is not replicated, we get a high volume of remote data
accesses.
If the relation is replicated, we get unnecessary replications, which
cause problems in executing updates and waste disk space
Might be an OK solution, if queries need all the data in the relation
and data stays only at the sites that use the data
Fragments of relation as unit of distribution:
Application views are usually subsets of relations
Thus, locality of accesses of applications is deﬁned on subsets of
relations
Permits a number of transactions to execute concurrently, since they
will access different portions of a relation
Parallel execution of a single query (intra-query concurrency)
However, semantic data control (especially integrity enforcement) is
more difﬁcult
⇒ Fragments of relations are (usually) appropriate unit of distribution.
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Fragmentation/2 Fragmentation/3
Types of Fragmentation
Horizontal: partitions a relation along its tuples
Fragmentation aims to improve: Vertical: partitions a relation along its attributes
Mixed/hybrid: a combination of horizontal and vertical fragmentation
Reliability
Performance
Balanced storage capacity and costs
Communication costs
Security
The following information is used to decide fragmentation:
Quantitative information: cardinality of relations, frequency of queries, (a) Horizontal Fragmentation
site where query is run, selectivity of the queries, etc.
Qualitative information: predicates in queries, types of access of
data, read/write, etc.

(b) Vertical Fragmentation (c) Mixed Fragmentation

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Fragmentation/4 Fragmentation/5
Example of database instance
Example (contd.): Horizontal fragmentation of PROJ relation
PROJ1: projects with budgets less than 200 000
PROJ2: projects with budgets greater than or equal to 200 000

Data
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Fragmentation/6 Correctness Rules of Fragmentation
Example (contd.): Vertical fragmentation of PROJ relation
PROJ1: information about project budgets Completeness
PROJ2: information about project names and locations Decomposition of relation R into fragments R1 , R2 , . . . , Rn is complete
iff each data item in R can also be found in some Ri .
Reconstruction
If relation R is decomposed into fragments R1 , R2 , . . . , Rn , then there
should exist some relational operator that reconstructs R from its
fragments, i.e., R = R1 . . . Rn
Union to combine horizontal fragments
Join to combine vertical fragments
Disjointness
If relation R is decomposed into fragments R1 , R2 , . . . , Rn and data
item di appears in fragment Rj , then di should not appear in any other
fragment Rk , k j (exception: primary key attribute for vertical
fragmentation)
For horizontal fragmentation, data item is a tuple
For vertical fragmentation, data item is an attribute

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Idea of Horizontal Fragmentation Information Requirements/1

Database information:
Links between relations (a link models a 1:N relationship between
Intuition behind horizontal fragmentation relations that are related to each other by an equality join)
Every site should hold all information that is used to query the site
The information at the site should be fragmented so the queries of the
site run faster
Horizontal fragmentation is deﬁned as selection operation, σp (R )
Example:
σBUDGET <200K (PROJ )
σBUDGET ≥200K (PROJ )
Links (one to many)
Cardinality of relations: card(R)

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Information Requirements/2 02.1 Information Requirements/3 02.2

Application information:
Simple predicates used in queries
Relation R [A1 , A2 , ..., An ]
Ai θvi a simple predicate (θ ∈ {=, <, ≤, >, ≥, }, vi ∈ Di , Di is the Application information (cnt’d):
domain of Ai ) Minterm selectivities
For relation R we deﬁne Pr = {p1 , p2 , ..., pm } The number of tuples of the relation that would be accessed by a user
Example: PNAME = ”Maintenance”, BUDGET < 200000 query, which is speciﬁed according to a given minterm predicate mi .
Minterm predicates Access frequencies
minterm predicates are conjunctions of simple predicates The frequency with which a user application qi accesses data.
Relation R, Pr = {p1 , p2 , ..., pm }
M = {m1 , m2 , ..., mr } such that

M = {mi | mi = ∧pj ∈Pr pj ∗}, 1 ≤ j ≤ m, 1 ≤ i ≤ z

where pj ∗ = pj or pj ∗ = ¬pj

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Horizontal Fragmentation/1 Horizontal Fragmentation/2

A horizontal fragment Ri of relation R consists of all the tuples of R that
We write Pr to denote the complete and minimal set of simple satisfy a predicate Fj :
predicates generated from Pr:
Rj = σFj (R ), 1 ≤ j ≤ w
The set of predicates is complete if and only if any two tuples in the
same fragment are referenced with the same probability by any where Fj is a selection formula, which is (preferably) a minterm predicate.
application
The set of predicates is minimal if and only if each simple predicate Computing the horizontal fragmentation:
is relevant for determining the fragmentation and for each pair of 1. Determine Pr (80/20 rule)
fragments there is at least one query that accesses the fragments 2. Compute Pr’
differently
3. Determine M
4. Minimize M (eliminating impossible minterms)

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Horizontal Fragmentation/3 Horizontal Fragmentation/4

Example: Fragmentation of the PROJ relation
Consider the following query: Find the name and budget of projects
given their location. If access is only according to the location, the above set of
The query is issued at all three locations predicates is complete
Fragmentation based on LOC, using the set of predicates i.e., in each fragment PROJi each tuple has the same probability of
{LOC = ‘Montreal ‘, LOC = ‘NewYork ‘, LOC = ‘Paris ‘} being accessed
If there is a second query/application that accesses only those
PROJ1 = σLOC =‘Montreal ‘ (PROJ )
PROJ2 = σLOC =‘NewYork ‘ (PROJ ) project tuples where the budget is less than $200K, the set of
PNO PNAME BUDGET LOC
PNO PNAME BUDGET LOC predicates is not complete.
P2 DB Develop. 135000 New York P2 in PROJ2 has higher probability to be accessed
P1 Instrument. 150000 Montreal
P3 CAD/CAM 250000 New York
PROJ3 = σLOC =‘Paris ‘ (PROJ )
PNO PNAME BUDGET LOC
P4 Maintenance 310000 Paris

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Horizontal Fragmentation/5 Horizontal Fragmentation/6 02.3

Example (contd.): Example (contd.): Now, PROJi , i = 1, 2, 3 will be split in two
Add BUDGET < 200K and BUDGET ≥ 200K to the set of predicates fragments
to make it complete.
⇒ {LOC = ‘Montreal ‘, LOC = ‘NewYork ‘, LOC = ‘Paris ‘, PROJ1 = σLOC =‘Montr ‘∧BDGT <200K (PROJ ) PROJ2 = σLOC =‘NY ‘∧BDGT <200K (PROJ )
BUDGET ≥ 200K , BUDGET < 200K } is a complete set PNO PNAME BUDGET LOC PNO PNAME BUDGET LOC
Minterms to fragment the relation are given as follows: P1 Instrumnt 150000 Montreal P2 DB Devel 135000 New York

PROJ3 = σLOC =‘Paris ‘∧BDGT ≥200K (PROJ ) PROJ2 = σLOC =‘NY ‘∧BDGT ≥200K (PROJ )
(LOC = ‘Montreal ‘) ∧ (BUDGET ≤ 200K )
PNO PNAME BUDGET LOC PNO PNAME BUDGET LOC
(LOC = ‘Montreal ‘) ∧ (BUDGET > 200K ) P4 Maintenance 310000 Paris P3 CAD/CAM 250000 New York
(LOC = ‘NewYork ‘) ∧ (BUDGET ≤ 200K )
(LOC = ‘NewYork ‘) ∧ (BUDGET > 200K ) Note that the following fragments are empty:
(LOC = ‘Paris ‘) ∧ (BUDGET ≤ 200K ) σLOC =‘Paris ‘∧BDGT <200K (PROJ )
σLOC =‘Montreal ‘∧BDGT ≥200K (PROJ )
(LOC = ‘Paris ‘) ∧ (BUDGET > 200K )

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Derived Horizontal Fragmentation/1 02.5 Derived Horizontal Fragmentation/2

Defined on a member (target) relation of a link according to a
selection operation specified on its owner (source).
Given a link L where owner (L ) = S and member (L ) = R, the
Each link is an equijoin.
derived horizontal fragments of R are defined as
Fragments of member relation can be defined with a semijoin on
fragments of owner relation. Ri = R |>< Si , 1 ≤ i ≤ w

where w is the maximum number of fragments that will be defined
on R and
Si = σFi (S )
where Fi is the formula according to which the primary horizontal
fragment Si is defined.

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Derived Horizontal Fragmentation/3 02.7 Vertical Fragmentation/1
Objective of vertical fragmentation is to partition a relation into a set
of smaller relations so that many of the applications will run on only
one fragment.
Given link L 1 where Vertical fragmentation of a relation R produces fragments
owner (L 1) = PAY and R1 , R2 , . . . , each of which contains a subset of R’s attributes.
member (L 1) = EMP Vertical fragmentation is defined using the projection operation of
and the relational algebra:
PAY 1 = σSAL ≤30000 (PAY ) πA1 ,A2 ,...,An (R )
PAY 2 = σSAL >30000 (PAY )
we get
EMP1 = EMP |>< PAY 1 Example:
EMP2 = EMP |>< PAY 2
PROJ1 = πPNO ,BUDGET (PROJ )
PROJ2 = πPNO ,PNAME ,LOC (PROJ )
Vertical fragmentation has also been studied for (centralized) DBMS
Smaller relations, and hence less page accesses
e.g., MONET system
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Vertical Fragmentation/2 Vertical Fragmentation/3

Two types of heuristics for vertical fragmentation exist:
Vertical fragmentation is more complicated than horizontal
Grouping: assign each attribute to one fragment, and at each step,
fragmentation
join some of the fragments until some criteria is satisfied.
In horizontal partitioning: for n simple predicates, the number of
Bottom-up approach
possible minterms is 2n ; some of them can be ruled out by existing
Splitting: starts with a relation and decides on beneficial partitionings
implications/constraints.
based on the access behaviour of applications to the attributes.
In vertical partitioning: for m non-primary key attributes, the number
of possible fragments is equal to B (m) (= the mth Bell number), i.e., Top-down approach
the number of partitions of a set with m members. Results in non-overlapping fragments
For large numbers, B (m) ≈ mm (e.g., B (15) = 109 ) Optimal solution is probably closer to the full relation than to a set of
small relations with only one attribute

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Given are the user queries/applications Q = (q1 , . . . , qq ) that will
Application information: The major information required as input run on relation R (A1 , . . . , An )
for vertical fragmentation is related to applications (queries) Attribute Usage Matrix: Denotes which query uses which attribute:
Since vertical fragmentation places in one fragment those attributes
usually accessed together, there is a need for some measure that 1 iff qi uses Aj
would define more precisely the notion of “togetherness”, i.e., how use (qi , Aj ) =
0 otherwise
closely related the attributes are.
This information is obtained from queries and collected in the
Attribute Usage Matrix and Attribute Affinity Matrix. The use (qi , •) vectors for each application are easy to define if the
designer knows the applications that willl run on the DB (consider
also the 80-20 rule)

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Vertical Fragmentation/6 02.8 Vertical Fragmentation/7
Example: Consider relation PROJ (PNO , PNAME , BUDGET , LOC )
and queries:

q1 = SELECT BUDGET FROM PROJ WHERE PNO=Value Attribute Affinity Matrix: Denotes the frequency of two attributes Ai
q2 = SELECT PNAME,BUDGET FROM PROJ and Aj with respect to a set of queries Q = (q1 , . . . , qn ):
q3 = SELECT PNAME FROM PROJ WHERE LOC=Value
aff (Ai , Aj ) = ( ref l (qk ) ∗ acc l (qk ))
q4 = SELECT SUM(BUDGET) FROM PROJ WHERE LOC =Value use (q ,A )=1,
k : use (qk ,Ai )=1 sites l
k j
Abbreviations:
A1 = PNO , A2 = PNAME , A3 = BUDGET , A4 = LOC where
Attribute Usage Matrix ref l (qk ) is the cost (= number of accesses to (Ai , Aj )) of query qK at
site l
acc l (qk ) is the frequency of query qk at site l

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Example (contd.): Let the cost of each query be ref l (qk ) = 1, and
the frequency acc l (qk ) of the queries be as follows:
Site1 Site2 Site3
acc1 (q1 ) = 15 acc2 (q1 ) = 20 acc3 (q1 ) = 10 Idea: Take the attribute affinity matrix (AA) and reorganize the
acc1 (q2 ) = 5 acc2 (q2 ) = 0 acc3 (q2 ) = 0 attribute orders to form clusters where the attributes in each cluster
acc1 (q3 ) = 25 acc2 (q3 ) = 25 acc3 (q3 ) = 25 demonstrate high affinity to one another.
acc1 (q4 ) = 3 acc2 (q4 ) = 0 acc3 (q4 ) = 0 Bond energy algorithm (BEA) has been suggested for that
purpose for several reasons:
Attribute affinity matrix aff (Ai , Aj ) = It is designed specifically to determine groups of similar items as
opposed to a linear ordering of the items.
The final groupings are insensitive to the order in which items are
presented.
The computation time is reasonable (O (n2 ), where n is the number of
attributes)

e.g., aff (A1 , A3 ) = 1 =1 3=1 acc l (qk ) =
k l
acc 1 (q1 ) + acc 2 (q1 ) + acc 3 (q1 ) = 45
(q1 is the only query to access both A1 and A3 )
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Example (contd.): Cluster Affinity Matrix CA after running BEA

BEA:
Input: AA matrix
Output: Clustered AA matrix (CA)
Permutation is done in such a way to maximize the following global
affinity mesaure (affinity of Ai and Aj with their neighbors):
n n
Elements with similar values are grouped together, and two clusters
can be identified
AM = aff(Ai , Aj )[aff(Ai , Aj −1 ) + aff(Ai , Aj +1 ) +
An additional partitioning algorithm is needed to identify the clusters
i =1 j =1
in CA
aff(Ai −1 , Aj ) + aff(Ai +1 , Aj )] Usually more clusters and more than one candidate partitioning, thus
additional steps are needed to select the best clustering.
The resulting fragmentation after partitioning (PNO is added in
PROJ2 explicilty as key):
PROJ1 = {PNO , BUDGET }
PROJ2 = {PNO , PNAME , LOC }
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BEA Algorithm/1 BEA Algorithm/2
In order to determine the best placement we define the contribution of a
placement.
Contribution of a placement:
Input: The AA matrix
Output: Clustered affinity matrix CA which is a permutation of AA cont (Ai , Ak , Aj ) =
2 ∗ bond (Ai , Ak )+
Initialization: Place and fix one of the columns of AA in CA (we 2 ∗ bond (Ak , Aj )−
choose column 1 in our example) 2 ∗ bond (Ai , Aj )
Iteration: Place the remaining n-i columns in the remaining i+1
positions in the CA matrix. For each column choose the placement
that makes the largest contribution to the global affinity measure. Bond between a pair of attributes is defined as:
Row order: Order the rows according to the column ordering.
bond (Ax , Ay ) =
n
aff (Az , Ax ) ∗ aff (Az , Ay )
z =1

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BEA Algorithm/3 BEA Algorithm/4
Ordering (0-3-1) :
Consider the following AA matrix and the corresponding CA matrix where cont (A 0, A 3, A 1)
A1 and A2 have been placed.
= 2 ∗ bond (A 0, A 3) + 2 ∗ bond (A 3, A 1)–2 ∗ bond (A 0, A 1)
A1 A2 A3 A4 A1 A2 = 2 ∗ 0 + 2 ∗ 4410 – 2 ∗ 0 = 8820
A1 45 0 45 0 A1 45 0
Ordering (1-3-2) :
AA = A2 0 80 5 75 CA = A2 0 80
A3 45 5 53 3 A3 45 5 cont (A 1, A 3, A 2)
A4 0 75 3 78 A4 0 75 = 2 ∗ bond (A 1, A 3) + 2 ∗ bond (A 3, A 2)–2 ∗ bond (A 1, A 2)
= 2 ∗ 4410 + 2 ∗ 890 – 2 ∗ 225 = 10150
bond (A3 , A1 ) = 45 ∗ 45 + 5 ∗ 0 + 53 ∗ 45 + 3 ∗ 0 = 4410 Ordering (2-3-4) :
bond (A3 , A2 ) = 45 ∗ 0 + 5 ∗ 80 + 53 ∗ 5 + 3 ∗ 75 = 890
cont (A 2, A 3, A 4)
= 2 ∗ bond (A 2, A 3) + 2 ∗ bond (A 3, A 4)–2 ∗ bond (A 2, A 4)
The next step is to place A3 . = 1780

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BEA Algorithm/5 02.9 BEA Algorithm/6
After adding A3 the CA matrix has the form
The ﬁnal step is to divide a set of clustered attributes {A1 , ..., An } into
A1 A3 A2 two sets {A1 , ..., Ai } and {Ai +1 , ..., An } such that the costs of queries
45 45 0 that access both sets are minimized.
CA = 0 5 80 The appropriate split point on the diagonal must be determined:
45 53 5
0 3 75 A1 A3 A2 A4
A1 45 45 0 0
After adding A4 and ordering rows the CA matrix has the form A3 45 53 5 3
A2 0 5 80 75
A1 A3 A2 A4 A4 0 3 75 78
A1 45 45 0 0
This is an optimization problem: the optimal point on the diagonal
CA = A3 45 53 5 3
must be determined.
A2 0 5 80 75
Note: Generalizations are needed to deal with
A4 0 3 75 78
partitions located in the middle of the matrix
multiple split points

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BEA Algorithm/7 Correctness of Vertical Fragmentation

Relation R is decomposed into fragments R1 , R2 , . . . , Rn
AQ (qi ) = {Aj | use (qi , Aj ) = 1} attributes accessed by qi e.g., PROJ = {PNO , BUDGET , PNAME , LOC } into
PROJ1 = {PNO , BUDGET } and PROJ2 = {PNO , PNAME , LOC }
TQ = {qi | AQ (qi ) ⊆ TA } queries that access TA only
BQ = {qi | AQ (qi ) ⊆ BA } queries that access BA only Completeness
OQ = Q − {TQ ∪ BQ } queries that access TA and BA Guaranteed by the partitioning algortihm, which assigns each
attribute in A to one partition
CQ = q i ∈Q ∀Sj refj (qi ) ∗ accj (qi ) cost of all queries Reconstruction
CTQ = refj (qi ) ∗ accj (qi ) cost of TQ queries Join to reconstruct vertical fragments
qi ∈TQ ∀Sj
R = R1 · · · Rn = PROJ1 PROJ2
CBQ = qi ∈BQ ∀Sj refj (qi ) ∗ accj (qi ) cost of BQ queries
Disjointness
COQ = qi ∈OQ ∀Sj refj (qi ) ∗ accj (qi ) cost of other queries Attributes have to be disjoint in VF. Two cases are distinguished:
If tuple IDs are used, the fragments are really disjoint
CTQ ∗ CBQ − COQ 2 maximize Otherwise, key attributes are replicated automatically by the system
e.g., PNO in the above example

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Mixed Fragmentation Replication and Allocation

In most cases simple horizontal or vertical fragmentation of a DB Replication: Which fragements shall be stored as multiple copies?
schema will not be sufﬁcient to satisfy the requirements of the Complete Replication
applications. Complete copy of the database is maintained in each site
Mixed fragmentation (hybrid fragmentation): Consists of a Selective Replication
horizontal fragment followed by a vertical fragmentation, or a vertical Selected fragments are replicated in some sites
fragmentation followed by a horizontal fragmentation Allocation: On which sites to store the various fragments?
Fragmentation is deﬁned using the selection and projection Centralized
Consists of a single DB and DBMS stored at one site with users
operations of relational algebra:
distributed across the network
Partitioned
σp (πA1 ,...,An (R ))
Database is partitioned into disjoint fragments, each fragment assigned
πA1 ,...,An (σp (R )) to one site

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Replication/1 Replication/2

Comparison of replication alternatives

Replicated DB
fully replicated: each fragment at each site
partially replicated: each fragment at some of the sites
Non-replicated DB (= partitioned DB)
partitioned: each fragment resides at only one site
Rule of thumb:
read only queries
If ≥ 1, then replication is advantageous, otherwise
update queries
replication may cause problems

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Fragment Allocation/1 Fragment Allocation/2

Fragment allocation problem Required information
Given are: Database Information
– fragments F = {F1 , F2 , ..., Fn } selectivity of fragments
– network sites S = {S1 , S2 , ..., Sm } size of a fragment
– and applications Q = {q1 , q2 , ..., ql } Application Information
Find: the ”optimal” distribution of F to S RRij : number of read accesses of a query qi to a fragment Fj
URij : number of update accesses of query qi to a fragment Fj
Optimality uij : a matrix indicating which queries updates which fragments,
Minimal cost rij : a similar matrix for retrievals
originating site of each query
Communication + storage + processing (read and update)
Cost in terms of time (usually) Site Information
Performance USCk : unit cost of storing data at a site Sk
LPCk : cost of processing one unit of data at a site Sk
Response time and/or throughput
Network Information
Constraints
communication cost/frame between two sites
Per site constraints (storage and processing)
frame size

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The total cost function has two components: storage and query
We discuss an allocation model which attempts to processing.
minimize the total cost of processing and storage
meet certain response time restrictions TOC = STCjk + QPCi
General Form: Sk ∈S Fj ∈F q i ∈Q
min(Total Cost)

Storage cost of fragment Fj at site Sk :
subject to
response time constraint STCjk = USCk ∗ size (Fj ) ∗ xij
storage constraint
processing constraint where USCk is the unit storage cost at site k
Decision variable xij
Query processing cost for a query qi is composed of two
components:
1 if fragment Fi is stored at site Sj
xij = composed of processing cost (PC) and transmission cost (TC)
0 otherwise
QPCi = PCi + TCi

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Processing cost is a sum of three components: The transmission cost is composed of two components:
access cost (AC), integrity contraint cost (IE), concurency control cost Cost of processing updates (TCU) and cost of processing retrievals
(CC) (TCR)

PCi = ACi + IEi + CCi TCi = TCUi + TCRi

Access cost: Cost of updates:
Inform all the sites that have replicas + a short conﬁrmation message
ACi = (URij + RRij ) ∗ xij ∗ LPCk back
sk ∈S Fj ∈F

TCUi = uij ∗ (update message cost + acknowledgment cost)
where LPCk is the unit process cost at site k
Sk ∈S Fj ∈F
Integrity and concurrency costs:
Can be similarly computed, though depends on the speciﬁc constraints Retrieval cost:
Send retrieval request to all sites that have a copy of fragments that are
Note: ACi assumes that processing a query involves decomposing needed + sending back the results from these sites to the originating
it into a set of subqueries, each of which works on a fragment, ..., site.
This is a very simplistic model
Does not take into consideration different query costs depending on TCRi = min (xjk ∗ (cost retrieval request + cost sending back result))
S k ∈S
F j ∈F
the operator or different algorithms that are applied
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Solution Methods
Modeling the constraints The complexity of this allocation model/problem is NP-complete
Correspondence between the allocation problem and similar
Response time constraint for a query qi
problems in other areas
execution time of qi ≤ max. allowable response time for qi Plant location problem in operations research
Knapsack problem
Storage constraints for a site Sk Network flow problem
Hence, solutions from these areas can be re-used
storage requirement of Fj at Sk ≤ storage capacity of Sk Use different heuristics to reduce the search space
F j ∈F Assume that all candidate partitionings have been determined together
with their associated costs and benefits in terms of query processing.
Processing constraints for a site Sk The problem is then reduced to find the optimal partitioning and
placement for each relation
processing load of qi at site Sk ≤ processing capacity ofSk Ignore replication at the first step and find an optimal non-replicated
qi ∈Q solution
Replication is then handeled in a second step on top of the previous
non-replicated solution.

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Conclusion/1 Conclusion/2

Distributed design decides on the placement of (parts of the) data
and programs across the sites of a computer network
There are two key technical questions: fragmentation and
allocation/replication of data Allocation/Replication of data
Horizontal fragmentation is defined via the selection operation Type of replication: no replication, partial replication, full replication
σp (R ) Optimal allocation/replication modelled as a cost function under a set
of constraints
Rewrites the queries of each site in the conjunctive normal form and
The complexity of the problem is NP-complete
finds a minimal and complete set of conjunctions to determine
Use of different heuristics to reduce the complexity
fragmentation
Vertical fragmentation via the projection operation πA (R )
Computes the attribute affinity matrix and groups “similar” attributes
together
Mixed fragmentation is a combination of both approaches

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