Schema Normalization, Concluded Zachary G. Ives University of Pennsylvania CIS 550 – Database &...
-
date post
22-Dec-2015 -
Category
Documents
-
view
219 -
download
3
Transcript of Schema Normalization, Concluded Zachary G. Ives University of Pennsylvania CIS 550 – Database &...
Schema Normalization, Concluded
Zachary G. IvesUniversity of Pennsylvania
CIS 550 – Database & Information Systems
October 11, 2005
Some slide content courtesy of Susan Davidson & Raghu Ramakrishnan
2
Announcements
Decide on 3-person project groups by 1 week from Thursday (10/20)
Homework 2 answers posted on Web Homework 3 due Thursday No class next Tuesday (Fall Break) Midterm: Thursday 10/20
3
Not All Designs are Equally Good
Why is this a poor schema design?
And why is this one better?
Stuff(sid, name, serno, subj, cid, exp-grade)
Student(sid, name)Course(serno, cid)Subject(cid, subj)Takes(sid, serno, exp-grade)
4
Functional DependenciesDescribe “Key-Like” Relationships
A key is a set of attributes where:If keys match, then the tuples match
A functional dependency (FD) is a generalization:If an attribute set determines another, written X ! Y
then if two tuples agree on attribute set X, they must agree on X:
sid ! name
What other FDs are there in this data? FDs are independent of our schema design
choice
5
Formal Definition of FD’s
Def. Given a relation schema R and subsets X, Y of R:An instance r of R satisfies FD X Y if,
for any two tuples t1, t2 2 r, t1[X ] = t2[X] implies t1[Y] = t2[Y]
For an FD to hold for schema R, it must hold for every possible instance of r
(Can a DBMS verify this? Can we determine this by looking at an instance?)
6
General Thoughts on Good Schemas
We want all attributes in every tuple to be determined by the tuple’s key attributes, i.e. part of a superkey (for key X Y, a superkey is a “non-minimal” X)What does this say about redundancy?
But: What about tuples that don’t have keys (other
than the entire value)? What about the fact that every attribute
determines itself?
7
Armstrong’s Axioms: Inferring FDs
Some FDs exist due to others; can compute using Armstrong’s axioms:
Reflexivity: If Y X then X Y (trivial dependencies)
name, sid name
Augmentation: If X Y then XW YWserno subj so serno, exp-grade subj, exp-grade
Transitivity: If X Y and Y Z then X Zserno cid and cid subj
so serno subj
8
Armstrong’s Axioms Lead to…
Union: If X Y and X Z then X YZ
Pseudotransitivity: If X Y and WY Z then XW Z
Decomposition: If X Y and Z Y then X Z
Let’s prove a few of these from Armstrong’s Axioms
9
Closure of a Set of FD’s
Defn. Let F be a set of FD’s. Its closure, F+, is the set of all FD’s:
{X Y | X Y is derivable from F by Armstrong’s Axioms}
Which of the following are in the closure of our Student-Course FD’s?name name
cid subj
serno subj
cid, sid subj
cid sid
10
Attribute Closures: Is SomethingDependent on X?
Defn. The closure of an attribute set X, X+, is:
X+ = {Y | X Y F +} This answers the question “is Y determined
(transitively) by X?”; compute X+ by:
Does sid, serno subj, exp-grade?
closure := X;repeat until no change {
if there is an FD U V in F such that U is in closure then add V to closure}
11
Equivalence of FD sets
Defn. Two sets of FD’s, F and G, are equivalent if
their closures are equivalent, F + = G +
e.g., these two sets are equivalent: {XY Z, X Y} and {X Z, X Y}
F + contains a huge number of FD’s (exponential in the size of the schema)
Would like to have smallest “representative” FD set
12
Minimal Cover
Defn. A FD set F is minimal if:1. Every FD in F is of the form X A,
where A is a single attribute2. For no X A in F is:
F – {X A } equivalent to F3. For no X A in F and Z X is: F – {X A } {Z A } equivalent to FDefn. F is a minimum cover for G if F is minimal
and is equivalent to G.e.g.,
{X Z, X Y} is a minimal cover for{XY Z, X Z, X Y}
in a sense,each FD is“essential”to the cover
we expresseach FD insimplest form
13
More on Closures
If F is a set of FD’s and X Y F + then for some attribute A Y, X A F +
Proof by counterexample. Assume otherwise and let Y = {A1,..., An} Since we assume X A1, ..., X An are in F +
then X A1 ... An is in F + by union rule,
hence, X Y is in F + which is a contradiction
14
Why Armstrong’s Axioms?Why are Armstrong’s axioms (or an
equivalent rule set) appropriate for FD’s? They are: Consistent: any relation satisfying FD’s in F will
satisfy those in F +
Complete: if an FD X Y cannot be derived by Armstrong’s axioms from F, then there exists some relational instance satisfying F but not X Y
In other words, Armstrong’s axioms derive all the FD’s that should hold
What is the goal of using these axioms?
15
Decomposition
Consider our original “bad” attribute set
We could decompose it into:
But this decomposition loses information about the relationship between students and courses. Why?
Stuff(sid, name, serno, subj, cid, exp-grade)
Student(sid, name)Course(serno, cid)Subject(cid, subj)
16
Lossless Join Decomposition
R1, … Rk is a lossless join decomposition of R w.r.t. an FD set F if for every instance r of R that satisfies F,
R1(r) ⋈ ... ⋈ Rk(r) = r
Consider:
What if we decompose on (sid, name) and (serno, subj, cid, exp-grade)?
sid
name serno subj
cid exp-grade
1 Sam 570103
AI 570 B
23 Nitin 550103
DB 550 A
17
Testing for Lossless Join
R1, R2 is a lossless join decomposition of R with respect to F iff at least one of the following dependencies is in F+
(R1 R2) R1 – R2
(R1 R2) R2 – R1
So for the FD set:sid nameserno cid, exp-gradecid subj
Is (sid, name) and (serno, subj, cid, exp-grade) a lossless decomposition?
18
Dependency Preservation
Ensures we can check whether a FD X Y is violated during DB updates, without using a join:
FZ, the projection of FD set F onto attribute set Z, is:
{X Y | X Y F +, X Y Z}i.e., it is those FDs only applicable to Z’s attributes
A decomposition R1, …, Rk is dependency preserving if F + = (FR1 ... FRk)+ (note we need an extra closure!)
We don’t lose the ability to test the “cover” of our FDs in a single table, just because we decompose
19
Example 1
For Schema R(sid, name, serno, cid, subj, exp-grade) and FD set:sid name serno cidcid subj sid, serno exp-grade
Is R1(sid, name) and R2(serno, subj, cid, exp-grade): A lossless decomposition? Is it dependency-preserving?
How about R1(sid, name) and R2(sid, serno, subj, cid, exp-grade)?
20
Example 2
Given schema R(name, street, city, st, zip, item, price),
FD set name street, city street, city ststreet, city zip name, item
priceand decomposition
R1(name, street, city, st, zip) and R2(name, item, price)
Is it lossless? Is it dependency preserving?
What if we replaced the first FD with name, street city?
21
A More Disturbing Example…
Given schema R(sid, fid, subj)and FD set: fid subj sid, subj fid
Consider the decomposition R1(sid, fid) and R2(fid, subj)
Is it lossless? Is it dependency preserving?
If it isn’t, can you think of a decomposition that is? Can you do this non-redundantly?
22
Redundancy vs. FDs
Ideally, we want a design s.t. for each nontrivial dependency X Y, X is a superkey for some relation schema in R
We just saw that this isn’t always possible in a non-redundant way…
Thus we have two kinds of normal forms, Boyce-Codd and Third Normal Form
23
Two Important Normal Forms
Boyce-Codd Normal Form (BCNF). For every relation scheme R and for every X A that holds over R,
either A X (it is trivial) ,oror X is a superkey for R
Third Normal Form (3NF). For every relation scheme R and for every X A that holds over R,
either A X (it is trivial), or X is a superkey for R, or A is a member of some key for R
24
Normal Forms Compared
BCNF is preferable, but sometimes in conflict with the goal of dependency preservation
It’s strictly stronger than 3NF
Let’s see algorithms to obtain: A BCNF lossless join decomposition
(nondeterministic) A 3NF lossless join, dependency preserving
decomposition
25
BCNF Decomposition Algorithm(from Korth et al.; our book gives a recursive version)
result := {R}compute F+while there is a relation schema Ri in result that isn’t in BCNF{
let A B be a nontrivial FD on Ri
s.t. A Ri is not in F+ and A and B are disjoint
result:= (result – Ri) {(Ri - B), (A,B)}}
26
3NF Decomposition Algorithm
Let F be a minimal coveri:=0for each FD A B in F { if none of the schemas Rj, 1 j i, contains AB { increment i Ri := (A, B) }}if no schema Rj, 1 j i contains a candidate key for R { increment i Ri := any candidate key for R}return (R1, …, Ri)
Build dep.-preservingdecomp.
Ensurelosslessdecomp.
27
Summary of Normalization
We can always decompose into 3NF and get: Lossless join Dependency preservation
But with BCNF we are only guaranteed lossless joins
BCNF is stronger than 3NF: every BCNF schema is also in 3NF
The BCNF algorithm is nondeterministic, so there is not a unique decomposition for a given schema R
29
Why XML?
XML is the confluence of several factors: The Web needed a more declarative format for data Documents needed a mechanism for extended tags Database people needed a more flexible interchange
format “Lingua franca” of data It’s parsable even if we don’t know what it means!
Original expectation: The whole web would go to XML instead of HTML
Today’s reality: Not so… But XML is used all over “under the covers”
30
Why DB People Like XML
Can get data from all sorts of sources Allows us to touch data we don’t own! This was actually a huge change in the DB community
Interesting relationships with DB techniques Useful to do relational-style operations Leverages ideas from object-oriented, semistructured
data
Blends schema and data into one format Unlike relational model, where we need schema first … But too little schema can be a drawback, too!
31
XML Anatomy<?xml version="1.0" encoding="ISO-8859-1" ?> <dblp> <mastersthesis mdate="2002-01-03" key="ms/Brown92"> <author>Kurt P. Brown</author> <title>PRPL: A Database Workload Specification Language</title> <year>1992</year> <school>Univ. of Wisconsin-Madison</school> </mastersthesis> <article mdate="2002-01-03" key="tr/dec/SRC1997-018"> <editor>Paul R. McJones</editor> <title>The 1995 SQL Reunion</title> <journal>Digital System Research Center Report</journal> <volume>SRC1997-018</volume> <year>1997</year> <ee>db/labs/dec/SRC1997-018.html</ee> <ee>http://www.mcjones.org/System_R/SQL_Reunion_95/</ee> </article>
Processing Instr.
Element
Attribute
Close-tag
Open-tag
32
Well-Formed XML
A legal XML document – fully parsable by an XML parser All open-tags have matching close-tags (unlike
so many HTML documents!), or a special:<tag/> shortcut for empty tags (equivalent to
<tag></tag>
Attributes (which are unordered, in contrast to elements) only appear once in an element
There’s a single root element XML is case-sensitive