XML Design for Relational Storagesolmaz/docs/ · database management systems. Questions: • How do...

XML Design for Relational Storage

Solmaz KolahiUniversity of Toronto

Leonid LibkinUniversity of Edinburgh

16th International World Wide Web Conference (WWW 2007)

Kolahi, Libkin @WWW 2007 – p.1/17

Motivation

Like relational data, XML documents can contain redundantinformation due to functional dependencies.

416

CityToronto

416 416 416

City City CityToronto TorontoToronto

AreaCode AreaCodeAreaCode AreaCode

Functional Dependency: @AreaCode → @City


Motivation

This redundancy is reflected in the relational storage of XMLdocuments.

AreaCode416

CityToronto

CityAreaCode

XML functional dependency

@AreaCode → @City AreaCode → City

regular functional dependency

416

416

416

416

Toronto

Toronto

Toronto

Toronto

XML Document Relational Storage


Motivation


AreaCode

416

416

416

416

City

Toronto

Toronto

Toronto

TorontoAreaCode416

CityToronto



equality-generating dependency

on

@AreaCode → @City ∀ R1(x, a) ∧ R2(y, c) ∧

R1(x′, a) ∧ R2(y′, c′) → c = c′


Motivation


AreaCode

416

416

416

416

City

Toronto

Toronto

Toronto

TorontoAreaCode416

CityToronto



equality-generating dependency

on

@AreaCode → @City ∀ R1(x, a) ∧ R2(y, c) ∧

R1(x′, a) ∧ R2(y′, c′) → c = c′

The more redundant the data, the more prone to updateanomalies.


Motivation

Solution: normalizing data to eliminate redundancies.

Normal forms for relational data:• BCNF eliminates all redundancies w.r.t. functional

dependencies but may lose dependencies.• 3NF eliminates some redundancies but preserves

dependencies.

Normal forms for XML documents:• XNF eliminates redundancies w.r.t. XML functional

dependencies.

If nontrivial FD p1, . . . , pn → q.@l holds then p1, . . . , pn → qshould also hold.


Motivation

Many XML documents are stored and queried using relationaldatabase management systems.

Questions:• How do XML constraints translate to the relational storage?• What is the best XML design to have a redundancy-free

relational storage?• What is a good XML design to have a low-redundancy

relational storage?

We use an information-theoretic technique to measure theredundancy of data.


Outline

• Overview of the information-theoretic measure.• Storing XML in relations and constraint translation.• Designing XML to achieve low redundancy in relational

storage.• Concluding remarks.


Measure of Information Content

• Proposed by Arenas & Libkin in PODS’03.• Used to measure the redundancy of a data value in a

database instance with respect to a set of constraints.• Intuitively, RICI(p|Σ) measures the relative information

content of position p in instance I w.r.t. constraints Σ.• Independent of data models and query languages.






Σ = {A → C}

RICI(P |Σ)

0.875

A B C D

1 2 3 4

1 2 3 5

Σ = {A → C, B → C}

RICI(P |Σ)

0.781






Σ = {A → C}

RICI(P |Σ)

0.875

0.781

A B C D

1 2 3 4

1 2 3 5

1 2 3 6

Σ = {A → C, B → C}

RICI(P |Σ)

0.781

0.629






Σ = {A → C}

RICI(P |Σ)

0.875

0.781

0.711

A B C D

1 2 3 4

1 2 3 5

1 2 3 6

1 2 3 7

Σ = {A → C, B → C}

RICI(P |Σ)

0.781

0.629

0.522






Σ = {A → C}

RICI(P |Σ)

0.875

0.781

0.711

0.658

A B C D

1 2 3 4

1 2 3 5

1 2 3 6

1 2 3 7

1 2 3 8

Σ = {A → C, B → C}

RICI(P |Σ)

0.781

0.629

0.522

0.446



R(A,B,C) Σ = {A → B}

A B C

1 2 3

1 2 4

Pick k such that adom(I) ⊆ {1, . . . , k} (k = 7).For every X ⊆ Pos(I) − {p} compute probability distributionP (a|X) for every a ∈ {1, . . . , k}.



R(A,B,C) Σ = {A → B}

A B C

2 3

1 2


P (2|X) =



R(A,B,C) Σ = {A → B}

A B C

1 2 3

1 2 1


P (2|X) =



R(A,B,C) Σ = {A → B}

A B C

4 2 3

1 2 7


P (2|X) =



R(A,B,C) Σ = {A → B}

A B C

4 2 3

1 2 7


P (2|X) = 48/



R(A,B,C) Σ = {A → B}

A B C

a 3

1 2


P (2|X) = 48/(48 + 6 × 42) = 0.16P (a|X) = 42/(48 + 6 × 42) = 0.14 for every a 6= 2



R(A,B,C) Σ = {A → B}

A B C

1 2 3

1 2 4


P (2|X) = 48/(48 + 6 × 42) = 0.16P (a|X) = 42/(48 + 6 × 42) = 0.14 for every a 6= 2

Conditional entropy : 2.8057

Average over all possible X: RICkI = 2.4558



R(A,B,C) Σ = {A → B}

A B C

1 2 3

1 2 4


P (2|X) = 48/(48 + 6 × 42) = 0.16P (a|X) = 42/(48 + 6 × 42) = 0.14 for every a 6= 2

Conditional entropy : 2.8057

Average over all possible X: RICkI = 2.4558

RICI(p|Σ) = limk→∞

RICkI (p | Σ)

log k= 0.875


Measure and Normal Forms

Ideally, we want to achieve a well-designed database by havingthe maximum information content for the entire database.

If this is not achievable, we want to maximize the informationcontent for all positions to the possible extent by enforcing somedesign conditions.

Given a condition C, guaranteed information content (GIC(C)) isthe largest number g ∈ [0, 1] such that for all positions in allinstances of all schemas satisfying C, the information content isnot smaller than g.

instances of (S, Σ)

RICI(p|Σ) ≥ g

Schema satisfying condition C(S, Σ) =⇒


Measure and Normal Forms

Known results (Arenas&Libkin: PODS’03, Kolahi&Libkin:PODS’06):

• For relational schemas with functional dependencies:◦ BCNF is the only normal form that guarantees

well-design databases: GIC(BCNF) = 1.◦ a good 3NF normalization guarantees a minimum of 1/2

information content: GIC(3NF+) = 1/2.• For XML designs with functional dependencies:

◦ XNF is the only normal form that guaranteeswell-designed XML documents: GIC(XNF) = 1.

How do we design XML to achieve high information content forthe relational storage?


Storing XML in Relations

Inlining technique: Given a DTD, separate relations are createdfor the root and elements occurring under a Kleene star.

student

db

contact

addressphone

@streetNo@number @city @postalCode

@name

*

**

student(stID, name, conID)

address(addID, conID, postalCode, streetNo, city)

phone(phID, conID, number)




student

db

contact

addressphone


@name

*

**




address[conID] ⊆FK student[conID]

phone[conID] ⊆FK student[conID]




student

db

contact

addressphone


@name

*

**




student, @postalCode → address

address[conID] ⊆FK student[conID]

phone[conID] ⊆FK student[conID]

∀ student(s, n, c) ∧ address(a, c, pc, st, apt, ct) ∧

student(s, n, c) ∧ address(a′, c, pc, st′, apt′, ct′)

→ a = a′


Redundancy-Free Design

XML Design

• DTD• XML functional depen-

dencies

Relational Storage

• Relational schema• Keys, foreign keys• EGDs



XML Design


dencies

Relational Storage


RICI (P |Σ) =?



XML Design


dencies

Relational Storage


RICI (P |Σ) =?

Theorem: XML design in XNF ⇔ Max information content forpositions in the relational storage.


Non-XNF Designs

If there are non-XNF functional dependencies,• we can do XNF normalization, but• normalizing is not always efficient.

With no restriction on FDs, we can have high redundancy in therelational storage.

• For any ε > 0, the information content can get as close as εto zero.

Can we restrict XML functional dependencies to guarantee areasonable information content?

• like the restriction that 3NF enforces to guarantee 1/2information content.


Relative vs Absolute Functional Dependencies

student

db

contact

addressphone


@name

*

**

Relative functional dependency:holds within each student element.

student, @postalCode → address

Absolute functional dependency:holds globally.

@postalCode → @city

We are interested in FDs relative to an element that occursunder a Kleene star in the DTD.


Another Good Design

Theorem: If all XML functional dependencies are either XNF orrelative, then the information content for all data values in therelational storage is not less than 1/2.


Another Good Design

Theorem: If all XML functional dependencies are either XNF orrelative, then the information content for all data values in therelational storage is not less than 1/2.

RICI (P |Σ) =?

only XNF functional dependencies ⇔ RICI(p|Σ) = 1

XNF or relative functional dependencies ⇒ RICI(p|Σ) ≥ 1/2


Another Relational Storage for XML

We can treat XML documents as edge-labeled graphs andshred them into relations:

Edge(source, target, label) orValue(vid, val)

Blabel(source, target)

Value(vid, val)


Another Relational Storage for XML

We can treat XML documents as edge-labeled graphs andshred them into relations:

Edge(source, target, label) orValue(vid, val)

Blabel(source, target)

Value(vid, val)

Our results extend:

only XNF functional dependencies ⇔ RICI(p|Σ) = 1

XNF or relative functional dependencies ⇒ RICI(p|Σ) ≥ 1/2

But enforcing XML functional dependencies requires arbitrarilymany more joins.


Conclusions

Design tips to have a good relational storage:• try to have an XNF design to ensure a redundancy-free

relational storage.• organize XML elements so that there is no absolute FD to

ensure a low-redundancy relational storage.

Comparing inlining and edge relational representations:• they are equivalent in terms of redundancy.• edge can be significantly worse when enforcing constraints.

Future work:• is it always possible to avoid absolute FDs?


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