November20,2013 DanielDeutch PierreSenellart AntoineAmarilli M.LamineBa ComplexityofMergingOrderedDocuments .......

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

.

...

Complexity of Merging Ordered Documents

Antoine Amarilli¹ M. Lamine Ba¹ Daniel Deutch² Pierre Senellart¹

1Télécom ParisTech, Paris, France

2Tel Aviv University, Tel Aviv, Israel

November 20, 2013

1/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Version control

Version control software (VCS) for line-based textual data.

Also, tree-shaped documents:

⇒ Texts (sections, paragraphs)

⇒ XML

⇒ Code

⇒ ...

Main problem: solvingconflicts.

2/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

DAG of versions

..

V3a

.

V2b

.

V3b .

V1

.

V2a

.

V4 (merged)

.

Root

3/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Conflict resolution

We mustmerge conflicting versions:

..

..

A.

....

..

. C ..

..

B

⇝ ..

..

A...

..

B...

..

D

and

..

..

A.

....

..

C...

..

F .

....

..

. B ..

..

E

..

..

A.

....

..I .

....

..

G...

..

H .

....

..

. C ..

..

B

⇒ How can such conflicts beresolved?

4/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Importance of order

Use of probabilistic XML for versioning already investigated:

⇒ M. Lamine Ba, T. Abdessalem, P. Senellart.

Uncertain Version Control in Open Collaborative Editing of Tree-Structured Documents.

DocEng’13, Florence, Italy.

⇒ M. Lamine Ba, T. Abdessalem, P. Senellart.

Merging Uncertain Multi-Version XML Documents.

DChanges’13, Florence, Italy.

Does not take orderinto account on child nodes.

Important source of uncertainty!

....

Root...

..

New-1

....

Root...

..

New-2

⇝

....

Root. ....

..

New-2 .

..

..

New-1

or

....

Root. ....

..

New-1 .

..

..

New-2

5/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Representing the merge

The actual merge is chosenby the user.

Computing all possible mergesis unreasonable.

Ideally, derive astrong representation systemto represent the possible merges, merges of merges, etc.

Here: study the complexity of a decision problem:

Input. Set of documentsDto merge.

Possible worldWof the merge.

Output. IsW really a possible mergeof D?

We restrict the study to an ordered listfor now.

(List of childrenfor a tree of height 1.)

6/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Table of contents

1... Introduction

2... Merge model

3... Merge without IDs

4... Merge with move

5... Conclusion

7/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Edition

Base version Bwith items and unique IDs.

Insert operation to insert a new node with a fresh ID.

# Section 1 Introduction 2 Problem 3 Conclusion

# Section 1 Introduction 2 Problem 4 Extension 5 Related work 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 3 Conclusion

8/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Edition

Base version Bwith items and unique IDs.

Insert operation to insert a new node with a fresh ID.

# Section 1 Introduction 2 Problem 3 Conclusion

# Section 1 Introduction 2 Problem 4 Extension 5 Related work 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 3 Conclusion

8/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Edition

Base version Bwith items and unique IDs.

Insert operation to insert a new node with a fresh ID.

# Section 1 Introduction 2 Problem 3 Conclusion

# Section 1 Introduction 2 Problem 4 Extension 5 Related work 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 3 Conclusion

8/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Edition

Base version Bwith items and unique IDs.

Insert operation to insert a new node with a fresh ID.

# Section 1 Introduction 2 Problem 3 Conclusion

# Section 1 Introduction 2 Problem 4 Extension 5 Related work 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 3 Conclusion

8/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Merge

Union of the nodes: D₁∪D₂.

Consistent order with the individual documents.

# Section 1 Introduction 2 Problem 4 Extension 5 Related work 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 4 Extension 7 Extension 5 Related work 3 Conclusion

9/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Merge

Union of the nodes: D₁∪D₂.

Consistent order with the individual documents.

# Section 1 Introduction 2 Problem 4 Extension 5 Related work 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 4 Extension 7 Extension 5 Related work 3 Conclusion

9/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Merge

Union of the nodes: D₁∪D₂.

Consistent order with the individual documents.

# Section 1 Introduction 2 Problem 4 Extension 5 Related work 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 4 Extension 7 Extension 5 Related work 3 Conclusion

9/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Merge

Union of the nodes: D₁∪D₂.

Consistent order with the individual documents.

# Section 1 Introduction 2 Problem 4 Extension 5 Related work 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 4 Extension 7 Extension 5 Related work 3 Conclusion

9/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Possible worlds

Exponentiallymany merges (_|D1|+|D2|

|D1|

)

PTIME algorithm to check if a world is a possible merge.

Given possible worldW with IDs, do the following:

Verify thedomain.

For everypairx<y,

For everydocumentDcontaining bothxandy, Checkifx<yin D.

# Section 1 Introduction 2 Problem 4 Extension 5 Related work 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 5 Related work 4 Extension 3 Conclusion

10/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Possible worlds

Exponentiallymany merges (_|D1|+|D2|

|D1|

)

PTIME algorithm to check if a world is a possible merge.

Given possible worldW with IDs, do the following:

Verify thedomain.

For everypairx<y,

For everydocumentDcontaining bothxandy, Checkifx<yin D.

# Section 1 Introduction 2 Problem 4 Extension 5 Related work 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 5 Related work 4 Extension 3 Conclusion

10/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Possible worlds

Exponentiallymany merges (_|D1|+|D2|

|D1|

)

PTIME algorithm to check if a world is a possible merge.

Given possible worldW with IDs, do the following:

Verify thedomain.

For everypairx<y,

For everydocumentDcontaining bothxandy, Checkifx<yin D.

# Section 1 Introduction 2 Problem 4 Extension 5 Related work 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 5 Related work 4 Extension 3 Conclusion

10/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Possible worlds

Exponentiallymany merges (_|D1|+|D2|

|D1|

)

PTIME algorithm to check if a world is a possible merge.

Given possible worldW with IDs, do the following:

Verify thedomain.

For everypairx<y,

For everydocumentDcontaining bothxandy, Checkifx<yin D.

# Section 1 Introduction 2 Problem 4 Extension 5 Related work 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 5 Related work 4 Extension

3 Conclusion _10/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Possible worlds

Exponentiallymany merges (_|D1|+|D2|

|D1|

)

PTIME algorithm to check if a world is a possible merge.

Given possible worldW with IDs, do the following:

Verify thedomain.

For everypairx<y,

For everydocumentDcontaining bothxandy, Checkifx<yin D.

# Section 1 Introduction 2 Problem 4 Extension 5 Related work 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 5 Related work 4 Extension 3 Conclusion _10/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Table of contents

1... Introduction

2... Merge model

3... Merge without IDs

4... Merge with move

5... Conclusion

11/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Possible worlds, without IDs

We may not have the IDsin W (e.g., user input).

Ambiguity about how to match elements!

# Section 1 Introduction 2 Problem 4 Extension 5 Related work 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 3 Conclusion

Section Introduction Preliminaries Problem Extension Related work Extension Conclusion

12/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Possible worlds, without IDs

We may not have the IDsin W (e.g., user input).

Ambiguity about how to match elements!

# Section 1 Introduction 2 Problem 4 Extension 5 Related work 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 3 Conclusion

Section Introduction Preliminaries Problem Extension Related work Extension Conclusion

12/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Possible worlds, without IDs

We may not have the IDsin W (e.g., user input).

Ambiguity about how to match elements!

# Section 1 Introduction 2 Problem 4 Extension 5 Related work 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 3 Conclusion

Section Introduction Preliminaries Problem Extension Related work Extension Conclusion

12/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Possible worlds, without IDs

We may not have the IDsin W (e.g., user input).

Ambiguity about how to match elements!

# Section 1 Introduction 2 Problem 4 Extension 5 Related work 3 Conclusion

# Section 1 Introduction 6 Preliminaries 2 Problem 7 Extension 3 Conclusion

Section Introduction Preliminaries Problem Extension Related work Extension Conclusion

12/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Dynamic algorithm

Ambiguity makes it look like the problem is hard In fact a dynamic algorithmsolves it inO(n²).

Show it on anexample.

# Label

1 A

3 A

4 B

2 B

# Label

1 A

5 A

2 B

6 A

Label A A A B B A

13/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Dynamic algorithm

Ambiguity makes it look like the problem is hard In fact a dynamic algorithmsolves it inO(n²).

Show it on anexample.

# Label

1 A

3 A

4 B

2 B

# Label

1 A

5 A

2 B

6 A

Label A A A B B A

13/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Dynamic algorithm

Ambiguity makes it look like the problem is hard In fact a dynamic algorithmsolves it inO(n²).

Show it on anexample.

# Label

1 A

3 A

4 B

2 B

# Label

1 A

5 A

2 B

6 A

Label A A A B B A

13/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Dynamic algorithm

Ambiguity makes it look like the problem is hard In fact a dynamic algorithmsolves it inO(n²).

Show it on anexample.

# Label

1 A

3 A

4 B

2 B

# Label

1 A

5 A

2 B

6 A

Label A A A B B A

13/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Dynamic algorithm (example)

Set of documents D:

A1 A3B4 B2. A1 A5B2 A6.

Possible world W: A A A B B A.

A₁ A₃ B₄ B₂ ·

A₁ A₅ B₂ A₆

·

14/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Dynamic algorithm (example)

Set of documents D:

A1 A3B4 B2. A1 A5B2 A6.

Possible world W: A A A B B A.

A₁ A₃ B₄ B₂ ·

A₁ A₅ B₂ A₆

·

14/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Dynamic algorithm (example)

Set of documents D:

A1 A3B4 B2. A1 A5B2 A6.

Possible world W: A A A B B A.

A₁ A₃ B₄ B₂ ·

A₁ A

A₅ A A B

B₂ A B B

A₆ A

· ·

14/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Dynamic algorithm (example)

Set of documents D:

A1 A3B4 B2. A1 A5B2 A6.

Possible world W: A A A B B A.

A₁ A₃ B₄ B₂ ·

A₁ A

A₅ A A B

B₂ A B B

A₆ A

· ·– ·

14/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Dynamic algorithm (example)

Set of documents D:

A1 A3B4 B2. A1 A5B2 A6.

Possible world W: A A A B B A.

A₁ A₃ B₄ B₂ ·

A₁ A

A₅ A A B

B₂ A B B

A₆ A –↓

· ·– ·

14/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Dynamic algorithm (example)

Set of documents D:

A1 A3B4 B2. A1 A5B2 A6.

Possible world W: A A A B B A.

A₁ A₃ B₄ B₂ ·

A₁ A

A₅ A A B

B₂ A B B– ↘

A₆ A –↓

· ·– ·

14/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Dynamic algorithm (example)

Set of documents D:

A1 A3B4 B2. A1 A5B2 A6.

Possible world W: A A A B B A.

A₁ A₃ B₄ B₂ ·

A₁ A

A₅ A A B

B₂ A B –→ B– ↘

A₆ A –↓

· ·– ·

14/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Dynamic algorithm (example)

Set of documents D:

A1 A3B4 B2. A1 A5B2 A6.

Possible world W: A A A B B A.

A₁ A₃ B₄ B₂ ·

A₁ A

A₅ A A B

B₂ A– → B –→ B– ↘

A₆ A –↓

· ·– ·

14/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Dynamic algorithm (example)

Set of documents D:

A1 A3B4 B2. A1 A5B2 A6.

Possible world W: A A A B B A.

A₁ A₃ B₄ B₂ ·

A₁ A

A₅ A A B– •

B₂ A– → B –→ B– ↘

A₆ A –↓

· ·– ·

14/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Dynamic algorithm (example)

Set of documents D:

A1 A3B4 B2. A1 A5B2 A6.

Possible world W: A A A B B A.

A₁ A₃ B₄ B₂ ·

A₁ A

A₅ A A –↓ B– •

B₂ A– → B –→ B– ↘

A₆ A –↓

· ·– ·

14/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Dynamic algorithm (example)

Set of documents D:

A1 A3B4 B2. A1 A5B2 A6.

Possible world W: A A A B B A.

A₁ A₃ B₄ B₂ ·

A₁ A

A₅ A–→↓ A –↓ B– •

B₂ A– → B –→ B– ↘

A₆ A –↓

· ·– ·

14/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Dynamic algorithm (example)

Set of documents D:

A1 A3B4 B2. A1 A5B2 A6.

Possible world W: A A A B B A.

A₁ A₃ B₄ B₂ ·

A₁ A– ↘

A₅ A–→↓ A –↓ B– •

B₂ A– → B –→ B– ↘

A₆ A –↓

· ·– ·

14/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

More documents

If we havemultiple documents to merge (sayk).

Generalized dynamic algorithmis inO(n^k).

Hence, PTIME for fixedk.

What if kis not fixed?

Certainly it is still in NP.

In fact, it is NP-hard (reduction from MinSAT).

Thanks: http://cstheory.stackexchange.com/a/19081/

15/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Table of contents

1... Introduction

2... Merge model

3... Merge without IDs

4... Merge with move

5... Conclusion

16/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Conflicts

Move operation to move arbitrary nodes.

Documents can disagree.

# Section 1 Problem A 2 Problem B

# Section 1 Problem A 2 Problem B 3 Conclusion

# Section 4 Intro 2 Problem B 1 Problem A

17/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Conflicts

Move operation to move arbitrary nodes.

Documents can disagree.

# Section 1 Problem A 2 Problem B

# Section 1 Problem A 2 Problem B 3 Conclusion

# Section 4 Intro 2 Problem B 1 Problem A

17/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Conflicts

Move operation to move arbitrary nodes.

Documents can disagree.

# Section 1 Problem A 2 Problem B

# Section 1 Problem A 2 Problem B 3 Conclusion

# Section 4 Intro 2 Problem B 1 Problem A

17/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Conflicts

Move operation to move arbitrary nodes.

Documents can disagree.

# Section 1 Problem A 2 Problem B

# Section 1 Problem A 2 Problem B 3 Conclusion

# Section 4 Intro 2 Problem B 1 Problem A

17/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Merge

Order isarbitrary if documents disagree.

However, try to maintain order withindocuments.

# Section 1 Problem A 2 Problem B 3 Conclusion

# Section 4 Intro 2 Problem B 1 Problem A

# Section 4 Intro 1 Problem A 2 Problem B 3 Conclusion

⇒ Only onething to decide here.

18/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Merge

Order isarbitrary if documents disagree.

However, try to maintain order withindocuments.

# Section 1 Problem A 2 Problem B 3 Conclusion

# Section 4 Intro 2 Problem B 1 Problem A

# Section 4 Intro 1 Problem A 2 Problem B 3 Conclusion

⇒ Only onething to decide here.

18/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Merge

Order isarbitrary if documents disagree.

However, try to maintain order withindocuments.

# Section 1 Problem A 2 Problem B 3 Conclusion

# Section 4 Intro 2 Problem B 1 Problem A

# Section 4 Intro 1 Problem A 2 Problem B 3 Conclusion

⇒ Only onething to decide here.

18/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Merge

Order isarbitrary if documents disagree.

However, try to maintain order withindocuments.

# Section 1 Problem A 2 Problem B 3 Conclusion

# Section 4 Intro 2 Problem B 1 Problem A

# Section 4 Intro 1 Problem A 2 Problem B 3 Conclusion

⇒ Only onething to decide here.

18/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Merge

Order isarbitrary if documents disagree.

However, try to maintain order withindocuments.

# Section 1 Problem A 2 Problem B 3 Conclusion

# Section 4 Intro 2 Problem B 1 Problem A

# Section 4 Intro 1 Problem A 2 Problem B 3 Conclusion

⇒ Only onething to decide here.

18/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Weird cases

Sometimes weird!

# Section 1 Problem A 3 Transition A-B 2 Problem B

# Section 2 Problem B 4 Transition B-A 1 Problem A

# Section 2 Problem B 4 Transition B-A 1 Problem A 3 Transition A-B

⇒ Satisfy a maximal subsetof the original constraints.

⇒ It is still in NPto decide if something is a possible world.

⇒ It is NP-hard even for 2 documents (from Unary 3-partition).

⇒ Thanks: http://cstheory.stackexchange.com/a/19415/

19/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Weird cases

Sometimes weird!

# Section 1 Problem A 3 Transition A-B 2 Problem B

# Section 2 Problem B 4 Transition B-A 1 Problem A

# Section 2 Problem B 4 Transition B-A 1 Problem A 3 Transition A-B

⇒ Satisfy a maximal subsetof the original constraints.

⇒ It is still in NPto decide if something is a possible world.

⇒ It is NP-hard even for 2 documents (from Unary 3-partition).

⇒ Thanks: http://cstheory.stackexchange.com/a/19415/

19/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Weird cases

Sometimes weird!

# Section 1 Problem A 3 Transition A-B 2 Problem B

# Section 2 Problem B 4 Transition B-A 1 Problem A

# Section 2 Problem B 4 Transition B-A 1 Problem A 3 Transition A-B

⇒ Satisfy a maximal subsetof the original constraints.

⇒ It is still in NPto decide if something is a possible world.

⇒ It is NP-hard even for 2 documents (from Unary 3-partition).

⇒ Thanks: http://cstheory.stackexchange.com/a/19415/

19/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Weird cases

Sometimes weird!

# Section 1 Problem A 3 Transition A-B 2 Problem B

# Section 2 Problem B 4 Transition B-A 1 Problem A

# Section 2 Problem B 4 Transition B-A 1 Problem A 3 Transition A-B

⇒ Satisfy a maximal subsetof the original constraints.

⇒ It is still in NPto decide if something is a possible world.

⇒ It is NP-hard even for 2 documents (from Unary 3-partition).

⇒ Thanks: http://cstheory.stackexchange.com/a/19415/

19/27

. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Weird cases

Sometimes weird!

# Section 1 Problem A 3 Transition A-B 2 Problem B

# Section 2 Problem B 4 Transition B-A 1 Problem A

# Section 2 Problem B 4 Transition B-A 1 Problem A 3 Transition A-B

⇒ Satisfy a maximal subsetof the original constraints.

⇒ It is still in NPto decide if something is a possible world.

⇒ It is NP-hard even for 2 documents (from Unary 3-partition).

⇒ Thanks: http://cstheory.stackexchange.com/a/19415/

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. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Weird cases

Sometimes weird!

# Section 1 Problem A 3 Transition A-B 2 Problem B

# Section 2 Problem B 4 Transition B-A 1 Problem A

# Section 2 Problem B 4 Transition B-A 1 Problem A 3 Transition A-B

⇒ Satisfy a maximal subsetof the original constraints.

⇒ It is still in NPto decide if something is a possible world.

⇒ It is NP-hard even for 2 documents (from Unary 3-partition).

⇒ Thanks: http://cstheory.stackexchange.com/a/19415/

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. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Table of contents

1... Introduction

2... Merge model

3... Merge without IDs

4... Merge with move

5... Conclusion

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. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Nested hierarchies

Necessary for XML.

Hardness results still holds.

Dynamic algorithm can be extended:

When solving the problem for twoforests:

Solve foreach pairof children.

Use the dynamic algorithm to solve the childsequence.

When solving the problem for twotrees:

Check equality of theroot labels.

Solve theforestsof children.

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. . . . . Introduction

. . . Merge model

. . . .

Merge without IDs

. . .

Merge with move

. . Conclusion

Relational algebra

Add orderto relational databases.

Allow arbitraryrelational operations.

Trackprovenance of the data.

Use provenance for order uncertainty.

Connected with provenance foraggregates.

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. . . .

Supplementary material

Thanks!

Thanks for your attention!

(Work in progress, questions and feedback welcome.)

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. . . .

Supplementary material

Hardness of multiple documents

Reduction fromMinSAT:

Input. n clauses ±x_i∨ ±x_j,kvariables, integerd Output. can we avoid satisfying >d clauses?

IDs:

e⁺_i for+xi,e⁻_i for−xi

cifor clausei Labels:

Clause, for clauses, Vari for variablei Document setD:

e^±_i cj if±xioccurs incj

Possible world W:

Vari from 1 tok Clausen−dtimes Vari from 1 tok Clausedtimes

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. . . .

Supplementary material

Hardness of multiple documents (example)

MinSAT instance:

C1:x∨y C2:¬x∨ ¬y d= 0

Encoding (D, andW):

# Label e⁺_x Var_x c₁ Clause

# Label e⁺_y Var_y c₁ Clause

# Label e⁻_x Var_x c₂ Clause

# Label e⁻_y Var_y c₂ Clause

Label Var_x Var_y Clause Clause Var_x Var_y

Thanks: http://cstheory.stackexchange.com/a/19081/

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. . . .

Supplementary material

Hardness of multiple documents (example)

MinSAT instance:

C1:x∨y C2:¬x∨ ¬y d= 0

Encoding (D, andW):

# Label e⁺_x Var_x c₁ Clause

# Label e⁺_y Var_y c₁ Clause

# Label e⁻_x Var_x c₂ Clause

# Label e⁻_y Var_y c₂ Clause

Label Var_x Var_y Clause Clause Var_x Var_y

Thanks: http://cstheory.stackexchange.com/a/19081/

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. . . .

Supplementary material

Hardness of multiple documents (example)

MinSAT instance:

C1:x∨y C2:¬x∨ ¬y d= 0

Encoding (D, andW):

# Label e⁺_x Var_x c₁ Clause

# Label e⁺_y Var_y c₁ Clause

# Label e⁻_x Var_x c₂ Clause

# Label e⁻_y Var_y c₂ Clause

Label Var_x Var_y Clause Clause Var_x Var_y

Thanks: http://cstheory.stackexchange.com/a/19081/

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. . . .

Supplementary material

Hardness of multiple documents (example)

MinSAT instance:

C1:x∨y C2:¬x∨ ¬y d= 0

Encoding (D, andW):

# Label e⁺_x Var_x c₁ Clause

# Label e⁺_y Var_y c₁ Clause

# Label e⁻_x Var_x c₂ Clause

# Label e⁻_y Var_y c₂ Clause

Label Var_x Var_y Clause Clause Var_x Var_y

Thanks: http://cstheory.stackexchange.com/a/19081/

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. . . .

Supplementary material

Hardness of multiple documents (example)

MinSAT instance:

C1:x∨y C2:¬x∨ ¬y d= 0

Encoding (D, andW):

# Label e⁺_x Var_x c₁ Clause

# Label e⁺_y Var_y c₁ Clause

# Label e⁻_x Var_x c₂ Clause

# Label e⁻_y Var_y c₂ Clause

Label Var_x Var_y Clause Clause Var_x Var_y

Thanks: http://cstheory.stackexchange.com/a/19081/

25/27

. . . .

Supplementary material

Hardness of multiple documents (example)

MinSAT instance:

C1:x∨y C2:¬x∨ ¬y d= 0

Encoding (D, andW):

# Label e⁺_x Var_x c₁ Clause

# Label e⁺_y Var_y c₁ Clause

# Label e⁻_x Var_x c₂ Clause

# Label e⁻_y Var_y c₂ Clause

Label Var_x Var_y Clause Clause Var_x Var_y Thanks: http://cstheory.stackexchange.com/a/19081/

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. . . .

Supplementary material

Hardness

Merging two versions is already NP-complete.

Reduction fromUnary 3-partition:

Input. 3mintegersn_i in unary, and integerB.

Output. can we partition in triples with sumalwaysB?

Create two blocksper integern_i:

Bothstartwith an element labeledOpen.

Bothcontainnielements labeledItem:

⇒ First block in one order.

⇒ Second block in reverse order.

Bothendwith an element labeledClose.

Document setD:

D1: concatenate all blocks in one order.

D2: concatenate all blocks in the reverse order.

Possible world W:

Open³,Item^B,Close³. Repeatmtimes.

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