Extension: From Text to Trees - Enumerating Pattern Matches in Words and Trees

Pattern matching on trees

•

^The^data^Tis no longertextbut is now atree:

<body>1

<section>3

<p>5

<img>7

<img>₆

<h2>4

<div>2

•

^The^pattern^Pasks about thestructureof the tree: Is there

α:

anh2header and

β:

animagein the same section?

•

^Results:^hα^:^4,^β ^:⁶ⁱ^,^hα^:^4,^β^:⁷ⁱ

18/36

Pattern matching on trees

•

^The^data^Tis no longertextbut is now atree:

<body>1

<section>3

<p>5

<img>7

<img>₆

<h2>4

<div>2

•

^The^pattern^Pasks about thestructureof the tree:

Is there

α:

anh2header and

β:

animagein the same section?

•

^Results:^hα^:^4,^β ^:⁶ⁱ^,^hα^:^4,^β^:⁷ⁱ

18/36

Pattern matching on trees

•

^The^data^Tis no longertextbut is now atree:

<body>1

<section>3

<p>5

<img>7

<img>₆

<h2>4

<div>2

•

^The^pattern^Pasks about thestructureof the tree:

Is there

α:

anh2header and

β:

animagein the same section?

•

^Results:

hα:4,β :6i,hα:4,β:7i

18/36

Pattern matching on trees

•

^The^data^Tis no longertextbut is now atree:

<body>1

<section>3

<p>5

<img>7

<img>₆

<h2>4

<div>2

•

^The^pattern^Pasks about thestructureof the tree:

Is thereα: anh2header andβ: animagein the same section?

•

^Results:

hα:4,β :6i,hα:4,β:7i

18/36

Pattern matching on trees

•

^The^data^Tis no longertextbut is now atree:

<body>1

<section>3

<p>5

<img>7

<img>₆

<h2>4

<div>2

•

^The^pattern^Pasks about thestructureof the tree:

Is thereα: anh2header andβ: animagein the same section?

•

^Results:^hα^:^4,^β ^:⁶ⁱ^,^hα^:^4,^β^:⁷ⁱ

18/36

Definitions and Results on Trees

•

Tree patterns can be written as atree automatonwith captures

•

^{Like for}^text, we can enumerate the matches oftree automata... Theorem [Bagan, 2006]

We can find all matches on a treeTof a tree automatonA (with constantly many capture variables) with:

• PreprocessinglinearinT

(data)

• DelayconstantinT

(data)

•

Again, this is only indata complexity!

•

^We^conjecturethe following bounds for this task (ongoing work): Conjecture

• PreprocessinglinearinT(data) andpolynomialinAandT(combined)

• DelayconstantinT(data) andpolynomialinAandT(combined)

19/36

Definitions and Results on Trees

•

Tree patterns can be written as atree automatonwith captures

•

^{Like for}^text, we can enumerate the matches oftree automata...

Theorem [Bagan, 2006]

We can find all matches on a treeTof a tree automatonA (with constantly many capture variables) with:

• PreprocessinglinearinT

(data)

• DelayconstantinT

(data)

•

Again, this is only indata complexity!

•

^We^conjecturethe following bounds for this task (ongoing work): Conjecture

• PreprocessinglinearinT(data) andpolynomialinAandT(combined)

• DelayconstantinT(data) andpolynomialinAandT(combined)

19/36

Definitions and Results on Trees

•

Tree patterns can be written as atree automatonwith captures

•

^{Like for}^text, we can enumerate the matches oftree automata...

Theorem [Bagan, 2006]

We can find all matches on a treeTof a tree automatonA (with constantly many capture variables) with:

• PreprocessinglinearinT(data)

• DelayconstantinT(data)

•

Again, this is only indata complexity!

•

^We^conjecturethe following bounds for this task (ongoing work): Conjecture

• PreprocessinglinearinT(data) andpolynomialinAandT(combined)

• DelayconstantinT(data) andpolynomialinAandT(combined)

19/36

Definitions and Results on Trees

•

Tree patterns can be written as atree automatonwith captures

•

^{Like for}^text, we can enumerate the matches oftree automata...

Theorem [Bagan, 2006]

We can find all matches on a treeTof a tree automatonA (with constantly many capture variables) with:

• PreprocessinglinearinT(data)

• DelayconstantinT(data)

•

Again, this is only indata complexity!

•

^We^conjecturethe following bounds for this task (ongoing work): Conjecture

• PreprocessinglinearinT(data) andpolynomialinAandT(combined)

• DelayconstantinT(data) andpolynomialinAandT(combined)

19/36

Definitions and Results on Trees

•

Tree patterns can be written as atree automatonwith captures

•

^{Like for}^text, we can enumerate the matches oftree automata...

Theorem [Bagan, 2006]

We can find all matches on a treeTof a tree automatonA (with constantly many capture variables) with:

• PreprocessinglinearinT(data)

• DelayconstantinT(data)

•

Again, this is only indata complexity!

•

^We^conjecturethe following bounds for this task (ongoing work):

Conjecture

• PreprocessinglinearinT(data) andpolynomialinAandT(combined)

• DelayconstantinT(data) andpolynomialinAandT(combined)

19/36

Proof idea for trees: structure

Similar structureto the previous proof, but with acircuit:

•

Preprocessing:Compute acircuit representationof the answers

•

Enumeration:Apply ageneric algorithmon the circuit

Tree

Phase 1:

Preprocessing Data structure

Phase 2:

Enumeration

hα:4,β:6i, hα:4,β:7i

Results

∃ssection(s)∧

s α∧s β∧

h2(α)∧img(β)

Pattern

20/36

Proof idea for trees: structure

Similar structureto the previous proof, but with acircuit:

•

Preprocessing:Compute acircuit representationof the answers

•

Enumeration:Apply ageneric algorithmon the circuit Tree

Proof idea for trees: set circuits

Aset circuitrepresents aset of answersto a patternP(α,β)

•

^Singleton^α^:6^→“the variableαis mapped to node6”

•

^Tuple^hα^:4,^β^:⁶ⁱ: tuple of singletons

•

The circuit captures asetof tuples, e.g.,

hα:4,β:6i,hα:4,β:7i

Three kinds ofset-valued gates:

•

Variable gate α:4 :

→ captures hα:4i

•

^{Union gate} ^∪ ^:

→ union of sets of tuples

•

Product gate × :

→ relational product

21/36

Proof idea for trees: set circuits

Aset circuitrepresents aset of answersto a patternP(α,β)

•

^Singleton^α^:6^→“the variableαis mapped to node6”

•

^Tuple^hα^:4,^β^:⁶ⁱ: tuple of singletons

•

The circuit captures asetof tuples, e.g.,

hα:4,β:6i,hα:4,β:7i

Three kinds ofset-valued gates:

•

Variable gate α:4 :

→ captures hα:4i

•

^{Union gate} ^∪ ^:

→ union of sets of tuples

•

Product gate × :

→ relational product

21/36

Proof idea for trees: set circuits

Aset circuitrepresents aset of answersto a patternP(α,β)

•

^Singleton^α^:6^→“the variableαis mapped to node6”

•

^Tuple^hα^:4,^β^:⁶ⁱ: tuple of singletons

•

The circuit captures asetof tuples, e.g.,

hα:4,β:6i,hα:4,β:7i

Three kinds ofset-valued gates:

•

Variable gate α:4 :

→ captures hα:4i

•

^{Union gate} ^∪ ^:

→ union of sets of tuples

•

Product gate × :

→ relational product

21/36

Proof idea for trees: set circuits

Aset circuitrepresents aset of answersto a patternP(α,β)

•

^Singleton^α^:6^→“the variableαis mapped to node6”

•

^Tuple^hα^:4,^β^:⁶ⁱ: tuple of singletons

•

The circuit captures asetof tuples, e.g.,

hα:4,β:6i,hα:4,β:7i

Three kinds ofset-valued gates:

•

Variable gate α:4 :

→ captures hα:4i

•

^{Union gate} ^∪ ^:

→ union of sets of tuples

•

Product gate × :

→ relational product

21/36

Proof idea for trees: set circuits

Aset circuitrepresents aset of answersto a patternP(α,β)

•

^Singleton^α^:6^→“the variableαis mapped to node6”

•

^Tuple^hα^:4,^β^:⁶ⁱ: tuple of singletons

•

The circuit captures asetof tuples, e.g.,

hα:4,β:6i,hα:4,β:7i

Three kinds ofset-valued gates:

•

Variable gate α:4 :

→ captures hα:4i

•

^{Union gate} ^∪ ^:

→ union of sets of tuples

•

Product gate × :

→ relational product

21/36

Proof idea for trees: set circuits

Aset circuitrepresents aset of answersto a patternP(α,β)

•

^Singleton^α^:6^→“the variableαis mapped to node6”

•

^Tuple^hα^:4,^β^:⁶ⁱ: tuple of singletons

•

The circuit captures asetof tuples, e.g.,

hα:4,β:6i,hα:4,β:7i

Three kinds ofset-valued gates:

•

Variable gate α:4 :

→ captures hα:4i

•

^{Union gate} ^∪ ^:

→ union of sets of tuples

•

Product gate × :

→ relational product

21/36

Proof idea for trees: set circuits

Aset circuitrepresents aset of answersto a patternP(α,β)

•

^Singleton^α^:6^→“the variableαis mapped to node6”

•

^Tuple^hα^:4,^β^:⁶ⁱ: tuple of singletons

•

The circuit captures asetof tuples, e.g.,

hα:4,β:6i,hα:4,β:7i

Three kinds ofset-valued gates:

•

Variable gate α:4 :

→ captures hα:4i

•

^{Union gate} ^∪ ^:

→ union of sets of tuples

•

Product gate × :

→ relational product

21/36

Proof idea for trees: set circuits

Aset circuitrepresents aset of answersto a patternP(α,β)

•

^Singleton^α^:6^→“the variableαis mapped to node6”

•

^Tuple^hα^:4,^β^:⁶ⁱ: tuple of singletons

•

The circuit captures asetof tuples, e.g.,

hα:4,β:6i,hα:4,β:7i

Three kinds ofset-valued gates:

•

Variable gate α:4 :

→ captures hα:4i

•

^{Union gate} ^∪ ^:

→ union of sets of tuples

•

Product gate × :

→ relational product

21/36

Proof idea for trees: set circuits

Aset circuitrepresents aset of answersto a patternP(α,β)

•

^Singleton^α^:6^→“the variableαis mapped to node6”

•

^Tuple^hα^:4,^β^:⁶ⁱ: tuple of singletons

•

The circuit captures asetof tuples, e.g.,

hα:4,β:6i,hα:4,β:7i

Three kinds ofset-valued gates:

•

Variable gate α:4 :

→ captures hα:4i

•

^{Union gate} ^∪ ^:

→ union of sets of tuples

•

Product gate × :

→ relational product

21/36

Proof idea for trees: set circuits

Aset circuitrepresents aset of answersto a patternP(α,β)

•

^Singleton^α^:6^→“the variableαis mapped to node6”

•

^Tuple^hα^:4,^β^:⁶ⁱ: tuple of singletons

•

The circuit captures asetof tuples, e.g.,

hα:4,β:6i,hα:4,β:7i

Three kinds ofset-valued gates:

•

Variable gate α:4 :

→ captures hα:4i

•

^{Union gate} ^∪ ^:

→ union of sets of tuples

•

Product gate × :

→ relational product

21/36

Proof idea for trees: set circuits

Aset circuitrepresents aset of answersto a patternP(α,β)

•

^Singleton^α^:6^→“the variableαis mapped to node6”

•

^Tuple^hα^:4,^β^:⁶ⁱ: tuple of singletons

•

The circuit captures asetof tuples, e.g.,

hα:4,β:6i,hα:4,β:7i

Three kinds ofset-valued gates:

•

Variable gate α:4 :

→ captures hα:4i

•

^{Union gate} ^∪ ^:

→ union of sets of tuples

•

Product gate × :

→ relational product

21/36

Proof idea for trees: set circuits

Aset circuitrepresents aset of answersto a patternP(α,β)

•

^Singleton^α^:6^→“the variableαis mapped to node6”

•

^Tuple^hα^:4,^β^:⁶ⁱ: tuple of singletons

•

The circuit captures asetof tuples, e.g.,

hα:4,β:6i,hα:4,β:7i

Three kinds ofset-valued gates:

•

Variable gate α:4 :

→ captures hα:4i

•

^{Union gate} ^∪ ^:

→ union of sets of tuples

•

Product gate × :

→ relational product

21/36

Proof idea for trees: set circuits

Aset circuitrepresents aset of answersto a patternP(α,β)

•

^Singleton^α^:6^→“the variableαis mapped to node6”

•

^Tuple^hα^:4,^β^:⁶ⁱ: tuple of singletons

•

The circuit captures asetof tuples, e.g.,

hα:4,β:6i,hα:4,β:7i

Three kinds ofset-valued gates:

•

Variable gate α:4 :

→ captures hα:4i

•

^{Union gate} ^∪ ^:

→ union of sets of tuples

•

Product gate × :

→ relational product

21/36

Proof idea for trees: set circuit construction given atreeT, we can build inO(|T| × |A|)aset circuitcapturing exactly the set of tuples{hα₁ :n₁, . . . ,α_k:n_kiin the output ofAonT

22/36

Proof idea for trees: set circuit construction (details)

•

^Automaton:“Select all node pairs(α,β)”

•

^States:^{∅,^α,^β,^αβ}

α:n

∪ ∪ ∪ ∪

∅ α β αβ

∪ ∪ ∪ ∪

∅ α β αβ

∪ ∪ ∪ ∪

∅ α β αβ

23/36

Proof idea for trees: set circuit construction (details)

•

^Automaton:“Select all node pairs(α,β)”

•

^States:^{∅,^α,^β,^αβ}

α:n

∪ ∪ ∪ ∪

∅ α β αβ

∪ ∪ ∪ ∪

∅ α β αβ

∪ ∪ ∪ ∪

∅ α β αβ

23/36

Proof idea for trees: set circuit construction (details)

•

^Automaton:“Select all node pairs(α,β)”

•

^States:^{∅,^α,^β,^αβ}

α:n

∪ ∪ ∪ ∪

∅ α β αβ

∪ ∪ ∪ ∪

∅ α β αβ

∪ ∪ ∪ ∪

∅ α β αβ

23/36

Proof idea for trees: set circuit construction (details)

•

^Automaton:“Select all node pairs(α,β)”

•

^States:^{∅,^α,^β,^αβ}

α:n

∪ ∪ ∪ ∪

∅ α β αβ

∪ ∪ ∪ ∪

∅ α β αβ

∪ ∪ ∪ ∪

∅ α β αβ

23/36

Proof idea for trees: set circuit construction (details)

•

^Automaton:“Select all node pairs(α,β)”

•

^States:^{∅,^α,^β,^αβ}

α:n

∪ ∪ ∪ ∪

∅ α β αβ

∪ ∪ ∪ ∪

∅ α β αβ

∪ ∪ ∪ ∪

∅ α β αβ

23/36

Proof idea for trees: set circuit construction (details)

•

^Automaton:“Select all node pairs(α,β)”

•

^States:^{∅,^α,^β,^αβ}

α:n

∪ ∪ ∪ ∪

∅ α β αβ

∪ ∪ ∪ ∪

∅ α β αβ

∪ ∪ ∪ ∪

∅ α β αβ

23/36

Proof idea for trees: set circuit construction (details)

•

^Automaton:“Select all node pairs(α,β)”

•

^States:^{∅,^α,^β,^αβ}

α:n

∪ ∪ ∪ ∪

∅ α β αβ

∪ ∪ ∪ ∪

∅ α β αβ

∪ ∪ ∪ ∪

∅ α β αβ

23/36

Proof idea for trees: enumeration on set circuits

Given a set circuitsatisfying some conditions, we can enumerate all tuples that it captures with linear preprocessing and constant delay E.g., for

hα:4,β:6i,hα:4,β:7i : enumeratehα:4,β:6ithenhα:4,β:7i

24/36

Proof idea for trees: general enumeration approach

→ Enumerate the setT(g)captured by each gateg

→ Do it bytop-down inductionon the circuit

Base case: variable α:n : enumeratehα:niand stop

∪-gate

∪

g₁ g₂

Concatenation:enumerateT(g₁) and then enumerateT(g₂)

×-gate

g1 g2

Lexicographic product: for everyt₁inT(g₁):

for everyt2 inT(g2): outputt₁+t₂

25/36

Proof idea for trees: general enumeration approach

→ Enumerate the setT(g)captured by each gateg

→ Do it bytop-down inductionon the circuit

Base case: variable α:n :

enumeratehα:niand stop

∪-gate

∪

g₁ g₂

Concatenation:enumerateT(g₁) and then enumerateT(g₂)

×-gate

g1 g2

Lexicographic product: for everyt₁inT(g₁):

for everyt2 inT(g2): outputt₁+t₂

25/36

Proof idea for trees: general enumeration approach

→ Enumerate the setT(g)captured by each gateg

→ Do it bytop-down inductionon the circuit

Base case: variable α:n : enumeratehα:niand stop

∪-gate

∪

g₁ g₂

Concatenation:enumerateT(g₁) and then enumerateT(g₂)

×-gate

g1 g2

Lexicographic product: for everyt₁inT(g₁):

for everyt2 inT(g2): outputt₁+t₂

25/36

Proof idea for trees: general enumeration approach

→ Enumerate the setT(g)captured by each gateg

→ Do it bytop-down inductionon the circuit

Base case: variable α:n : enumeratehα:niand stop

∪-gate

∪

g1 g2

Concatenation:enumerateT(g₁) and then enumerateT(g₂)

×-gate

g1 g2

Lexicographic product: for everyt₁inT(g₁):

for everyt2 inT(g2): outputt₁+t₂

25/36

Proof idea for trees: general enumeration approach

→ Enumerate the setT(g)captured by each gateg

→ Do it bytop-down inductionon the circuit

Base case: variable α:n : enumeratehα:niand stop

∪-gate

∪

g1 g2

Concatenation:enumerateT(g₁) and then enumerateT(g₂)

×-gate

g1 g2

Lexicographic product:

for everyt₁inT(g₁): for everyt2 inT(g2):

outputt₁+t₂

25/36

Proof idea for trees: circuit conditions

Enumeration relies on someconditionson the input circuit (d-DNNF):

•

^∪ ^{are all}deterministic:

For any two inputsg1andg2of a∪-gate, the captured setsT(g₁)andT(g₂)aredisjoint (they have no tuple in common)

→ Avoidsduplicate tuples

•

^× ^{are all}decomposable:

For any two inputsg₁andg₂of a×-gate, no variable has a path to bothg₁ andg₂

→ Avoidsduplicate singletons

α:4

β:6

∪

β:7

•

Also an additionalupwards-determinismcondition

•

Our circuit satisfies these thanks toautomaton determinism

26/36

Proof idea for trees: circuit conditions

Enumeration relies on someconditionson the input circuit (d-DNNF):

•

^∪ ^{are all}deterministic:

For any two inputsg1andg2of a∪-gate, the captured setsT(g₁)andT(g₂)aredisjoint (they have no tuple in common)

→ Avoidsduplicate tuples

•

^× ^{are all}decomposable:

For any two inputsg₁andg₂of a×-gate, no variable has a path to bothg₁ andg₂

→ Avoidsduplicate singletons

α:4

β:6

∪

β:7

•

Also an additionalupwards-determinismcondition

•

Our circuit satisfies these thanks toautomaton determinism

26/36

Proof idea for trees: circuit conditions

Enumeration relies on someconditionson the input circuit (d-DNNF):

•

^∪ ^{are all}deterministic:

For any two inputsg1andg2of a∪-gate, the captured setsT(g₁)andT(g₂)aredisjoint (they have no tuple in common)

→ Avoidsduplicate tuples

•

^× ^{are all}decomposable:

For any two inputsg₁andg₂of a×-gate, no variable has a path to bothg₁ andg₂

→ Avoidsduplicate singletons

α:4

β:6

∪

β:7

•

Also an additionalupwards-determinismcondition

•

Our circuit satisfies these thanks toautomaton determinism

26/36

Proof idea for trees: circuit conditions

Enumeration relies on someconditionson the input circuit (d-DNNF):

•

^∪ ^{are all}deterministic:

For any two inputsg1andg2of a∪-gate, the captured setsT(g₁)andT(g₂)aredisjoint (they have no tuple in common)

→ Avoidsduplicate tuples

•

^× ^{are all}decomposable:

For any two inputsg₁andg₂of a×-gate, no variable has a path to bothg₁ andg₂

→ Avoidsduplicate singletons

α:4

β:6

∪

β:7

•

Also an additionalupwards-determinismcondition

•

Our circuit satisfies these thanks toautomaton determinism

26/36

Proof idea for trees: circuit conditions

Enumeration relies on someconditionson the input circuit (d-DNNF):

•

^∪ ^{are all}deterministic:

For any two inputsg1andg2of a∪-gate, the captured setsT(g₁)andT(g₂)aredisjoint (they have no tuple in common)

→ Avoidsduplicate tuples

•

^× ^{are all}decomposable:

For any two inputsg₁andg₂of a×-gate, no variable has a path to bothg₁ andg₂

→ Avoidsduplicate singletons

α:4

β:6

∪

β:7

•

Also an additionalupwards-determinismcondition

•

Our circuit satisfies these thanks toautomaton determinism _26/36

Proof idea for trees: enumeration subtleties

•

We must not waste time in gates capturing∅

→ Labelthem during the preprocessing

•

We must not waste time because of gates capturing hi

→ Homogenizationto set them aside

•

We must not waste time inhierarchies of∪-gates

→ Precompute areachability index(usesupwards-determinism)

27/36

Proof idea for trees: enumeration subtleties

•

We must not waste time in gates capturing∅

→ Labelthem during the preprocessing

•

We must not waste time because of gates capturing hi

→ Homogenizationto set them aside

•

We must not waste time inhierarchies of∪-gates

→ Precompute areachability index(usesupwards-determinism)

27/36

Proof idea for trees: enumeration subtleties

•

We must not waste time in gates capturing∅

→ Labelthem during the preprocessing

•

We must not waste time because of gates capturing hi

→ Homogenizationto set them aside

•

We must not waste time inhierarchies of∪-gates

→ Precompute areachability index(usesupwards-determinism)

27/36

Proof idea for trees: enumeration subtleties

•

We must not waste time in gates capturing∅

→ Labelthem during the preprocessing

•

We must not waste time because of gates capturing hi

→ Homogenizationto set them aside

•

We must not waste time inhierarchies of∪-gates

→ Precompute areachability index(usesupwards-determinism)

27/36

Proof idea for trees: enumeration subtleties

•

We must not waste time in gates capturing∅

→ Labelthem during the preprocessing

•

We must not waste time because of gates capturing hi

→ Homogenizationto set them aside

•

We must not waste time inhierarchies of∪-gates

→ Precompute areachability index(usesupwards-determinism)

27/36

Proof idea for trees: enumeration subtleties

•

We must not waste time in gates capturing∅

→ Labelthem during the preprocessing

•

We must not waste time because of gates capturing hi

→ Homogenizationto set them aside

•

We must not waste time inhierarchies of∪-gates

→ Precompute areachability index(usesupwards-determinism)

27/36

Dans le document Enumerating Pattern Matches in Words and Trees (Page 131-182)