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

Pattern Matching on Trees

•

^The^data^T is 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

Pattern Matching on Trees

•

^The^data^T is 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

Pattern Matching on Trees

•

^The^data^T is 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

Pattern Matching on Trees

•

^The^data^T is 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?

hα :4,β :6i,hα:4,β:7i

Pattern Matching on Trees

•

^The^data^T is 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

Definitions and Results on Trees

•

Tree patternsPcan be written as a kind oftree automaton...

•

Existing work has studied this problem and shown: Theorem [Bagan, 2006]

We can find all matches on a treeT of a tree patternP (with constantly many capture variables) with:

• PreprocessinglinearinT

and exponential in P

• DelayconstantinT

and exponential in P

•

Again, this only measures thecomplexity inT! We show: Theorem [Amarilli et al., 2019]

• Preprocessing inO(|T| ×Poly(P))

• DelaypolynomialinPandindependentfromT

Definitions and Results on Trees

•

Tree patternsPcan be written as a kind oftree automaton...

•

Existing work has studied this problem and shown:

Theorem [Bagan, 2006]

We can find all matches on a treeT of a tree patternP (with constantly many capture variables) with:

• PreprocessinglinearinT

and exponential in P

• DelayconstantinT

and exponential in P

•

Again, this only measures thecomplexity inT! We show: Theorem [Amarilli et al., 2019]

• Preprocessing inO(|T| ×Poly(P))

• DelaypolynomialinPandindependentfromT

Definitions and Results on Trees

•

Tree patternsPcan be written as a kind oftree automaton...

•

Existing work has studied this problem and shown:

Theorem [Bagan, 2006]

We can find all matches on a treeT of a tree patternP (with constantly many capture variables) with:

• PreprocessinglinearinT

and exponential in P

• DelayconstantinT

and exponential in P

•

Again, this only measures thecomplexity inT! We show: Theorem [Amarilli et al., 2019]

• Preprocessing inO(|T| ×Poly(P))

• DelaypolynomialinPandindependentfromT

Definitions and Results on Trees

•

Tree patternsPcan be written as a kind oftree automaton...

•

Existing work has studied this problem and shown:

Theorem [Bagan, 2006]

We can find all matches on a treeT of a tree patternP (with constantly many capture variables) with:

• PreprocessinglinearinTand exponential in P

• DelayconstantinTand exponential in P

•

Again, this only measures thecomplexity inT! We show:

Theorem [Amarilli et al., 2019]

• Preprocessing inO(|T| ×Poly(P))

• DelaypolynomialinPandindependentfromT

Definitions and Results on Trees

•

Tree patternsPcan be written as a kind oftree automaton...

•

Existing work has studied this problem and shown:

Theorem [Bagan, 2006]

We can find all matches on a treeT of a tree patternP (with constantly many capture variables) with:

• PreprocessinglinearinTand exponential in P

• DelayconstantinTand exponential in P

•

Again, this only measures thecomplexity inT! We show:

Theorem [Amarilli et al., 2019]

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

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 variableα is 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

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

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

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

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

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

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

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

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 × :

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

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 × :

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

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 × :

Proof Idea for Trees: Results given atreeT, we can build inO(|T| × |A|)aset circuitcapturing

Proof Idea for Trees: Results

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

Dans le document Enumerating Pattern Matches in Texts and Trees (Page 134-162)