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Dynamic Complexity of the Dyck Reachability Patricia Bouyer-Decitre - - PowerPoint PPT Presentation

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Dynamic Complexity of the Dyck Reachability Patricia Bouyer-Decitre & Vincent Jug CNRS, LSV & ENS Paris-Saclay 25/04/2017 P. Bouyer-Decitre & V. Jug Dynamic Complexity of the Dyck Reachability Contents Dynamic Complexity of

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SLIDE 1

Dynamic Complexity of the Dyck Reachability

Patricia Bouyer-Decitre & Vincent Jugé

CNRS, LSV & ENS Paris-Saclay

25/04/2017

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 2

Contents

1

Dynamic Complexity of Decision Problems

2

Reachability and its Variants

3

The Result

4

Conclusion and Future Work

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 3

Dynamic Complexity of Decision Problems

Modulo 3 Decision

Input: Bit vector b1 ¨ b2 ¨ . . . ¨ bn P Fn

3

Output: Yes if b1 ` b2 ` . . . ` bn “ 0 — No otherwise

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 4

Dynamic Complexity of Decision Problems

Modulo 3 Decision

Input: Bit vector b1 ¨ b2 ¨ . . . ¨ bn P Fn

3

Output: Yes if b1 ` b2 ` . . . ` bn “ 0 — No otherwise Solving this problem. . . Static world: membership in a regular language

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 5

Dynamic Complexity of Decision Problems

Modulo 3 Decision

Input: Bit vector b1 ¨ b2 ¨ . . . ¨ bn P Fn

3

Output: Yes if b1 ` b2 ` . . . ` bn “ 0 — No otherwise Solving this problem. . . Static world: membership in a regular language Dynamic world: what if some bit bk changes?

§ Maintain predicates Auxi ” pb1 ` b2 ` . . . ` bn “ iq for i P F3 § Update the values of Aux0, Aux1, Aux2 when bk changes § Use the new value of Aux0 and answer the problem

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 6

Dynamic Complexity of Decision Problems

Modulo 3 Decision

Input: Bit vector b1 ¨ b2 ¨ . . . ¨ bn P Fn

3

Output: Yes if b1 ` b2 ` . . . ` bn “ 0 — No otherwise Solving this problem. . . Static world: membership in a regular language Dynamic world: what if some bit bk changes?

§ Maintain predicates Auxi ” pb1 ` b2 ` . . . ` bn “ iq for i P F3 § Update the values of Aux0, Aux1, Aux2 when bk changes § Use the new value of Aux0 and answer the problem

How complex is it? Static world: linear time Dynamic world:

§ Easy initial instance pb1 “ b2 “ . . . “ bn “ 0q: constant time § Each update: constant time

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 7

Dynamic Complexity of Decision Problems

Reachability in DAGs

Input: Directed acyclic graph G “ pV , Eq & two vertices s, t P V Output: Yes if D path from s to t in G — No otherwise

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 8

Dynamic Complexity of Decision Problems

Reachability in DAGs

Input: Directed acyclic graph G “ pV , Eq & two vertices s, t P V Output: Yes if D path from s to t in G — No otherwise Solving this problem. . . Static world: use your favorite graph exploration algorithm Dynamic world: what if edge u Ñ v is inserted/deleted?

§ Maintain a predicate E‹px, yq ” pD path from x to y in Gq for x, y P V § Update the values of E‹px, yq when u Ñ v is inserted/deleted § Use the new value of E‹ps, tq and answer the problem

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 9

Dynamic Complexity of Decision Problems

Reachability in DAGs

Input: Directed acyclic graph G “ pV , Eq & two vertices s, t P V Output: Yes if D path from s to t in G — No otherwise Solving this problem. . . Static world: use your favorite graph exploration algorithm Dynamic world: what if edge u Ñ v is inserted/deleted?

§ Maintain a predicate E‹px, yq ” pD path from x to y in Gq for x, y P V § Update the values of E‹px, yq when u Ñ v is inserted/deleted § Use the new value of E‹ps, tq and answer the problem

How complex is it? Static world: linear time Dynamic world:

§ Easy initial edgeless instance: FO formulas § Each update: FO formulas

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 10

Dynamic Complexity of Decision Problems

Reachability in DAGs

Input: Directed acyclic graph G “ pV , Eq & two vertices s, t P V Output: Yes if D path from s to t in G — No otherwise Solving this problem. . . Static world: use your favorite graph exploration algorithm Dynamic world: what if edge u Ñ v is inserted/deleted?

§ Maintain a predicate E‹px, yq ” pD path from x to y in Gq for x, y P V § Update the values of E‹px, yq when u Ñ v is inserted/deleted § Use the new value of E‹ps, tq and answer the problem

How complex is it? Static world: linear time Dynamic world:

§ Easy initial edgeless instance: FO formulas (parallel «constant time) § Each update: FO formulas (parallel «constant time)

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 11

FO formulas ñ parallel «constant time φ “ Dx.@y.ψpx, yq_ψpy, xq

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 12

FO formulas ñ parallel «constant time φ “ Dx.@y.ψpx, yq_ψpy, xq

ψpe1, e1q ψpe1, e2q ψpe2, e1q ψpe2, e2q

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 13

FO formulas ñ parallel «constant time φ “ Dx.@y.ψpx, yq_ψpy, xq

ψpe1, e1q ψpe1, e2q ψpe2, e1q ψpe2, e2q

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 14

FO formulas ñ parallel «constant time φ “ Dx.@y.ψpx, yq_ψpy, xq _ _ _ _

x “e1 y “e1 x “e1 y “e2 x “e2 y “e1 x “e2 y “e2

ψpe1, e1q ψpe1, e2q ψpe2, e1q ψpe2, e2q

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 15

FO formulas ñ parallel «constant time φ “ Dx.@y.ψpx, yq_ψpy, xq _ _ _ _

x “e1 y “e1 x “e1 y “e2 x “e2 y “e1 x “e2 y “e2

^ ^

x “e1 x “e2

ψpe1, e1q ψpe1, e2q ψpe2, e1q ψpe2, e2q

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 16

FO formulas ñ parallel «constant time φ “ Dx.@y.ψpx, yq_ψpy, xq _ _ _ _

x “e1 y “e1 x “e1 y “e2 x “e2 y “e1 x “e2 y “e2

^ ^

x “e1 x “e2

_

ψpe1, e1q ψpe1, e2q ψpe2, e1q ψpe2, e2q

φ

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 17

Dynamic Complexity of Decision Problems

Reachability in DAGs with FO formulas

Initialization (on the edgeless graph): E‹px, yq Ð px “ yq E‹px, yq Ð E‹px, yq Ð E‹px, yq Ð E‹px, yq Ð

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 18

Dynamic Complexity of Decision Problems

Reachability in DAGs with FO formulas

Initialization (on the edgeless graph): Update after inserting the edge u Ñ v u v x y E‹px, yq Ð E‹px, yq E‹px, yq Ð E‹px, yq Ð E‹px, yq Ð E‹px, yq Ð

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 19

Dynamic Complexity of Decision Problems

Reachability in DAGs with FO formulas

Initialization (on the edgeless graph): Update after inserting the edge u Ñ v: u v x y E‹px, yq Ð E‹px, yq_ pE‹px, uq ^ E‹pv, yqq E‹px, yq Ð E‹px, yq Ð E‹px, yq Ð

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-20
SLIDE 20

Dynamic Complexity of Decision Problems

Reachability in DAGs with FO formulas

Initialization (on the edgeless graph): Update after inserting the edge u Ñ v: Update after deleting the edge u Ñ v u v x y E‹px, yq Ð pE‹px, yq ^ E‹px, uqq E‹px, yq Ð E‹px, yq Ð E‹px, yq Ð E‹px, yq Ð

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-21
SLIDE 21

Dynamic Complexity of Decision Problems

Reachability in DAGs with FO formulas

Initialization (on the edgeless graph): Update after inserting the edge u Ñ v: Update after deleting the edge u Ñ v u v x y E‹px, yq Ð pE‹px, yq ^ E‹px, uqq_ pE‹px, yq ^ E‹py, uqq E‹px, yq Ð E‹px, yq Ð E‹px, yq Ð

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 22

Dynamic Complexity of Decision Problems

Reachability in DAGs with FO formulas

Initialization (on the edgeless graph): Update after inserting the edge u Ñ v: Update after deleting the edge u Ñ v: u v a b x y E‹px, yq Ð pE‹px, yq ^ E‹px, uqq_ pE‹px, yq ^ E‹py, uqq_ pDa.Db.E‹px, aq ^ E‹pb, yq^ ppa Ñ bq ^ pa, bq ‰ pu, vq^ pE‹pa, uq ^ E‹pb, uqq

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 23

Dynamic Complexity of Decision Problems

Reachability in DAGs with FO formulas

Initialization (on the edgeless graph): Update after inserting the edge u Ñ v: Update after deleting the edge u Ñ v:

Definition (Patnaik & Immerman 97, Dong & Su & Topor 93)

A decision problem with updates is in C-DynFO if D predicates s.t.: every predicate can be initialized in C every predicate can be updated in FO

  • ne predicate is the goal predicate
  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-24
SLIDE 24

Dynamic Complexity of Decision Problems

Reachability in DAGs with FO formulas

Initialization (on the edgeless graph): Update after inserting the edge u Ñ v: Update after deleting the edge u Ñ v:

Definition (Patnaik & Immerman 97, Dong & Su & Topor 93)

A decision problem with updates is in DynFO if D predicates s.t.: every predicate can be initialized in FO every predicate can be updated in FO

  • ne predicate is the goal predicate
  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-25
SLIDE 25

Dynamic Complexity of Decision Problems

Reachability in DAGs with FO formulas

Initialization (on the edgeless graph): Update after inserting the edge u Ñ v: Update after deleting the edge u Ñ v:

Definition (Patnaik & Immerman 97, Dong & Su & Topor 93)

A decision problem with updates is in DynFO if D predicates s.t.: every predicate can be initialized in FO every predicate can be updated in FO

  • ne predicate is the goal predicate

$1000000 question: P ? “ NP

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-26
SLIDE 26

Dynamic Complexity of Decision Problems

Reachability in DAGs with FO formulas

Initialization (on the edgeless graph): Update after inserting the edge u Ñ v: Update after deleting the edge u Ñ v:

Definition (Patnaik & Immerman 97, Dong & Su & Topor 93)

A decision problem with updates is in DynFO if D predicates s.t.: every predicate can be initialized in FO every predicate can be updated in FO

  • ne predicate is the goal predicate

$1000000 question: P ? “ NP 1000000 kr. question: PTime-DynFO ? “ PTime

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 27

Dynamic Complexity of Decision Problems

Some more problems in DynFO Reachability in undirected graphs (Patnaik & Immerman 97) Integer multiplication (Patnaik & Immerman 97) Dyck reachability in DAGs (Weber & Schwentick 07) Context-free language membership (Gelade et al. 08) Distance in undirected graphs (Grädel & Siebertz 12) Reachability in directed graphs (Datta et al. 15) Context-free reachability in DAGs (Muñoz et al. 16)

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 28

Dynamic Complexity of Decision Problems

Some more problems in DynFO Reachability in undirected graphs (Patnaik & Immerman 97) Integer multiplication (Patnaik & Immerman 97) Dyck reachability in DAGs (Weber & Schwentick 07) Context-free language membership (Gelade et al. 08) Distance in undirected graphs (Grädel & Siebertz 12) Reachability in directed graphs (Datta et al. 15) Context-free reachability in DAGs (Muñoz et al. 16) Some problems that are probably not in PTime-DynFO Reachability in 2-player games (Patnaik & Immerman 97) Dyck reachability in (un)directed graphs (Bouyer & Jugé 17)

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 29

Contents

1

Dynamic Complexity of Decision Problems

2

Reachability and its Variants

3

The Result

4

Conclusion and Future Work

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-30
SLIDE 30

Reachability in 2-Player Games

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 31

Reachability in 2-Player Games

Moving a token on a finite directed graph

Input: Directed graph G “ pV , Eq, a partition VA Z VB “ V , two vertices s, t P V

§ A token is first placed in s § Alice controls VA, Barbara controls VB § Players move the token along edges of G (when they can)

Alice wins if either:

§ the token reaches a vertex x P VB without outgoing edge § the token reaches the vertex t

Output: Yes if Alice has a winning strategy — No otherwise

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 32

Reachability in 2-Player Games

Who wins?

t s

Barbara’s vertices Alice’s vertices

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 33

Reachability in 2-Player Games

Who wins?

t s

Barbara’s vertices Alice’s vertices Alice’s winning vertices

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 34

Reachability in 2-Player Games

Who wins?

t s

Barbara’s vertices Alice’s vertices Alice’s winning vertices

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 35

Reachability in 2-Player Games

Who wins?

t s

Barbara’s vertices Alice’s vertices Alice’s winning vertices

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 36

Reachability in 2-Player Games

Who wins?

t s

Barbara’s vertices Alice’s vertices Alice’s winning vertices

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 37

Reachability in 2-Player Games

Who wins? Alice !

t s

Barbara’s vertices Alice’s vertices Alice’s winning vertices

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-38
SLIDE 38

Reachability in 2-Player Games

Who wins? Alice !

t s

Barbara’s vertices Alice’s vertices Alice’s winning vertices

Theorem (Patnaik & Immerman 97)

Reachability in 2-player games is in PTime-DynFO iff PTime = PTime-DynFO

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 39

Reachability in 2-Player Games

Who wins? Alice !

t s

Barbara’s vertices Alice’s vertices Alice’s winning vertices

Theorem (Patnaik & Immerman 97)

Reachability in 2-player games is in PTime-DynFO iff PTime = PTime-DynFO PTime-complete for LogSpace reductions

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 40

Reachability in 2-Player Games

Who wins? Alice !

t s

Barbara’s vertices Alice’s vertices Alice’s winning vertices

Theorem (Patnaik & Immerman 97)

Reachability in 2-player games is in PTime-DynFO iff PTime = PTime-DynFO PTime-complete for LogSpace reductions PTime-completedyn for dynamic-adapted reductions

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 41

Reachability in 2-Player Games

Who wins? Alice !

t s

Barbara’s vertices Alice’s vertices Alice’s winning vertices

Theorem (Patnaik & Immerman 97)

Reachability in 2-player games is in PTime-DynFO iff PTime = PTime-DynFO PTime-complete for LogSpace reductions PTime-completedyn for dynamic-adapted reductions

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 42

Dyck Reachability

Dyck words = Well-parenthesized words Are these words Dyck? ( [ ( ) ] ( ) ) ( [ ( ] ) ) ( [ ( ) ] ( ) ]

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 43

Dyck Reachability

Dyck words = Well-parenthesized words Are these words Dyck? ( [ ( ) ] ( ) ) ( [ ( ] ) ) ( [ ( ) ] ( ) ]

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 44

Dyck Reachability

Dyck words = Well-parenthesized words Are these words Dyck? ( [ ( ) ] ( ) ) ( [ ( ] ) ) ( [ ( ) ] ( ) ]

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

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SLIDE 45

Dyck Reachability

Dyck words = Well-parenthesized words Are these words Dyck? ( [ ( ) ] ( ) ) ( [ ( ] ) ) ( [ ( ) ] ( ) ]

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-46
SLIDE 46

Dyck Reachability

Dyck words = Well-parenthesized words Are these words Dyck? ( [ ( ) ] ( ) ): ( [ ( ] ) ) ( [ ( ) ] ( ) ]

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-47
SLIDE 47

Dyck Reachability

Dyck words = Well-parenthesized words Are these words Dyck? ( [ ( ) ] ( ) ): ( [ ( ] ) ) ( [ ( ) ] ( ) ]

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-48
SLIDE 48

Dyck Reachability

Dyck words = Well-parenthesized words Are these words Dyck? ( [ ( ) ] ( ) ): ( [ ( ] ) ): ✗ ( [ ( ) ] ( ) ]

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-49
SLIDE 49

Dyck Reachability

Dyck words = Well-parenthesized words Are these words Dyck? ( [ ( ) ] ( ) ): ( [ ( ] ) ): ✗ ( [ ( ) ] ( ) ]

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-50
SLIDE 50

Dyck Reachability

Dyck words = Well-parenthesized words Are these words Dyck? ( [ ( ) ] ( ) ): ( [ ( ] ) ): ✗ ( [ ( ) ] ( ) ]

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-51
SLIDE 51

Dyck Reachability

Dyck words = Well-parenthesized words Are these words Dyck? ( [ ( ) ] ( ) ): ( [ ( ] ) ): ✗ ( [ ( ) ] ( ) ]

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-52
SLIDE 52

Dyck Reachability

Dyck words = Well-parenthesized words Are these words Dyck? ( [ ( ) ] ( ) ): ( [ ( ] ) ): ✗ ( [ ( ) ] ( ) ]: ✗

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-53
SLIDE 53

Dyck Reachability

Dyck words = Well-parenthesized words Are these words Dyck? ( [ ( ) ] ( ) ): ( [ ( ] ) ): ✗ ( [ ( ) ] ( ) ]: ✗ Dyck paths = Paths labeled with Dyck words

) ( [ [ ]

v1 v2 v3 v4

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-54
SLIDE 54

Dyck Reachability

Dyck words = Well-parenthesized words Are these words Dyck? ( [ ( ) ] ( ) ): ( [ ( ] ) ): ✗ ( [ ( ) ] ( ) ]: ✗ Dyck paths = Paths labeled with Dyck words

1 1 1

v1 v2 v3 v4

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-55
SLIDE 55

Dyck Reachability

Dyck words = Well-parenthesized words Are these words Dyck? ( [ ( ) ] ( ) ): ( [ ( ] ) ): ✗ ( [ ( ) ] ( ) ]: ✗ Dyck paths = Paths labeled with Dyck words

1 1 1

v1 v2 v3 v4 v4

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-56
SLIDE 56

Dyck Reachability

Dyck words = Well-parenthesized words Are these words Dyck? ( [ ( ) ] ( ) ): ( [ ( ] ) ): ✗ ( [ ( ) ] ( ) ]: ✗ Dyck paths = Paths labeled with Dyck words

1 1 1

v1 v2 v3 v4 v3

1

Ý Ñv4

1

Ý Ñv2

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-57
SLIDE 57

Dyck Reachability

Dyck words = Well-parenthesized words Are these words Dyck? ( [ ( ) ] ( ) ): ( [ ( ] ) ): ✗ ( [ ( ) ] ( ) ]: ✗ Dyck paths = Paths labeled with Dyck words

1 1 1

v1 v2 v3 v4 v2 Ý Ñv3

1

Ý Ñv4

1

Ý Ñv2 Ý Ñv1

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-58
SLIDE 58

Dyck Reachability

Dyck words = Well-parenthesized words Are these words Dyck? ( [ ( ) ] ( ) ): ( [ ( ] ) ): ✗ ( [ ( ) ] ( ) ]: ✗ Dyck paths = Paths labeled with Dyck words

1 1 1

v1 v2 v3 v4 v3

1

Ý Ñv4

1

Ý Ñv2 Ý Ñv3

1

Ý Ñv4

1

Ý Ñv2 Ý Ñv1

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-59
SLIDE 59

Dyck Reachability

Dyck words = Well-parenthesized words Are these words Dyck? ( [ ( ) ] ( ) ): ( [ ( ] ) ): ✗ ( [ ( ) ] ( ) ]: ✗ Dyck paths = Paths labeled with Dyck words

1 1 1

v1 v2 v3 v4 v3

1

Ý Ñv4

1

Ý Ñv2 Ý Ñv3

1

Ý Ñv4

1

Ý Ñv2 Ý Ñv1

Theorem (Weber & Schwentick 05)

Computing endpoints of Dyck paths in acyclic graphs is in DynFO.

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-60
SLIDE 60

Contents

1

Dynamic Complexity of Decision Problems

2

Reachability and its Variants

3

The Result

4

Conclusion and Future Work

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-61
SLIDE 61

Dyck Reachability in Directed Graphs is Hard

Theorem #1 (Bouyer & Jugé 17)

2-letter Dyck reachability in directed graphs is in PTime-DynFO iff PTime = PTime-DynFO

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-62
SLIDE 62

Dyck Reachability in Directed Graphs is PTime-Harddyn

Theorem #1 (Bouyer & Jugé 17)

2-letter Dyck reachability in directed graphs is in PTime-DynFO iff PTime = PTime-DynFO Use a dynamic-adapted reduction from Reachability in 2-player games!

2 3 4

Alice

1 5

Barbara s t

2 3 4 1 5

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-63
SLIDE 63

Dyck Reachability in Directed Graphs is PTime-Harddyn

Theorem #1 (Bouyer & Jugé 17)

2-letter Dyck reachability in directed graphs is in PTime-DynFO iff PTime = PTime-DynFO Use a dynamic-adapted reduction from Reachability in 2-player games!

2 3 4

Alice

1 5

Barbara s t

2 3 4 1 5 1 2 3 4 5 t s

2 3 5 1 5 1 4 4 3 2 4 2 5 2 4

1 2 3 4 5 t s

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-64
SLIDE 64

Dyck Reachability in Directed Graphs is PTime-Harddyn

Theorem #1 (Bouyer & Jugé 17)

2-letter Dyck reachability in directed graphs is in PTime-DynFO iff PTime = PTime-DynFO Use a dynamic-adapted reduction from Reachability in 2-player games!

2 3 4

Alice

1 5

Barbara s t

2 3 4 1 5 1 2 3 4 5 t s

2 3 5 1 5 1 4 4 3 2 4 2 5 2 4

1 2 3 4 5 t s Stack

1

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-65
SLIDE 65

Dyck Reachability in Directed Graphs is PTime-Harddyn

Theorem #1 (Bouyer & Jugé 17)

2-letter Dyck reachability in directed graphs is in PTime-DynFO iff PTime = PTime-DynFO Use a dynamic-adapted reduction from Reachability in 2-player games!

2 3 4

Alice

1 5

Barbara s t

2 3 4 1 5 1 2 3 4 5 t s

2 3 5 1 5 1 4 4 3 2 4 2 5 2 4

1 2 3 4 5 t s Stack

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-66
SLIDE 66

Dyck Reachability in Directed Graphs is PTime-Harddyn

Theorem #1 (Bouyer & Jugé 17)

2-letter Dyck reachability in directed graphs is in PTime-DynFO iff PTime = PTime-DynFO Use a dynamic-adapted reduction from Reachability in 2-player games!

2 3 4

Alice

1 5

Barbara s t

2 3 4 1 5 1 2 3 4 5 t s

2 3 5 1 5 1 4 4 3 2 4 2 5 2 4

1 2 3 4 5 t s Stack

2 5

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-67
SLIDE 67

Dyck Reachability in Directed Graphs is PTime-Harddyn

Theorem #1 (Bouyer & Jugé 17)

2-letter Dyck reachability in directed graphs is in PTime-DynFO iff PTime = PTime-DynFO Use a dynamic-adapted reduction from Reachability in 2-player games!

2 3 4

Alice

1 5

Barbara s t

2 3 4 1 5 1 2 3 4 5 t s

2 3 5 1 5 1 4 4 3 2 4 2 5 2 4

1 2 3 4 5 t s Stack

2

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-68
SLIDE 68

Dyck Reachability in Directed Graphs is PTime-Harddyn

Theorem #1 (Bouyer & Jugé 17)

2-letter Dyck reachability in directed graphs is in PTime-DynFO iff PTime = PTime-DynFO Use a dynamic-adapted reduction from Reachability in 2-player games!

2 3 4

Alice

1 5

Barbara s t

2 3 4 1 5 1 2 3 4 5 t s

2 3 5 1 5 1 4 4 3 2 4 2 5 2 4

1 2 3 4 5 t s Stack

2 2 4

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-69
SLIDE 69

Dyck Reachability in Directed Graphs is PTime-Harddyn

Theorem #1 (Bouyer & Jugé 17)

2-letter Dyck reachability in directed graphs is in PTime-DynFO iff PTime = PTime-DynFO Use a dynamic-adapted reduction from Reachability in 2-player games!

2 3 4

Alice

1 5

Barbara s t

2 3 4 1 5 1 2 3 4 5 t s

2 3 5 1 5 1 4 4 3 2 4 2 5 2 4

1 2 3 4 5 t s Stack

2 2

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-70
SLIDE 70

Dyck Reachability in Directed Graphs is PTime-Harddyn

Theorem #1 (Bouyer & Jugé 17)

2-letter Dyck reachability in directed graphs is in PTime-DynFO iff PTime = PTime-DynFO Use a dynamic-adapted reduction from Reachability in 2-player games!

2 3 4

Alice

1 5

Barbara s t

2 3 4 1 5 1 2 3 4 5 t s

2 3 5 1 5 1 4 4 3 2 4 2 5 2 4

1 2 3 4 5 t s Stack

2

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-71
SLIDE 71

Dyck Reachability in Directed Graphs is PTime-Harddyn

Theorem #1 (Bouyer & Jugé 17)

2-letter Dyck reachability in directed graphs is in PTime-DynFO iff PTime = PTime-DynFO Use a dynamic-adapted reduction from Reachability in 2-player games!

2 3 4

Alice

1 5

Barbara s t

2 3 4 1 5 1 2 3 4 5 t s

2 3 5 1 5 1 4 4 3 2 4 2 5 2 4

1 2 3 4 5 t s Stack

2 3

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-72
SLIDE 72

Dyck Reachability in Directed Graphs is PTime-Harddyn

Theorem #1 (Bouyer & Jugé 17)

2-letter Dyck reachability in directed graphs is in PTime-DynFO iff PTime = PTime-DynFO Use a dynamic-adapted reduction from Reachability in 2-player games!

2 3 4

Alice

1 5

Barbara s t

2 3 4 1 5 1 2 3 4 5 t s

2 3 5 1 5 1 4 4 3 2 4 2 5 2 4

1 2 3 4 5 t s Stack

Until, one day . . .

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-73
SLIDE 73

Dyck Reachability in Directed Graphs is PTime-Harddyn

Theorem #1 (Bouyer & Jugé 17)

2-letter Dyck reachability in directed graphs is in PTime-DynFO iff PTime = PTime-DynFO Use a dynamic-adapted reduction from Reachability in 2-player games!

2 3 4

Alice

1 5

Barbara s t

2 3 4 1 5 1 2 3 4 5 t s

2 3 5 1 5 1 4 4 3 2 4 2 5 2 4

1 2 3 4 5 t s Stack

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-74
SLIDE 74

Dyck Reachability in Directed Graphs is PTime-Harddyn

Theorem #1 (Bouyer & Jugé 17)

2-letter Dyck reachability in directed graphs is in PTime-DynFO iff PTime = PTime-DynFO Use a dynamic-adapted reduction from Reachability in 2-player games!

2 3 4

Alice

1 5

Barbara s t

2 3 4 1 5 1 2 3 4 5 t s

2 3 1 5 1 4 4 3 2 4 2 5 2 4 1

1 2 3 4 5 t s Stack

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-75
SLIDE 75

Dyck Reachability in Directed Graphs is PTime-Harddyn

Theorem #1 (Bouyer & Jugé 17)

2-letter Dyck reachability in directed graphs is in PTime-DynFO iff PTime = PTime-DynFO Use a dynamic-adapted reduction from Reachability in 2-player games!

2 3 4

Alice

1 5

Barbara s t

2 3 4 1 5 1 2 3 4 5 t s

2 3 1 5 1 4 4 3 2 4 2 5 2 4 1

1 2 3 4 5 t s Stack

Use binary label encoding & achieve 2-letter Dyck reachability

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-76
SLIDE 76

Dyck Reachability in Undirected Graphs is PTime-Harddyn

Theorem #2 (Bouyer & Jugé 17)

2-letter Dyck reachability in undirected graphs is in PTime-DynFO iff PTime = PTime-DynFO Transform directed labeled edges into undirected labeled paths!

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-77
SLIDE 77

Dyck Reachability in Undirected Graphs is PTime-Harddyn

Theorem #2 (Bouyer & Jugé 17)

2-letter Dyck reachability in undirected graphs is in PTime-DynFO iff PTime = PTime-DynFO Transform directed labeled edges into undirected labeled paths!

v w

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-78
SLIDE 78

Dyck Reachability in Undirected Graphs is PTime-Harddyn

Theorem #2 (Bouyer & Jugé 17)

2-letter Dyck reachability in undirected graphs is in PTime-DynFO iff PTime = PTime-DynFO Transform directed labeled edges into undirected labeled paths!

v w v w 1 1 1 1 1 1 1 v w

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-79
SLIDE 79

Dyck Reachability in Undirected Graphs is PTime-Harddyn

Theorem #2 (Bouyer & Jugé 17)

2-letter Dyck reachability in undirected graphs is in PTime-DynFO iff PTime = PTime-DynFO Transform directed labeled edges into undirected labeled paths!

v w v w 1 1 1 1 1 1 1 v w

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-80
SLIDE 80

Dyck Reachability in Undirected Graphs is PTime-Harddyn

Theorem #2 (Bouyer & Jugé 17)

2-letter Dyck reachability in undirected graphs is in PTime-DynFO iff PTime = PTime-DynFO Transform directed labeled edges into undirected labeled paths!

v w v w 1 1 1 1 1 1 1 1 v w

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-81
SLIDE 81

Dyck Reachability in Undirected Graphs is PTime-Harddyn

Theorem #2 (Bouyer & Jugé 17)

2-letter Dyck reachability in undirected graphs is in PTime-DynFO iff PTime = PTime-DynFO Transform directed labeled edges into undirected labeled paths!

v w v w 1 1 1 1 1 1 1 1 v w

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-82
SLIDE 82

Dyck Reachability in Undirected Graphs is PTime-Harddyn

Theorem #2 (Bouyer & Jugé 17)

2-letter Dyck reachability in undirected graphs is in PTime-DynFO iff PTime = PTime-DynFO Transform directed labeled edges into undirected labeled paths!

v w v w v w 1 v w 1

Similar transformations

v w 1 1 1 1 1 1 1 v w v w 1 1 1 1 1 1 1 v w v w 1 1 1 v w v w 1 1 1 1 1

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-83
SLIDE 83

And With One Letter Only?

Dynamic complexity of Dyck reachability problems With ě 2 letters: PTime-completedyn

in (un)directed graphs

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-84
SLIDE 84

And With One Letter Only?

Dynamic complexity of Dyck reachability problems With ě 2 letters: PTime-completedyn

in (un)directed graphs

With 1 letter:

§ in DynFO (and not NLogSpace-harddyn)

in undirected graphs

§ in NLogSpace

in directed graphs

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-85
SLIDE 85

Contents

1

Dynamic Complexity of Decision Problems

2

Reachability and its Variants

3

The Result

4

Conclusion and Future Work

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-86
SLIDE 86

Future work

Some problems to investigate: 1-letter Dyck reachability in directed graphs Dyck reachability in Cayley graphs

1 1

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability

slide-87
SLIDE 87

Future work

Some problems to investigate: 1-letter Dyck reachability in directed graphs Dyck reachability in Cayley graphs

1 1

  • P. Bouyer-Decitre & V. Jugé

Dynamic Complexity of the Dyck Reachability