Courcelles Theorem Made Dynamic Patricia Bouyer-Decitre 1,2 , Vincent - PowerPoint PPT Presentation

Courcelle’s Theorem Made Dynamic Patricia Bouyer-Decitre 1,2 , Vincent Jugé 1,2,3 & Nicolas Markey 1,2,4 1: CNRS — 2: ENS Paris-Saclay (LSV) — 3: UPEM (LIGM) — 4: Rennes (IRISA) 03/10/2017 P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

Contents Dynamic Complexity of Decision Problems 1 Courcelle’s Theorem 2 Making Courcelle’s Theorem Dynamic 3 P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

Dynamic Complexity of Decision Problems Modulo 3 Decision Input: Elements x 1 , x 2 , . . . , x n of F 3 Output: Yes if x 1 ` x 2 ` . . . ` x n “ 0 — No otherwise P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

Dynamic Complexity of Decision Problems Modulo 3 Decision Input: Elements x 1 , x 2 , . . . , x n of F 3 Output: Yes if x 1 ` x 2 ` . . . ` x n “ 0 — No otherwise Solving this problem. . . Static world : membership in a regular language P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

Dynamic Complexity of Decision Problems Modulo 3 Decision Input: Elements x 1 , x 2 , . . . , x n of F 3 Output: Yes if x 1 ` x 2 ` . . . ` x n “ 0 — No otherwise Solving this problem. . . Static world : membership in a regular language Dynamic world : what if some element x k changes? § Maintain predicates S i ” “ x 1 ` x 2 ` . . . ` x n “ i ” for i P F 3 § Update the values of S 0 , S 1 , S 2 when x k changes § Use the new value of S 0 and answer the problem P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

Dynamic Complexity of Decision Problems Modulo 3 Decision Input: Elements x 1 , x 2 , . . . , x n of F 3 Output: Yes if x 1 ` x 2 ` . . . ` x n “ 0 — No otherwise Solving this problem. . . Static world : membership in a regular language Dynamic world : what if some element x k changes? § Maintain predicates S i ” “ x 1 ` x 2 ` . . . ` x n “ i ” for i P F 3 § Update the values of S 0 , S 1 , S 2 when x k changes § Use the new value of S 0 and answer the problem How complex is it? Static world : linear time Dynamic world : § Easy initial instance p x 1 “ x 2 “ . . . “ x n “ 0 q : constant time § Each update: constant time P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

Dynamic Complexity of Decision Problems Reachability in DAGs Input: Directed acyclic graph G “ p V , E q & two vertices s , t P V Output: Yes if D path from s to t in G — No otherwise P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

Dynamic Complexity of Decision Problems Reachability in DAGs Input: Directed acyclic graph G “ p V , E q & 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 R p x , y q ” pD path from x to y in G q for x , y P V § Update the values of R p x , y q when u Ñ v is inserted/deleted § Use the new value of R p s , t q and answer the problem P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

Dynamic Complexity of Decision Problems Reachability in DAGs Input: Directed acyclic graph G “ p V , E q & 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 R p x , y q ” pD path from x to y in G q for x , y P V § Update the values of R p x , y q when u Ñ v is inserted/deleted § Use the new value of R p s , t q and answer the problem How complex is it? Static world : linear time Dynamic world : § Easy initial edgeless instance: FO formulæ § Each update: FO formulæ P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

Dynamic Complexity of Decision Problems Reachability in DAGs Input: Directed acyclic graph G “ p V , E q & 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 R p x , y q ” pD path from x to y in G q for x , y P V § Update the values of R p x , y q when u Ñ v is inserted/deleted § Use the new value of R p s , t q and answer the problem How complex is it? Static world : linear time Dynamic world : § Easy initial edgeless instance: FO formulæ ( parallel constant time) § Each update: FO formulæ ( parallel constant time) P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

FO formulæ ñ parallel « constant time φ “ D x . @ y .ψ p x , y q_ ψ p y , x q P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

FO formulæ ñ parallel « constant time φ “ D x . @ y .ψ p x , y q_ ψ p y , x q ψ p e 1 , e 1 q ψ p e 1 , e 2 q ψ p e 2 , e 1 q ψ p e 2 , e 2 q P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

FO formulæ ñ parallel « constant time φ “ D x . @ y .ψ p x , y q_ ψ p y , x q ψ p e 1 , e 1 q ψ p e 1 , e 2 q ψ p e 2 , e 1 q ψ p e 2 , e 2 q P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

FO formulæ ñ parallel « constant time φ “ D x . @ y .ψ p x , y q_ ψ p y , x q ψ p e 1 , e 1 q ψ p e 1 , e 2 q ψ p e 2 , e 1 q ψ p e 2 , e 2 q x “ e 1 x “ e 1 x “ e 2 x “ e 2 _ _ _ _ y “ e 1 y “ e 2 y “ e 1 y “ e 2 P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

FO formulæ ñ parallel « constant time φ “ D x . @ y .ψ p x , y q_ ψ p y , x q ψ p e 1 , e 1 q ψ p e 1 , e 2 q ψ p e 2 , e 1 q ψ p e 2 , e 2 q x “ e 1 x “ e 1 x “ e 2 x “ e 2 _ _ _ _ y “ e 1 y “ e 2 y “ e 1 y “ e 2 ^ ^ x “ e 1 x “ e 2 P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

FO formulæ ñ parallel « constant time φ “ D x . @ y .ψ p x , y q_ ψ p y , x q ψ p e 1 , e 1 q ψ p e 1 , e 2 q ψ p e 2 , e 1 q ψ p e 2 , e 2 q x “ e 1 x “ e 1 x “ e 2 x “ e 2 _ _ _ _ y “ e 1 y “ e 2 y “ e 1 y “ e 2 ^ ^ x “ e 1 x “ e 2 _ φ P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

Dynamic Complexity of Decision Problems Reachability in DAGs with FO formulæ Initialization (on the edgeless graph): � R p x , y q Ð p x “ y q R p x , y q Ð R p x , y q Ð R p x , y q Ð R p x , y q Ð P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

Dynamic Complexity of Decision Problems Reachability in DAGs with FO formulæ Initialization (on the edgeless graph): � Update after inserting the edge u Ñ v R p x , y q Ð R p x , y q u R p x , y q Ð v R p x , y q Ð x R p x , y q Ð y R p x , y q Ð P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

Dynamic Complexity of Decision Problems Reachability in DAGs with FO formulæ Initialization (on the edgeless graph): � Update after inserting the edge u Ñ v : � x R p x , y q Ð R p x , y q_ u p R p x , u q ^ R p v , y qq v R p x , y q Ð y R p x , y q Ð R p x , y q Ð P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

Dynamic Complexity of Decision Problems Reachability in DAGs with FO formulæ Initialization (on the edgeless graph): � Update after inserting the edge u Ñ v : � Update after deleting the edge u Ñ v R p x , y q Ð p R p x , y q ^ � R p x , u qq u R p x , y q Ð v R p x , y q Ð R p x , y q Ð y x R p x , y q Ð P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

Dynamic Complexity of Decision Problems Reachability in DAGs with FO formulæ Initialization (on the edgeless graph): � Update after inserting the edge u Ñ v : � Update after deleting the edge u Ñ v R p x , y q Ð p R p x , y q ^ � R p x , u qq_ u p R p x , y q ^ R p y , u qq y v R p x , y q Ð x R p x , y q Ð R p x , y q Ð P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

Dynamic Complexity of Decision Problems Reachability in DAGs with FO formulæ Initialization (on the edgeless graph): � Update after inserting the edge u Ñ v : � Update after deleting the edge u Ñ v : � R p x , y q Ð p R p x , y q ^ � R p x , u qq_ u p R p x , y q ^ R p y , u qq_ a v pD a . D b . R p x , a q ^ R p b , y q^ x b pp a Ñ b q ^ p a , b q ‰ p u , v q^ y p R p a , u q ^ � R p b , u qq P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic

Dynamic Complexity of Decision Problems Reachability in DAGs with FO formulæ Initialization (on the edgeless graph): � Update after inserting the edge u Ñ v : � Update after deleting the edge u Ñ v : � ñ You can even maintain paths from s to t ! P. Bouyer-Decitre, V. Jugé & N. Markey Courcelle’s Theorem Made Dynamic