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

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

Contents Dynamic Complexity of Decision Problems 1 Reachability and its Variants 2 The Result 3 Conclusion and Future Work 4 P. Bouyer-Decitre & V. Jugé Dynamic Complexity of the Dyck Reachability

Dynamic Complexity of Decision Problems Modulo 3 Decision Input: Bit vector b 1 ¨ b 2 ¨ . . . ¨ b n P F n 3 Output: Yes if b 1 ` b 2 ` . . . ` b n “ 0 — No otherwise P. Bouyer-Decitre & V. Jugé Dynamic Complexity of the Dyck Reachability

Dynamic Complexity of Decision Problems Modulo 3 Decision Input: Bit vector b 1 ¨ b 2 ¨ . . . ¨ b n P F n 3 Output: Yes if b 1 ` b 2 ` . . . ` b n “ 0 — No otherwise Solving this problem. . . Static world : membership in a regular language P. Bouyer-Decitre & V. Jugé Dynamic Complexity of the Dyck Reachability

Dynamic Complexity of Decision Problems Modulo 3 Decision Input: Bit vector b 1 ¨ b 2 ¨ . . . ¨ b n P F n 3 Output: Yes if b 1 ` b 2 ` . . . ` b n “ 0 — No otherwise Solving this problem. . . Static world : membership in a regular language Dynamic world : what if some bit b k changes? § Maintain predicates Aux i ” p b 1 ` b 2 ` . . . ` b n “ i q for i P F 3 § Update the values of Aux 0 , Aux 1 , Aux 2 when b k changes § Use the new value of Aux 0 and answer the problem P. Bouyer-Decitre & V. Jugé Dynamic Complexity of the Dyck Reachability

Dynamic Complexity of Decision Problems Modulo 3 Decision Input: Bit vector b 1 ¨ b 2 ¨ . . . ¨ b n P F n 3 Output: Yes if b 1 ` b 2 ` . . . ` b n “ 0 — No otherwise Solving this problem. . . Static world : membership in a regular language Dynamic world : what if some bit b k changes? § Maintain predicates Aux i ” p b 1 ` b 2 ` . . . ` b n “ i q for i P F 3 § Update the values of Aux 0 , Aux 1 , Aux 2 when b k changes § Use the new value of Aux 0 and answer the problem How complex is it? Static world : linear time Dynamic world : § Easy initial instance p b 1 “ b 2 “ . . . “ b n “ 0 q : constant time § Each update: constant time P. Bouyer-Decitre & V. Jugé Dynamic Complexity of the Dyck Reachability

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 Dynamic Complexity of the Dyck Reachability

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 E ‹ p x , y q ” pD path from x to y in G q for x , y P V § Update the values of E ‹ p x , y q when u Ñ v is inserted/deleted § Use the new value of E ‹ p s , t q and answer the problem P. Bouyer-Decitre & V. Jugé Dynamic Complexity of the Dyck Reachability

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 E ‹ p x , y q ” pD path from x to y in G q for x , y P V § Update the values of E ‹ p x , y q when u Ñ v is inserted/deleted § Use the new value of E ‹ p s , t q 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

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 E ‹ p x , y q ” pD path from x to y in G q for x , y P V § Update the values of E ‹ p x , y q when u Ñ v is inserted/deleted § Use the new value of E ‹ p s , t q 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

FO formulas ñ parallel « constant time φ “ D x . @ y .ψ p x , y q_ ψ p y , x q P. Bouyer-Decitre & V. Jugé Dynamic Complexity of the Dyck Reachability

FO formulas 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 Dynamic Complexity of the Dyck Reachability

FO formulas 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 Dynamic Complexity of the Dyck Reachability

FO formulas 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 Dynamic Complexity of the Dyck Reachability

FO formulas 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 Dynamic Complexity of the Dyck Reachability

FO formulas 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 Dynamic Complexity of the Dyck Reachability

Dynamic Complexity of Decision Problems Reachability in DAGs with FO formulas Initialization (on the edgeless graph): � E ‹ p x , y q Ð p x “ y q E ‹ p x , y q Ð E ‹ p x , y q Ð E ‹ p x , y q Ð E ‹ p x , y q Ð P. Bouyer-Decitre & V. Jugé Dynamic Complexity of the Dyck Reachability

Dynamic Complexity of Decision Problems Reachability in DAGs with FO formulas Initialization (on the edgeless graph): � Update after inserting the edge u Ñ v E ‹ p x , y q Ð E ‹ p x , y q u E ‹ p x , y q Ð v E ‹ p x , y q Ð x E ‹ p x , y q Ð y E ‹ p x , y q Ð P. Bouyer-Decitre & V. Jugé Dynamic Complexity of the Dyck Reachability

Dynamic Complexity of Decision Problems Reachability in DAGs with FO formulas Initialization (on the edgeless graph): � Update after inserting the edge u Ñ v : � x E ‹ p x , y q Ð E ‹ p x , y q_ u p E ‹ p x , u q ^ E ‹ p v , y qq v E ‹ p x , y q Ð y E ‹ p x , y q Ð E ‹ p x , y q Ð P. Bouyer-Decitre & V. Jugé Dynamic Complexity of the Dyck Reachability

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 E ‹ p x , y q Ð p E ‹ p x , y q ^ � E ‹ p x , u qq u E ‹ p x , y q Ð v E ‹ p x , y q Ð E ‹ p x , y q Ð y x E ‹ p x , y q Ð P. Bouyer-Decitre & V. Jugé Dynamic Complexity of the Dyck Reachability

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 E ‹ p x , y q Ð p E ‹ p x , y q ^ � E ‹ p x , u qq_ u p E ‹ p x , y q ^ E ‹ p y , u qq y v E ‹ p x , y q Ð x E ‹ p x , y q Ð E ‹ p x , y q Ð P. Bouyer-Decitre & V. Jugé Dynamic Complexity of the Dyck Reachability

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 : � E ‹ p x , y q Ð p E ‹ p x , y q ^ � E ‹ p x , u qq_ u p E ‹ p x , y q ^ E ‹ p y , u qq_ a v pD a . D b . E ‹ p x , a q ^ E ‹ p b , y q^ x b pp a Ñ b q ^ p a , b q ‰ p u , v q^ y p E ‹ p a , u q ^ � E ‹ p b , u qq P. Bouyer-Decitre & V. Jugé Dynamic Complexity of the Dyck Reachability

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 one predicate is the goal predicate P. Bouyer-Decitre & V. Jugé Dynamic Complexity of the Dyck Reachability