Linear Time Algorithm for Update Games via Strongly-Trap-Connected Components

An arena is a finite directed graph whose vertices are divided into two classes, i.e., V = V ∪ V#; this forms the basic playground for a plethora of 2-player infinite pebble-games. We introduce and study a refined notion of reachability for arenas, named trap-reachability, where Player attempts to reach vertices without leaving a prescribed subset V ′ ⊆ V , while Player # works against. It is shown that every arena decomposes into strongly-trap-connected components (STCCs). Our main result is a linear time algorithm for computing this unique decomposition. Both the graph structures and the algorithm generalize the classical decomposition of a directed graph into its strongly-connected components (SCCs). The algorithm builds on a generalization of the depth-first search (DFS), taking inspiration from Tarjan’s SCCs classical algorithm. The structures of palm-trees and jungles described in Tarjan’s original paper need to be revisited and generalized (i.e., tr-palm-trees and tr-jungles) in order to handle the 2-player infinite pebble-game setting. This theory has direct applications in solving Update Games (UGs) faster. Dinneen and Khoussainov showed in 1999 that deciding who’s the winner in a given UG costs O(mn) time, where n is the number of vertices and m is that of arcs. We solve that problem in Θ(m + n) linear time and space. The result is obtained by observing that the UG is a win for Player if and only if the arena comprises one single STCC. It is also observed that the tr-palm-tree returned by the algorithm encodes routing information that an Θ(n)-space agent can consult to win the UG in O(1) time per move. Finally, the polynomial-time complexity for deciding Explicit McNaughton-Müller Games is also improved, from cubic to quadratic.