Searching constant width mazes captures the AC^0 hierarchy

摘要

We show that searching a width k maze is complete for Pi_k, i.e., for the k'th level of the AC^0 hierarchy. Equivalently, st-connectivity for width k grid graphs is complete for Pi_k. As an application, we show that there is a data structure solving dynamic st-connectivity for constant width grid graphs with time bound O(log log n) per operation on a random access machine. The dynamic algorithm is derived from the parallel one in an indirect way using algebraic tools.