Searching Constant Width Mazes Captures the AC0 Hierarchy

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Abstract

We show that searching a width k maze is complete for Π k, i.e., for the k'th level of the AC 0 hierarchy. Equivalently, st-connectivity for width k grid graphs is complete for Π k. As an application, we show that there is a data structure solving dynamic st-connectivity for con stant 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.