On the capability of finite automata in 2 and 3 dimensional space

The paper describes two algorithms for threading unknown, finite directed Eulerian mazes. Each of these algorithms is performed by a traveling robot whose control is a finite-state automaton. It is assumed that each vertex has a circular list of its outgoing edges. The items of this list are called exits. Each of the algorithms puts in one of the exits of each vertex a scan pebble. These pebbles can be used by a simple robot as traffic signals, which allow it to traverse an Eulerian cycle of the maze. For a directed graph (maze) G(V, E), the simple algorithm performs O(|V | · |E|) edge traversals, while the advanced algorithm traverses every edge three times. Let dout(v) be the out-degree of vertex v. The algorithms use, at each vertex v, a local memory of size O(log dout(v)). Communicated by S. Khuller: submitted January 2002; revised June 2002 Work by S. Even supported by the Fund for the Promotion of Research at the Technion. S. Bhatt et al., Traversing Eulerian Mazes, JGAA, 6(2) 157–173 (2002) 158