On the power of the compass (or, why mazes are easier to search than graphs)

Abstract

About the same time, A.N. Shah gave a finite autom~ton with 5 pebbles which could search an arbitrary maze (the autonlaton may drop a pebble on a cell it is visiting, then upon returning to that cell later on can sense the pebble's presence, and if desired pick it up and move it to a new cell). Shah also conjectured that fewer than 5 pebbles would not suffice. The first of our two main results is that, contrary to Shah's conjecture, the search can be implemented with just two pebbles. The question is still open whether a finite automaton with just one pebble can search any maze.