  The rotor router model is a popular deterministic analogue of a random walk
on a graph. Instead of moving to a random neighbor, the neighbors are served in
a fixed order. We examine how fast this "deterministic random walk" covers all
vertices (or all edges). We present general techniques to derive upper bounds
for the vertex and edge cover time and derive matching lower bounds for several
important graph classes. Depending on the topology, the deterministic random
walk can be asymptotically faster, slower or equally fast as the classic random
walk. We also examine the short term behavior of deterministic random walks,
that is, the time to visit a fixed small number of vertices or edges.
