  A memoryless routing algorithm is one in which the decision about the next
edge on the route to a vertex t for a packet currently located at vertex v is
made based only on the coordinates of v, t, and the neighbourhood, N(v), of v.
The current paper explores the limitations of such algorithms by showing that,
for any (randomized) memoryless routing algorithm A, there exists a convex
subdivision on which A takes Omega(n^2) expected time to route a message
between some pair of vertices. Since this lower bound is matched by a random
walk, this result implies that the geometric information available in convex
subdivisions is not helpful for this class of routing algorithms. The current
paper also shows the existence of triangulations for which the Random-Compass
algorithm proposed by Bose etal (2002,2004) requires 2^{\Omega(n)} time to
route between some pair of vertices.
