  The routing number is a graph invariant introduced by Alon, Chung, and Graham
in 1994, and it has been studied for trees and other classes of graphs such as
hypercubes. It gives the minimum number of routing steps needed to sort a set
of distinct tokens, placed one on each vertex, where each routing step swaps a
set of disjoint pairs of adjacent tokens. Our main theorem generalizes the
known estimate that a rectangular grid graph R with width w(R) and height h(R)
has routing number rt(R) in O(w(R)+h(R)). We show that for the subgraph P of
the infinite square lattice enclosed by any convex polygon, its routing number
rt(P) is in O(w(P)+h(P)).
