The cyclic string alignment problem asks for the optimal alignment
between two "cyclic" strings.  For example suppose we want to compare
two bead necklesses.  The beads are the characters and each neckless
forms a cyclic arrangement of beads.  

We can also think of a cyclic string as an equivalence class of
strings that are identical up to cyclic permutations.  So "hello"
represents the same cyclic string as "elloh" or "llohe" and so on.  In
particular the cyclic string "hello" can be transformed into the
cyclic string "lluhe" by substituting a single character (o for u).

Let A(X,Y) be the standard definition of the alignment cost between
two non-cyclic strings X and Y (see Section 6.6 in the textbook).
Define the cyclic alignment cost C(X,Y) to be the minimum over
A(X',Y') where X' ranges over cyclic permutations of X and Y' ranges
over cyclic permutations of Y.

Let |X| = n and |Y| = m.  Assume n >= m.  The standard alignment
problem can be solved in O(nm) time.  As a warm up, give an algorithm
that computes the cyclic alignment cost in O(nm^2) time.

Can you find an algorithm that computes the cyclic alignment cost
in O(nm log m) time?