Consider every pair of consecutive words such that neither word is a prefix of the other. Let $k$ be the first position at which the words differ. Let $a$ be the $k^{\text{th}}$ letter of the word that appears later in the input, and $b$ the $k^{\text{th}}$ letter of the other word. It follows that $a$ comes after $b$ in alphabetical order.

Let's define the binary relation greater than on the given set of letters. We'll say that $a$ is greater than $b$ if $a$ comes after $b$ in alphabetical order. Notice that this relation is transitive (if $a>b$ and $b>c$, then $a>c$). We can easily compute its transitive closure using the Floyd-Warshall algorithm.

If the transitive closure suggests that $a>a$ for some letter $a$, the ordering does not exist.

Next, if there are two letters $a$ and $b$ such that neither $a>b$ nor $b>a$ holds, the ordering is not unique.

Otherwise, the ordering does exist and it is unique. Let the number $k$ be equal to the number of letters $b$ such that $a>b$ for some letter $a$. The letter $a$ will take the $k^{\text{th}}$ place in the sorted sequence of all letters in the alphabet, indexed from zero.