Variation to edit distance depending on position still a metric?

Question

The edit distance between two strings $d(w_1, w_2)$ sums up the cost of edit operations that transform one string into the other. If done right, the edit distance is a metric.

In particular, the cost of an edit operation may depend on the character inserted or deleted, meaning we have costs $w_\text{insert}(a)$ and $w_\text{delete}(a)$ for all characters $a$.

Question: If the cost also depends on the position in the input string where it is applied, will the result still be a metric?

As an example, consider a spell checker that assumes mistakes in the first character of a word are rather seldom as compared to other positions, so when searching for corrected words, a change in the first position would have high cost.

Answer 1

Here is why the usual edit distance is a metric:

1. $d(w,w) = 0$ since no operations need to be performed to get from $w$ to $w$.

2. $d(x,y) = d(y,x)$ since given a sequence of operations for transforming $x$ to $y$, we can perform it in reverse to transform $y$ to $x$ at the same cost. This shows that $d(y,x) \leq d(x,y)$, and similarly $d(x,y) \leq d(y,x)$.

3. $d(x,y) \leq d(x,z) + d(z,y)$ since if $\alpha$ transforms $x$ to $z$ and $\beta$ transforms $z$ to $y$ then $\alpha\beta$ transforms $x$ to $y$, and moreover the cost of $\alpha\beta$ is the sum of the costs of $\alpha$ and $\beta$.

Now you're in a position to answer your question yourself.