Consider the normal edit distance DP with time complexity $O(|S||T|)$, i.e.
If you analysis it carefully, it's unnecessary to consider all states with edit distance $> K$, thus for each $i$, we only need to consider $dp[i][j]$ where $i - K \leq j \leq i + K$, which reduce the complexity into $O(|S|K)$.
Consider the normal edit distance DP with time complexity $O(|S||T|)$, i.e.
If you analysis it carefully, it's unnecessary to consider all states with edit distance $> K$, thus for each $i$, we only need to consider $dp[i][j]$ where $i - K \leq j \leq i + K$, which reduce the complexity into $O(|S|K)$.