LeetCode 516 Longest Palindromic Subsequence - Transition

The recurrence has two cases based on whether the endpoints match. If $s[i] = s[j]$: Both characters can extend a palindrome. Look at $dp[i+1][j-1]$ (the inner substring) and add $2$.

So $dp[i][j] = dp[i+1][j-1] + 2$. If $s[i] \neq s[j]$: At least one endpoint isn't part of the LPS. Take the better of excluding $s[i]$ or excluding $s[j]$: $dp[i][j] = \max(dp[i+1][j], dp[i][j-1])$. Notice the dependency: $dp[i][j]$ needs $dp[i+1][...]$ and $dp[...][j-1]$. These are shorter ranges. Iterate by increasing interval length so subproblems are ready.