LeetCode 198 House Robber - Transition

$2.$ Find the transition (the formula to compute each state)

Consider house at $i$. You have two choices:

• If do not rob house $i$, then the best answer using houses up to $i$ is the same as using houses up to $i - 1$: $dp[i] = dp[i - 1].$

• If do rob house $i$, then you gain $\text{a}[i]$ money, and you are not allowed to take money from house $i - 1$. The best rob from the remaining houses is $dp[i - 2]$, so this option gives $dp[i] = dp[i - 2] + \text{a}[i].$

So the recurrence is $dp[i] = \max\bigl(dp[i - 1],\ dp[i - 2] + \text{a}[i]\bigr).$

This way, calculate all the dp values. The only question left is: what is the final answer? You have established the recurrence. Now let us see the base cases.