LeetCode 198 House Robber - Defining the DP

$1.$ Define the state: Let $dp[i]$ = the maximum money you can steal from houses $1$ through $i$. With this definition, you know the best outcome at each position without committing to whether you rob the last house.

For $a = [2, 7, 9, 3, 1]$: $dp[1] = 2$, $dp[2] = 7$ (take house $2$ alone), $dp[3] = 11$ (take houses $1$ and $3$: $2 + 9 = 11$). The state at position $i$ depends only on earlier positions, so you can fill the table left to right using a simple recurrence.