What Is Multi-Dimensional DP?

Multi-dimensional DP (dynamic programming) uses states with two or more parameters. Instead of $dp[i]$, you have $dp[i][j]$ or even $dp[i][j][k]$. A classic example: counting paths in a grid. You need to track both your row and column to know where you are. One number is not enough. $dp[r][c]$ stores the number of paths to reach cell $(r, c)$.

Each dimension captures one aspect of the problem state. More dimensions mean more information tracked, but also more memory and computation. The skill is identifying which dimensions you need.