790. Domino and Tromino Tiling

Time: O(n)

Space: O(n)

Approach

DP with recurrence dp[i] = 2·dp[i-1] + dp[i-3]; base cases dp[1]=1, dp[2]=2, dp[3]=5.

Key Techniques

Dynamic programming solves problems by breaking them into overlapping sub-problems and storing results to avoid redundant work. The key steps are: define state, write a recurrence relation, set base cases, and choose top-down (memoization) or bottom-up (tabulation). DP often yields O(n²) → O(n) time improvements over brute force.

Matrix

Matrix problems often involve BFS/DFS flood fill, dynamic programming on 2D grids, or spiral/diagonal traversal. For row × column DP, break it into 1D sub-problems column by column. Common pitfalls: boundary checks and modifying the input matrix in-place.

790.cs

// Approach: DP with recurrence dp[i] = 2·dp[i-1] + dp[i-3]; base cases dp[1]=1, dp[2]=2, dp[3]=5.
// Time: O(n) Space: O(n)

public class Solution
{
    public int NumTilings(int n)
    {
        int MOD = 1_000_000_007;
        long[] dp = new long[1001];

        if (n >= 1)
            dp[1] = 1;
        if (n >= 2)
            dp[2] = 2;
        if (n >= 3)
            dp[3] = 5;

        for (int i = 4; i <= n; ++i)
            dp[i] = (2 * dp[i - 1] + dp[i - 3]) % MOD;

        return (int)dp[n];
    }
}

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