LeetCode //C - 790. Domino and Tromino Tiling

790. Domino and Tromino Tiling

You have two types of tiles: a 2 x 1 domino shape and a tromino shape. You may rotate these shapes.

Given an integer n, return the number of ways to tile an 2 x n board. Since the answer may be very large, return it modulo $10^9+7$.

In a tiling, every square must be covered by a tile. Two tilings are different if and only if there are two 4-directionally adjacent cells on the board such that exactly one of the tilings has both squares occupied by a tile.
 

Example 1:

Input: n = 3
Output: 5
Explanation: The five different ways are show above.

Example 2:

Input: n = 1
Output: 1

Constraints:

• 1 <= n <= 1000

From: LeetCode
Link: 790. Domino and Tromino Tiling

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Solution:

Ideas：

1. Define a recurrence relation to calculate the number of tilings for a board of width n.
2. The base cases will be small widths for which we can manually count the number of tilings.
3. For larger widths, we build up the solution from the base cases, using the recurrence relation.
4. We need to consider the last column which could be filled by:
   • A vertical domino, which leaves the subproblem of tiling a 2 x (n-1) board.
   • Two horizontal dominos, which leaves the subproblem of tiling a 2 x (n-2) board.
   • A tromino along with a domino, which will lead to two subproblems: tiling a 2 x (n-2) board and a 2 x (n-3) board.
5. Since the answer can be very large, we will return it modulo $10^9+7$.

Caode:

int numTilings(int n) {
    if (n == 1) return 1;
    if (n == 2) return 2;
    if (n == 3) return 5;

    long dp[n+1];
    dp[0] = 1; dp[1] = 1; dp[2] = 2; dp[3] = 5;

    for (int i = 4; i <= n; ++i) {
        dp[i] = (2 * dp[i-1] % 1000000007 + dp[i-3]) % 1000000007; // Main recurrence relation
    }

    return (int) dp[n];
}