790. Domino and Tromino Tiling

We have two types of tiles: a 2x1 domino shape, and an "L" tromino shape. These shapes may be rotated.

XX  <- domino

XX  <- "L" tromino
X

Given N, how many ways are there to tile a 2 x N board? Return your answer 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:
Input: 3
Output: 5
Explanation: 
The five different ways are listed below, different letters indicates different tiles:
XYZ XXZ XYY XXY XYY
XYZ YYZ XZZ XYY XXY

 

Note:

• N  will be in range [1, 1000].

 

Approach #1: DP. [C++]

class Solution {
public:
    int numTilings(int N) {
        constexpr int mod = 1000000007;
        vector<vector<long>> dp(N+1, vector<long>(2, 0));
        
        dp[0][0] = dp[1][0] = 1;
        
        for (int i = 2; i <= N; ++i) {
            dp[i][0] = (dp[i-1][0] + dp[i-2][0] + 2 * dp[i-1][1]) % mod;
            dp[i][1] = (dp[i-2][0] + dp[i-1][1]) % mod;
        }
        
        return dp[N][0];
    }
};

　　

Analysis:

http://zxi.mytechroad.com/blog/dynamic-programming/leetcode-790-domino-and-tromino-tiling/