【LeetCode 790】 Domino and Tromino Tiling

题目描述

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].

思路

画了很久很久…想要投机取巧，未果。
一直在想半块的是什么规律。。
原来也是可以状态转换的。。。
这种动态规划都能想出来。。。
真是。。。惊了。

代码

class Solution {
public:
    int numTilings(int N) {
        vector<vector<long long>> dp(N+2, vector<long long>(3, 0));
        
        int MOD = 1e9+7;
        
        dp[1][0] = 1;
        dp[2][0] = 2;
        dp[2][1] = 1;
        dp[2][2] = 1;
        
        for (int i=3; i<=N; ++i) {
            dp[i][0] = (dp[i-1][0] + dp[i-2][0] + dp[i-1][1] + dp[i-1][2])%MOD;
            dp[i][1] = (dp[i-2][0] + dp[i-1][2])%MOD;
            dp[i][2] = (dp[i-2][0] + dp[i-1][1])%MOD;
        }
        
        return dp[N][0];
    }
};