Leetcode
Dynamic Programming
790 Domino and Tromino Tiling

1. Question

We have two types of tiles: a 2x1 domino shape, and an "L" tromino shape. These shapes may be rotated.
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XX <- domino
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XX <- "L" tromino
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X
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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.)
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Example:
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Input: 3
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Output:5
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Explanation:
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The five different ways are listed below, different letters indicates different tiles:
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XYZ XXZ XYY XXY XYY
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XYZ YYZ XZZ XYY XXY
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Note:
  • N will be in range[1, 1000].

2. Implementation

(1) DP
思路: 从n = 1到n = 5先画出能拼出tile的,然后推导出递归公式。具体推导可以看这里。数学推导可以参考这个
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class Solution {
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public int numTilings(int N) {
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int MOD = 1000000007;
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long[][] dp = new long[N + 1][3];
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dp[0][0] = 1;
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dp
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[1][0] = 1;
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for (int i = 2; i <= N; i++) {
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dp[i][0] = (dp[i - 1][0] + dp[i - 2][0] + dp[i - 1][1] + dp[i - 1][2]) % MOD;
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dp[i][1] = (dp[i - 2][0] + dp[i - 1][2]) % MOD;
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dp[i][2] = (dp[i - 2][0] + dp[i - 1][1]) % MOD;
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}
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return (int)dp[N][0];
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}
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}
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3. Time & Space Complexity

DP:时间复杂度O(n), 空间复杂度O(n)