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 1 0 9 + 7 10^9 + 7 109+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


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 1 0 9 + 7 10^9+7 109+7.
Caode:
c 复制代码
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];
}
相关推荐
JohnFF4 分钟前
48. 旋转图像
数据结构·算法·leetcode
bbc1212264 分钟前
AT_abc306_b [ABC306B] Base 2
算法
生锈的键盘12 分钟前
推荐算法实践:movielens数据集
算法
董董灿是个攻城狮13 分钟前
Transformer 通关秘籍9:词向量的数值实际上是特征
算法
代码AC不AC22 分钟前
【数据结构】队列
c语言·数据结构·学习·队列·深度讲解
林泽毅22 分钟前
SwanLab x EasyR1:多模态LLM强化学习后训练组合拳,让模型进化更高效
算法·llm·强化学习
小林熬夜学编程24 分钟前
【高并发内存池】第八弹---脱离new的定长内存池与多线程malloc测试
c语言·开发语言·数据结构·c++·算法·哈希算法
刚入门的大一新生30 分钟前
归并排序延伸-非递归版本
算法·排序算法
独好紫罗兰35 分钟前
洛谷题单3-P1980 [NOIP 2013 普及组] 计数问题-python-流程图重构
开发语言·python·算法
独好紫罗兰40 分钟前
洛谷题单3-P1009 [NOIP 1998 普及组] 阶乘之和-python-流程图重构
开发语言·python·算法