Domino and Tromino Tiling
- Review: Dynamic Programming Problems
- Review: mod Problems
- Tag: #inspiring, #mod, #dynamicprogramming
We have two types of tiles: a 2×1 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.)
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
N will be in range [1, 1000].
Credits To: leetcode.com
Leave me comments, if you have better ways to solve.
## Blog link: https://code.dennyzhang.com/domino-and-tromino-tiling ## Basic Ideas: dynamic programming ## ## Complexity: Time O(n), Space O(1) class Solution: def numTilings(self, N): """ :type N: int :rtype: int """ if N==1: return 1 if N==2: return 2 if N==3: return 5 mod_num = pow(10, 9)+7 v3,v2,v1 = 5,2,1 sum_v=4 for i in range(4, N+1): v = (v3+v2+sum_v) % mod_num v3,v2,v1 = v,v3,v2 sum_v=sum_v+v1*2 return v3 # s = Solution() # print(s.numTilings(4)) # 11 # print(s.numTilings(5)) # 24 # print(s.numTilings(6)) # 53