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domino-and-tromino-tiling.py
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domino-and-tromino-tiling.py
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# Time: O(logn)
# Space: O(1)
import itertools
class Solution(object):
def numTilings(self, N):
"""
:type N: int
:rtype: int
"""
M = int(1e9+7)
def matrix_expo(A, K):
result = [[int(i==j) for j in xrange(len(A))] \
for i in xrange(len(A))]
while K:
if K % 2:
result = matrix_mult(result, A)
A = matrix_mult(A, A)
K /= 2
return result
def matrix_mult(A, B):
ZB = zip(*B)
return [[sum(a*b for a, b in itertools.izip(row, col)) % M \
for col in ZB] for row in A]
T = [[1, 0, 0, 1], # #(|) = #(|) + #(=)
[1, 0, 1, 0], # #(「) = #(|) + #(L)
[1, 1, 0, 0], # #(L) = #(|) + #(「)
[1, 1, 1, 0]] # #(=) = #(|) + #(「) + #(L)
return matrix_mult([[1, 0, 0, 0]], matrix_expo(T, N))[0][0] # [a0, a(-1), a(-2), a(-3)] * T^N
# Time: O(n)
# Space: O(1)
class Solution2(object):
def numTilings(self, N):
"""
:type N: int
:rtype: int
"""
# Prove:
# dp[n] = dp[n-1](|) + dp[n-2](=) + 2*(dp[n-3](「」) + ... + d[0](「 = ... = 」))
# = dp[n-1] + dp[n-2] + dp[n-3] + dp[n-3] + 2*(dp[n-4] + ... + d[0])
# = dp[n-1] + dp[n-3] + (dp[n-2] + dp[n-3] + 2*(dp[n-4] + ... + d[0])
# = dp[n-1] + dp[n-3] + dp[n-1]
# = 2*dp[n-1] + dp[n-3]
M = int(1e9+7)
dp = [1, 1, 2]
for i in xrange(3, N+1):
dp[i%3] = (2*dp[(i-1)%3]%M + dp[(i-3)%3])%M
return dp[N%3]