mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-11-23 21:11:08 +00:00
bfcb95b297
* fixup! Format Python code with psf/black push * Create codespell.yml * fixup! Format Python code with psf/black push
101 lines
2.5 KiB
Python
101 lines
2.5 KiB
Python
"""Matrix Exponentiation"""
|
|
|
|
import timeit
|
|
|
|
"""
|
|
Matrix Exponentiation is a technique to solve linear recurrences in logarithmic time.
|
|
You read more about it here:
|
|
http://zobayer.blogspot.com/2010/11/matrix-exponentiation.html
|
|
https://www.hackerearth.com/practice/notes/matrix-exponentiation-1/
|
|
"""
|
|
|
|
|
|
class Matrix:
|
|
def __init__(self, arg):
|
|
if isinstance(arg, list): # Initializes a matrix identical to the one provided.
|
|
self.t = arg
|
|
self.n = len(arg)
|
|
else: # Initializes a square matrix of the given size and set the values to zero.
|
|
self.n = arg
|
|
self.t = [[0 for _ in range(self.n)] for _ in range(self.n)]
|
|
|
|
def __mul__(self, b):
|
|
matrix = Matrix(self.n)
|
|
for i in range(self.n):
|
|
for j in range(self.n):
|
|
for k in range(self.n):
|
|
matrix.t[i][j] += self.t[i][k] * b.t[k][j]
|
|
return matrix
|
|
|
|
|
|
def modular_exponentiation(a, b):
|
|
matrix = Matrix([[1, 0], [0, 1]])
|
|
while b > 0:
|
|
if b & 1:
|
|
matrix *= a
|
|
a *= a
|
|
b >>= 1
|
|
return matrix
|
|
|
|
|
|
def fibonacci_with_matrix_exponentiation(n, f1, f2):
|
|
# Trivial Cases
|
|
if n == 1:
|
|
return f1
|
|
elif n == 2:
|
|
return f2
|
|
matrix = Matrix([[1, 1], [1, 0]])
|
|
matrix = modular_exponentiation(matrix, n - 2)
|
|
return f2 * matrix.t[0][0] + f1 * matrix.t[0][1]
|
|
|
|
|
|
def simple_fibonacci(n, f1, f2):
|
|
# Trivial Cases
|
|
if n == 1:
|
|
return f1
|
|
elif n == 2:
|
|
return f2
|
|
|
|
fn_1 = f1
|
|
fn_2 = f2
|
|
n -= 2
|
|
|
|
while n > 0:
|
|
fn_1, fn_2 = fn_1 + fn_2, fn_1
|
|
n -= 1
|
|
|
|
return fn_1
|
|
|
|
|
|
def matrix_exponentiation_time():
|
|
setup = """
|
|
from random import randint
|
|
from __main__ import fibonacci_with_matrix_exponentiation
|
|
"""
|
|
code = "fibonacci_with_matrix_exponentiation(randint(1,70000), 1, 1)"
|
|
exec_time = timeit.timeit(setup=setup, stmt=code, number=100)
|
|
print("With matrix exponentiation the average execution time is ", exec_time / 100)
|
|
return exec_time
|
|
|
|
|
|
def simple_fibonacci_time():
|
|
setup = """
|
|
from random import randint
|
|
from __main__ import simple_fibonacci
|
|
"""
|
|
code = "simple_fibonacci(randint(1,70000), 1, 1)"
|
|
exec_time = timeit.timeit(setup=setup, stmt=code, number=100)
|
|
print(
|
|
"Without matrix exponentiation the average execution time is ", exec_time / 100
|
|
)
|
|
return exec_time
|
|
|
|
|
|
def main():
|
|
matrix_exponentiation_time()
|
|
simple_fibonacci_time()
|
|
|
|
|
|
if __name__ == "__main__":
|
|
main()
|