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