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@ -7,6 +7,8 @@ the Binet's formula function because the Binet formula function uses floats
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NOTE 2: the Binet's formula function is much more limited in the size of inputs
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that it can handle due to the size limitations of Python floats
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NOTE 3: the matrix function is the fastest and most memory efficient for large n
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See benchmark numbers in __main__ for performance comparisons/
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https://en.wikipedia.org/wiki/Fibonacci_number for more information
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@ -17,6 +19,9 @@ from collections.abc import Iterator
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from math import sqrt
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from time import time
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import numpy as np
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from numpy import ndarray
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def time_func(func, *args, **kwargs):
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"""
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@ -230,6 +235,88 @@ def fib_binet(n: int) -> list[int]:
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return [round(phi**i / sqrt_5) for i in range(n + 1)]
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def matrix_pow_np(m: ndarray, power: int) -> ndarray:
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"""
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Raises a matrix to the power of 'power' using binary exponentiation.
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Args:
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m: Matrix as a numpy array.
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power: The power to which the matrix is to be raised.
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Returns:
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The matrix raised to the power.
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Raises:
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ValueError: If power is negative.
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>>> m = np.array([[1, 1], [1, 0]], dtype=int)
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>>> matrix_pow_np(m, 0) # Identity matrix when raised to the power of 0
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array([[1, 0],
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[0, 1]])
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>>> matrix_pow_np(m, 1) # Same matrix when raised to the power of 1
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array([[1, 1],
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[1, 0]])
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>>> matrix_pow_np(m, 5)
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array([[8, 5],
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[5, 3]])
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>>> matrix_pow_np(m, -1)
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Traceback (most recent call last):
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...
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ValueError: power is negative
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"""
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result = np.array([[1, 0], [0, 1]], dtype=int) # Identity Matrix
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base = m
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if power < 0: # Negative power is not allowed
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raise ValueError("power is negative")
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while power:
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if power % 2 == 1:
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result = np.dot(result, base)
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base = np.dot(base, base)
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power //= 2
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return result
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def fib_matrix_np(n: int) -> int:
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"""
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Calculates the n-th Fibonacci number using matrix exponentiation.
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https://www.nayuki.io/page/fast-fibonacci-algorithms#:~:text=
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Summary:%20The%20two%20fast%20Fibonacci%20algorithms%20are%20matrix
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Args:
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n: Fibonacci sequence index
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Returns:
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The n-th Fibonacci number.
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Raises:
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ValueError: If n is negative.
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>>> fib_matrix_np(0)
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0
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>>> fib_matrix_np(1)
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1
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>>> fib_matrix_np(5)
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5
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>>> fib_matrix_np(10)
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55
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>>> fib_matrix_np(-1)
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Traceback (most recent call last):
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...
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ValueError: n is negative
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"""
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if n < 0:
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raise ValueError("n is negative")
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if n == 0:
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return 0
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m = np.array([[1, 1], [1, 0]], dtype=int)
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result = matrix_pow_np(m, n - 1)
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return int(result[0, 0])
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if __name__ == "__main__":
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from doctest import testmod
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@ -242,3 +329,4 @@ if __name__ == "__main__":
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time_func(fib_memoization, num) # 0.0100 ms
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time_func(fib_recursive_cached, num) # 0.0153 ms
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time_func(fib_recursive, num) # 257.0910 ms
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time_func(fib_matrix_np, num) # 0.0000 ms
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@ -26,6 +26,15 @@ def quick_sort(collection: list) -> list:
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[]
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>>> quick_sort([-2, 5, 0, -45])
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[-45, -2, 0, 5]
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Time Complexity:
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Best Case: O(n log n)
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Average case: O(n log n)
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Worst Case: O(n^2)
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Space Complexity:
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Best Case: O(log n)
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Worst Case: O(n)
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"""
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# Base case: if the collection has 0 or 1 elements, it is already sorted
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if len(collection) < 2:
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