mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-11-30 16:31:08 +00:00
3ad90cea83
* add a yiled method to fibonaci * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * fibonaci * Update fibonacci.py * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * Update fibonacci.py --------- Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com> Co-authored-by: Christian Clauss <cclauss@me.com>
245 lines
6.6 KiB
Python
245 lines
6.6 KiB
Python
"""
|
|
Calculates the Fibonacci sequence using iteration, recursion, memoization,
|
|
and a simplified form of Binet's formula
|
|
|
|
NOTE 1: the iterative, recursive, memoization functions are more accurate than
|
|
the Binet's formula function because the Binet formula function uses floats
|
|
|
|
NOTE 2: the Binet's formula function is much more limited in the size of inputs
|
|
that it can handle due to the size limitations of Python floats
|
|
|
|
See benchmark numbers in __main__ for performance comparisons/
|
|
https://en.wikipedia.org/wiki/Fibonacci_number for more information
|
|
"""
|
|
|
|
import functools
|
|
from collections.abc import Iterator
|
|
from math import sqrt
|
|
from time import time
|
|
|
|
|
|
def time_func(func, *args, **kwargs):
|
|
"""
|
|
Times the execution of a function with parameters
|
|
"""
|
|
start = time()
|
|
output = func(*args, **kwargs)
|
|
end = time()
|
|
if int(end - start) > 0:
|
|
print(f"{func.__name__} runtime: {(end - start):0.4f} s")
|
|
else:
|
|
print(f"{func.__name__} runtime: {(end - start) * 1000:0.4f} ms")
|
|
return output
|
|
|
|
|
|
def fib_iterative_yield(n: int) -> Iterator[int]:
|
|
"""
|
|
Calculates the first n (1-indexed) Fibonacci numbers using iteration with yield
|
|
>>> list(fib_iterative_yield(0))
|
|
[0]
|
|
>>> tuple(fib_iterative_yield(1))
|
|
(0, 1)
|
|
>>> tuple(fib_iterative_yield(5))
|
|
(0, 1, 1, 2, 3, 5)
|
|
>>> tuple(fib_iterative_yield(10))
|
|
(0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55)
|
|
>>> tuple(fib_iterative_yield(-1))
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: n is negative
|
|
"""
|
|
if n < 0:
|
|
raise ValueError("n is negative")
|
|
a, b = 0, 1
|
|
yield a
|
|
for _ in range(n):
|
|
yield b
|
|
a, b = b, a + b
|
|
|
|
|
|
def fib_iterative(n: int) -> list[int]:
|
|
"""
|
|
Calculates the first n (0-indexed) Fibonacci numbers using iteration
|
|
>>> fib_iterative(0)
|
|
[0]
|
|
>>> fib_iterative(1)
|
|
[0, 1]
|
|
>>> fib_iterative(5)
|
|
[0, 1, 1, 2, 3, 5]
|
|
>>> fib_iterative(10)
|
|
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
|
|
>>> fib_iterative(-1)
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: n is negative
|
|
"""
|
|
if n < 0:
|
|
raise ValueError("n is negative")
|
|
if n == 0:
|
|
return [0]
|
|
fib = [0, 1]
|
|
for _ in range(n - 1):
|
|
fib.append(fib[-1] + fib[-2])
|
|
return fib
|
|
|
|
|
|
def fib_recursive(n: int) -> list[int]:
|
|
"""
|
|
Calculates the first n (0-indexed) Fibonacci numbers using recursion
|
|
>>> fib_iterative(0)
|
|
[0]
|
|
>>> fib_iterative(1)
|
|
[0, 1]
|
|
>>> fib_iterative(5)
|
|
[0, 1, 1, 2, 3, 5]
|
|
>>> fib_iterative(10)
|
|
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
|
|
>>> fib_iterative(-1)
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: n is negative
|
|
"""
|
|
|
|
def fib_recursive_term(i: int) -> int:
|
|
"""
|
|
Calculates the i-th (0-indexed) Fibonacci number using recursion
|
|
>>> fib_recursive_term(0)
|
|
0
|
|
>>> fib_recursive_term(1)
|
|
1
|
|
>>> fib_recursive_term(5)
|
|
5
|
|
>>> fib_recursive_term(10)
|
|
55
|
|
>>> fib_recursive_term(-1)
|
|
Traceback (most recent call last):
|
|
...
|
|
Exception: n is negative
|
|
"""
|
|
if i < 0:
|
|
raise ValueError("n is negative")
|
|
if i < 2:
|
|
return i
|
|
return fib_recursive_term(i - 1) + fib_recursive_term(i - 2)
|
|
|
|
if n < 0:
|
|
raise ValueError("n is negative")
|
|
return [fib_recursive_term(i) for i in range(n + 1)]
|
|
|
|
|
|
def fib_recursive_cached(n: int) -> list[int]:
|
|
"""
|
|
Calculates the first n (0-indexed) Fibonacci numbers using recursion
|
|
>>> fib_iterative(0)
|
|
[0]
|
|
>>> fib_iterative(1)
|
|
[0, 1]
|
|
>>> fib_iterative(5)
|
|
[0, 1, 1, 2, 3, 5]
|
|
>>> fib_iterative(10)
|
|
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
|
|
>>> fib_iterative(-1)
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: n is negative
|
|
"""
|
|
|
|
@functools.cache
|
|
def fib_recursive_term(i: int) -> int:
|
|
"""
|
|
Calculates the i-th (0-indexed) Fibonacci number using recursion
|
|
"""
|
|
if i < 0:
|
|
raise ValueError("n is negative")
|
|
if i < 2:
|
|
return i
|
|
return fib_recursive_term(i - 1) + fib_recursive_term(i - 2)
|
|
|
|
if n < 0:
|
|
raise ValueError("n is negative")
|
|
return [fib_recursive_term(i) for i in range(n + 1)]
|
|
|
|
|
|
def fib_memoization(n: int) -> list[int]:
|
|
"""
|
|
Calculates the first n (0-indexed) Fibonacci numbers using memoization
|
|
>>> fib_memoization(0)
|
|
[0]
|
|
>>> fib_memoization(1)
|
|
[0, 1]
|
|
>>> fib_memoization(5)
|
|
[0, 1, 1, 2, 3, 5]
|
|
>>> fib_memoization(10)
|
|
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
|
|
>>> fib_iterative(-1)
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: n is negative
|
|
"""
|
|
if n < 0:
|
|
raise ValueError("n is negative")
|
|
# Cache must be outside recursuive function
|
|
# other it will reset every time it calls itself.
|
|
cache: dict[int, int] = {0: 0, 1: 1, 2: 1} # Prefilled cache
|
|
|
|
def rec_fn_memoized(num: int) -> int:
|
|
if num in cache:
|
|
return cache[num]
|
|
|
|
value = rec_fn_memoized(num - 1) + rec_fn_memoized(num - 2)
|
|
cache[num] = value
|
|
return value
|
|
|
|
return [rec_fn_memoized(i) for i in range(n + 1)]
|
|
|
|
|
|
def fib_binet(n: int) -> list[int]:
|
|
"""
|
|
Calculates the first n (0-indexed) Fibonacci numbers using a simplified form
|
|
of Binet's formula:
|
|
https://en.m.wikipedia.org/wiki/Fibonacci_number#Computation_by_rounding
|
|
|
|
NOTE 1: this function diverges from fib_iterative at around n = 71, likely
|
|
due to compounding floating-point arithmetic errors
|
|
|
|
NOTE 2: this function doesn't accept n >= 1475 because it overflows
|
|
thereafter due to the size limitations of Python floats
|
|
>>> fib_binet(0)
|
|
[0]
|
|
>>> fib_binet(1)
|
|
[0, 1]
|
|
>>> fib_binet(5)
|
|
[0, 1, 1, 2, 3, 5]
|
|
>>> fib_binet(10)
|
|
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
|
|
>>> fib_binet(-1)
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: n is negative
|
|
>>> fib_binet(1475)
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: n is too large
|
|
"""
|
|
if n < 0:
|
|
raise ValueError("n is negative")
|
|
if n >= 1475:
|
|
raise ValueError("n is too large")
|
|
sqrt_5 = sqrt(5)
|
|
phi = (1 + sqrt_5) / 2
|
|
return [round(phi**i / sqrt_5) for i in range(n + 1)]
|
|
|
|
|
|
if __name__ == "__main__":
|
|
from doctest import testmod
|
|
|
|
testmod()
|
|
# Time on an M1 MacBook Pro -- Fastest to slowest
|
|
num = 30
|
|
time_func(fib_iterative_yield, num) # 0.0012 ms
|
|
time_func(fib_iterative, num) # 0.0031 ms
|
|
time_func(fib_binet, num) # 0.0062 ms
|
|
time_func(fib_memoization, num) # 0.0100 ms
|
|
time_func(fib_recursive_cached, num) # 0.0153 ms
|
|
time_func(fib_recursive, num) # 257.0910 ms
|