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A collection of prime finding algorithms implemented in pure python
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| def sieve_of_atkin(limit): | |
| """The Sieve of Atkin prime finding algorithm, [1]_ | |
| References | |
| ---------- | |
| .. [1] A. O. L. Atkin, D. J. Bernstein, 2003 | |
| """ | |
| P = [2,3] | |
| sieve=[False]*(limit+1) | |
| sqrt_limit = int(limit ** 0.5) | |
| for x in range(1,sqrt_limit+1): | |
| for y in range(1,sqrt_limit+1): | |
| h = x ** 2 | |
| k = y ** 2 | |
| n = 4*h + k | |
| if n<=limit and (n%12==1 or n%12==5) : sieve[n] = not sieve[n] | |
| n = 3*h+k | |
| if n<= limit and n%12==7 : sieve[n] = not sieve[n] | |
| n = 3*h - k | |
| if x>y and n<=limit and n%12==11 : sieve[n] = not sieve[n] | |
| for n in range(5,sqrt_limit): | |
| if sieve[n]: | |
| z = n ** 2 | |
| for k in range(z,limit+1,z): | |
| sieve[k] = False | |
| for n in range(5,limit): | |
| if sieve[n]: P.append(n) | |
| return P | |
| def sieve_of_eratosthenes(n): | |
| """The Sieve of Eratosthenes prime finding algorithm, [1]_ | |
| References | |
| ---------- | |
| .. [1] Nicomachus of Gerasa, 2nd c. AD | |
| Introduction to Arithmetic | |
| """ | |
| sieve = [True] * (n+1) | |
| for x in range(2, int(n**0.5) + 1): | |
| if sieve[x]: | |
| for i in range(x*x, n+1, x): | |
| sieve[i] = False | |
| return [i for i in range(2, n) if sieve[i]] | |
| def segmented_sieve_of_eratosthenes(n): | |
| """The page-segmented Sieve of Eratosthenes prime finding algorithm, [1]_ | |
| Notes | |
| ----- | |
| The algorithm trades space efficiency for time efficiency to the non-segmented algorithm. | |
| References | |
| ---------- | |
| .. [1] C. Bays, R. H. Hudson, 1977 | |
| The Segmented Sieve of Eratosthenes and Primes in Arithmetic Progressions to 10^12 | |
| """ | |
| limit = math.isqrt(n) + 1 | |
| primes = [] | |
| sieve = [True] * limit | |
| for p in range(2, limit): | |
| if sieve[p]: | |
| primes.append(p) | |
| for i in range(p*p, limit, p): | |
| sieve[i] = False | |
| low_limit = limit | |
| while low_limit < n: | |
| high_limit = min(low_limit + limit, n) | |
| segments = [True] * (high_limit - low_limit) | |
| for p in primes: | |
| start = max(p*p, (low_limit // p) * p) | |
| if start < low_limit: | |
| start += p | |
| for i in range(start, high_limit, p): | |
| segments[i - low_limit] = False | |
| for i in range(low_limit, high_limit): | |
| if segments[i - low_limit]: | |
| primes.append(i) | |
| low_limit = high_limit | |
| return primes | |
| def mixed_sieve(n): | |
| """A prime finding algorithm mixed via rule-of-thumb""" | |
| if n < 1_000: return sieve_of_atkin(n) | |
| return sieve_of_eratosthenes(n) |
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| if __name__ == "__main__": | |
| # Initialize the results | |
| results = { | |
| 'sieve_of_eratosthenes': {'time': [], 'space': []}, | |
| 'sieve_of_atkin': {'time': [], 'space': []}, | |
| 'mixed_sieve': {'time': [], 'space': []}, | |
| # 'segmented_sieve_of_eratosthenes': {'time': [], 'space': []}, # FIXME: too slow | |
| } | |
| for limit in tqdm(limits): | |
| print(f"Computing primes up to {limit}...") | |
| for func in [sieve_of_eratosthenes, sieve_of_atkin, mixed_sieve]: | |
| elapsed_time, mem_usage = measure(func, limit) | |
| results[func.__name__]['time'].append(elapsed_time) | |
| results[func.__name__]['space'].append(mem_usage) | |
| plot_spacetime_efficiency(limits, results) |
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| import matplotlib.pyplot as plt | |
| def plot_spacetime_efficiency(limits, results): | |
| plt.figure(figsize=(12, 6)) | |
| print(len(limits), results) | |
| plt.subplot(1, 2, 1) | |
| for func in results: | |
| plt.plot(limits, results[func]['time'], label=func) | |
| plt.xlabel('Limit') | |
| plt.ylabel('Time (seconds)') | |
| plt.xscale('log') | |
| plt.yscale('log') | |
| plt.legend() | |
| plt.title('Time Efficiency') | |
| plt.subplot(1, 2, 2) | |
| for func in results: | |
| plt.plot(limits, results[func]['space'], label=func) | |
| plt.xlabel('Limit') | |
| plt.ylabel('Space (MiB)') | |
| plt.xscale('log') | |
| plt.yscale('log') | |
| plt.legend() | |
| plt.title('Space Efficiency') | |
| plt.tight_layout() | |
| plt.show() |
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| from memory_profiler import memory_usage | |
| def measure(func, *args, **kwargs): | |
| start_time = time.time() | |
| mem_usage = memory_usage((func, args, kwargs), interval=0.1, max_usage=True) | |
| elapsed_time = time.time() - start_time | |
| return elapsed_time, mem_usage |
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| matplotlib | |
| memory-profiler | |
| tqdm |
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