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POOptimizing Divisor Sieve
 

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    <p>I have two sieves that I wrote in python and would like help optimizing them if at all possible. The divisorSieve calculates the divisors of all numbers up to <code>n</code>. Each index of the list contains a list of its divisors. The numDivisorSieve just counts the number of divisors each index has but doesn't store the divisors themselves. These sieves work in a similar way as you would do a Sieve of Eratosthenes to calculate all prime numbers up to <code>n</code>.</p> <pre><code>def divisorSieve(n): divs = [[1] for x in xrange(0, n + 1)] divs[0] = [0] for i in xrange(2, n + 1): for j in xrange(1, n / i + 1): divs[i * j] += [i] return divs def numDivisorSieve(n): divs = [1] * (n + 1) divs[0] = 0 for i in xrange(2, n + 1): for j in xrange(1, n / i + 1): divs[i * j] += 1 return divs #Timer test for function if __name__=='__main__': from timeit import Timer n = ... t1 = Timer(lambda: divisorSieve(n)) print n, t1.timeit(number=1) </code></pre> <p>Results:</p> <pre><code> -----n-----|--time(divSieve)--|--time(numDivSieve)-- 100,000 | 0.452978167569 | 0.187762331281 200,000 | 0.98932170181 | 0.362314797537 300,000 | 1.6338203645 | 0.55124339118 400,000 | 2.17913634385 | 0.748340797412 500,000 | 2.84024755862 | 0.959312993718 600,000 | 3.46395907197 | 1.17777010636 700,000 | 4.11840766312 | 1.38268800149 800,000 | 4.77613733906 | 1.62560614543 900,000 | 5.4634148162 | 1.83002270324 1,000,000 | 6.11017136282 | 2.10247496423 2,000,000 | 13.0507389438 | 4.59150618897 3,000,000 | 20.6068050631 | 7.24799900479 4,000,000 | 28.3432841948 | 10.1484527586 5,000,000 | 36.4113970268 | 12.7670585308 6,000,000 | 44.4017867139 | 15.4226118057 7,000,000 | 52.4697580394 | 18.2902677738 8,000,000 | 61.4056966474 | 21.1247001928 9,000,000 | 69.7948510294 | 23.8988925173 10,000,000 | 78.639434285 | 26.8588813211 20,000,000 | N/A | 56.2527237669 (waited for a while but divisorSieve didn't finish) 30,000,000 | MemoryError | 86.8917332214 40,000,000 | MemoryError | 118.457179822 50,000,000 | MemoryError | 149.526622815 60,000,000 | MemoryError | 181.627320396 70,000,000 | MemoryError | 214.17467749 80,000,000 | MemoryError | 246.23677614 90,000,000 | MemoryError | 279.53308422 100,000,000 | MemoryError | 314.813166014 </code></pre> <p>Results are pretty good and I'm happy I was able to get it this far, but I'm looking to get it even faster. If at all possible, I'd like to get <em><strong>100,000,000</strong></em> at a reasonable speed with the divisorSieve. Although this also brings into the issue that anything over <em><strong>30,000,000+</strong></em> throws a <code>MemoryError</code> at <code>divs = [[1] for x in xrange(0, n + 1)]</code>) in divisorSieve. numDivisorSieve does allow the full <em><strong>100,000,000</strong></em> to run.</p> <p>I've tried replacing numDivisorSieve's <code>divs = [1] * (n + 1)</code> with both <code>divs = array.array('i', [1] * (n + 1))</code> and <code>divs = numpy.ones((n + 1), dtype='int')</code> but both resulted in a loss of speed (slight difference for array, much larger difference for numpy). I expect that since numDivisorSieve had a loss in efficiency, then so would divisorSieve. Of course there's always the chance I'm using one or both of these incorrectly since I'm not used to either of them.</p> <p>I would appreciate any help you can give me. I hope i have provided enough details. Thank you.</p>
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    1. COYou may want to also try your luck on http://codereview.stackexchange.com
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    2. COif anyone sees this and is interested, I have taken newfurniturey's advice and continued here http://codereview.stackexchange.com/questions/18346/optimizing-divisor-sieve
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