For those enjoying the fun with prime finding, I've updated the source at http://www.cs.hmc.edu/~oneill/code/haskell-primes.zip I've tweaked my code a little to improve its space behavior when finding primes up to some limit, added an up-to-limit version of the Naive Primes algorithm, and added Oleg's prime finding code too. I also got a chance to look at space usage more generally. I won't reproduce a table here, but the conclusions were more-or-less what you'd expect. The "unlimited list" algorithms used O(n) space to find n primes (except for Runciman's algorithm, which appeared to be much worse), and the "primes up to a limit" algorithms used O(sqrt (n)) space to find the nth prime. Both of these are better than the classic C algorithm, which uses O(n log n) space to find the nth prime. For example, heap profiling shows that my own O(sqrt(n)) algorithm uses only 91200 bytes to find the 10^7th prime, whereas the classic C algorithm needs at least 11214043 bytes for its array -- a factor of more than 100 different, and one that gets worse for larger n. Lennart Augustsson wrote:
Another weird thing is that much of the Haskell code seems to work with Integer whereas the C code uses int.
Originally, I was comparing Haskell with Haskell, and for that purpose I wanted to have a level playing field, so going with Integer everywhere made sense.
That doesn't seem fair.
Actually, to the extent that any of the comparisons are "fair", I think this one is too. After all, typical Haskell code uses Integer and typical C code uses int. I could use arrays in my Haskell code and never use laziness, but when I program in Haskell, I'm not trying to exactly recreate C programs, but rather write their Haskell equivalents. For example, to me, producing a lazy list was essential for a true Haskell feel. For some people, the "Haskell feel" also includes treating the language as a declarative specification language where brevity is everything -- but for me, other things (like fundamental algorithmic efficiency and faithfulness to the core ideas that make the Sieve of Eratosthenes an *efficient* algorithm) are universal and ought to be common to both C and Haskell versions. But to allow a better comparison with C, I've added a run for an Int version of my algorithm. With that change, my code is closer to the speed of the C code. More interestingly, for larger n, I seem to be narrowing the gap. At 10^6, my code runs nearly 30 times slower than the classic C version, but at 10^8, I'm only about 20 times slower. This is especially interesting to me there was some (reasonable looking) speculation from apfelmus several days ago, that suggested that my use of a priority queue incurred an extra log(n) overhead, from which you would expect a worse asymptotic complexity, not equivalent or better. Melissa. Enc. (best viewed with a fixed-width font) ------------------------------------------------------------------ Time (in seconds) for Number of Primes ---------------------------------------------------- Algorithm 10^3 10^4 10^5 10^6 10^7 10^8 ------------------------------------------------------------------ C-Sieve 0.00 0.00 0.01 0.29 5.12 88.24 O'Neill (#3) 0.01 0.04 0.55 8.34 122.62 1779.18 O'Neill (#2) 0.01 0.06 0.95 13.85 194.96 2699.61 O'Neill (#1) 0.01 0.07 1.07 15.95 230.11 - Bromage 0.02 0.39 6.50 142.85 - - "sieve" (#3) 0.01 0.25 7.28 213.19 - - Naive (#2) 0.02 0.59 14.70 386.40 - - Naive (#1) 0.32 0.66 16.04 419.22 - - Runciman 0.02 0.74 29.25 - - - Reinke 0.04 1.21 41.00 - - - Zilibowitz 0.02 2.50 368.33 - - - Gale (#1) 0.12 17.99 - - - - "sieve" (#1) 0.16 32.59 - - - - "sieve" (#2) 0.01 32.76 - - - - Oleg 0.18 68.40 - - - - Gale (#2) 1.36 268.65 - - - - ------------------------------------------------------------------ - The dashes in the table mean "I gave up waiting" (i.e., > 500 seconds) - "sieve" (#1) is the classic example we're all familiar with - "sieve" (#2) is the classic example, but sieving a list without multiples of 2,3,5, or 7 -- notice how it makes no real difference - "sieve" (#3) is the classic example, but generating a lazy-but- finite list (see below) - O'Neill (#1) is basically the algorithm of mine discussed in http:// www.cs.hmc.edu/~oneill/papers/Sieve-JFP.pdf, with a few minor tweaks - O'Neill (#2) is a variant of that algorithm that generates a lazy- but-finite list of primes. - O'Neill (#3) is a variant of that algoritm that uses Ints when it can get away with it. - Naive (#1) is the the non-sieve-based "divide by every prime up to the square root" algorithm for finding primes (called SimplePrimes in the source) - Naive (#2) is the same algorithm, with a limit on the number of primes - Runciman is Colin Runciman's algorithm, from his _Lazy Wheel Sieves and Spirals of Primes_ paper - Reinke is the ``applyAt'' algorithm Claus Reinke posted here - Gale (#1) is Yitz Gale's deleteOrd algorithm - Gale (#2) is Yitz Gale's crossOff algorithm - Oleg is oleg@pobox.com's algoirthm, as posted to Haskell Cafe - Zilibowitz is Ruben Zilibowitz's GCD-based primes generator, as posted on Haskell-Cafe - Bromage is Andrew Bromage's implementation of the Atkin-Bernstein sieve. Like O'Neill (#2) and "sieve" (#3), asks for some upper limit on the number of primes it generates. Unlike O'Neill (#2) and "sieve" (#3), it uses arrays, and the upper limit causes a large initial array allocation. Also, unlike the other Haskell algorithms, it does not produce a lazy list; no output is produced until sieving is complete - C-Sieve is a "typical" simple implementation of the sieve in C found with Google; it skips multiples of 2 and uses a bit array. Also, obviously, it doesn't produce incremental output.