I read this interesting paper, The Genuine Sieve of Eratosthenes (PDF) by Melissa O'Neill, when it was discussed on Lambda the Ultimate. I came back to look at it again today because I wanted to improve my Haskell fluency, which I only seem to exercise when I read papers like this. I decided to try to correctly construe the Haskell code in the paper by implementing the algorithms in Python. I focused on reproducing the laziness but not the functional style because I could substitute iterators for lazy lists but had to use conventional control structures for all the pattern matching and tail calls.

#!/usr/bin/env python

# Author: Jared Brothers

#

# A Python version of the Haskell code from

# "The Genuine Sieve of Eratosthenes"

# www.cs.hmc.edu/~oneill/papers/Sieve-JFP.pdf

#

import heapq, itertools, operator

def sieve(xs):

"""Generate the prime numbers, given an iterable of candidate numbers.

Cross off multiples of prime numbers incrementally using iterators."""

table = []

while 1:

x = xs.next()

if table == [] or x < table[0][0]:

yield x

xs, ys = itertools.tee(xs)

timesx = (lambda x: lambda y: x*y)(x)

heapq.heappush(table, (x**2, itertools.imap(timesx, ys)))

else:

while table[0][0] <= x:

heapq.heapreplace(table, (table[0][1].next(), table[0][1]))

def wheel(factors=[2, 3, 5, 7], next=11):

"""Generate the distances between numbers not divisible by a list of small

primes, from the next prime up to the product of the list."""

circumference = reduce(operator.mul, factors)

prev = next

next += 1

end = next + circumference

while next < end:

if not any(next % factor == 0 for factor in factors):

yield next - prev

prev = next

next += 1

def spin(factors=[2, 3, 5, 7], next=11):

"""Generate candidates by making a wheel and cycling through it."""

for gap in itertools.cycle(wheel(factors, next)):

yield next

next += gap

def primes(k=5):

"""Generate primes with the sieve and wheel factorization, which filters

multiples of the first k primes."""

smallprimes = list(itertools.islice(sieve(itertools.count(2)), k + 1))

factors = smallprimes[:-1]

next = smallprimes[-1]

return itertools.chain(factors, sieve(spin(factors, next)))

Labels: haskell, lazy, papers, python

My brother is writing a combinatorics paper about playing with dominoes. Pure mathematics is a lot of fun that way, but with more Hamiltonian paths of graphs embedded on a torus. Here is a Python script I wrote for him last night to solve a counting problem.

#!/usr/bin/env python

# Author: Jared Brothers

# Find the orderings of a set of dominoes such that

# adjacent dominoes share a number and the other two

# numbers differ by one.

def search(start):

""" Depth first search, generating all solutions. """

fringe = [start]

while fringe:

s = fringe.pop()

if s.check():

yield s

fringe.extend(s.successors())

class state():

""" A state in the search space. """

def __init__(self, d, r):

""" The ordered dominoes are in done (list),

and the unordered dominoes are in rest (set). """

self.done = d

self.rest = r

def __str__(self):

return str(self.done)

def check(self):

""" Is this a solution? """

return not self.rest

def successors(self):

""" The states you can get to by moving a domino from rest to done. """

if self.done:

# Try only dominoes adjacent to the previous one,

a,b = self.done[-1]

a1 = (a + 1) % n

a2 = (a - 1) % n

b1 = (b + 1) % n

b2 = (b - 1) % n

adjs = set([(a,b1), (b1,a), (a,b2), (b2,a), (b,a1), (a1,b), (b,a2), (a2,b)])

for x in self.rest.intersection(adjs):

d = self.done[:]

d.append(x)

r = self.rest.copy()

r.discard(x)

yield state(d, r)

else:

# or all the rest if nothing has been done yet.

for x in sorted(self.rest):

d = [x]

r = self.rest.copy()

r.discard(x)

yield state(d, r)

if __name__ == '__main__':

n = 7 # The number of numbers. The number of dominoes is sum(range(n + 1)).

start = state([], set((a, b) for a in range(n) for b in range(a + 1)))

for s in search(start):

print(s)

Labels: combinatorics, python

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