Related
First of all, I apologise for the title, I did not know how to put my problem in words. Well, here it is:
For an integer a greater than 1, let F be a sorted list of prime factors of a. I need to find all tuples c (filled with whole numbers), such that length of each tuple is equal to the size of F and (F[0] ** c[0]) * (F[1] ** c[1]) * (...) < a. I should add that I write in Python.
Example:
a = 60
F = [2,3,5]
# expected result:
C = {(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 1, 0), (0, 1, 1), (0, 2, 0),
(0, 2, 1), (0, 3, 0), (1, 0, 0), (1, 0, 1), (1, 0, 2), (1, 1, 0), (1, 1, 1),
(1, 2, 0), (1, 3, 0), (2, 0, 0), (2, 0, 1), (2, 1, 0), (2, 2, 0), (3, 0, 0),
(3, 0, 1), (3, 1, 0), (4, 0, 0), (4, 1, 0), (5, 0, 0)}
I generated this result using itertools.product(), specifically:
m = math.floor(round(math.log(a, min(F)), 12))
for i in itertools.product(range(m + 1), repeat=len(F)):
if math.prod([F[j] ** i[j] for j in range(len(F))]) < a: print(i)
I think it works but it's inefficient. For example number 5 appears only in one tuple, but was checked many more times! Is there any way to make it faster? I would use multiple while loops (with break statements) but since I don't know what is the length of F, I don't think that is possible.
You base all your range limits on just min(F). Let's customize each to the log(a, factor) to reduce the cases:
from math import ceil, log, prod
from itertools import product
a = 60
F = [2, 3, 5]
ranges = [range(0, ceil(log(a, factor))) for factor in F]
C = []
for powers in product(*ranges):
if prod(F[i] ** power for i, power in enumerate(powers)) < a:
C.append(powers)
print(C)
By my measure, your code generates 216 test cases to come up with 25 results, but the above code only generates 1/3 of those test cases.
You could iterate over all the "valid" tuples with a generator, like so:
def exponent_tuples(prime_factors, limit):
def next_tuple(t):
n = math.prod(f ** tt for f, tt in zip(prime_factors, t))
for idx, (f, tt) in enumerate(zip(prime_factors, t)):
n *= f
if n < limit:
return (0,) * idx + (tt + 1,) + t[idx + 1 :]
n //= f**(tt+1)
return None
t = (0,) * len(prime_factors)
while t is not None:
yield t
t = next_tuple(t)
for t in exponent_tuples([2, 3, 5], 60):
print(t)
The idea here is to basically increment the tuple entries like digits of a number and have the respective digit roll over to zero and carry the 1 whenever you reach the defined limit.
I'm pretty sure this does exactly what you want, except for maybe the order in which it yields the tuples (can be adjusted by modifying the next_tuple function)
EDIT: Simplified the code a bit
The almost cooked proposition would go like this (shell execution)
>>> max_exponents(42,[2,3,7])
[5, 3, 1]
>>> #pick 2
>>> max_exponents(42//2**2,[3,7])
[2, 1]
>>> #pick 1
>>> max_exponents(42//(2**2*3**1),[7])
[0]
I'm almost done. This will adapt to any number of factors !
Somehow your proposition reduces to this (more readable form ?)
import math as m
import pprint
a = 60
prime_factors = [2,3,5]
exponents =list(map(lambda x:m.floor(m.log(a,x)),prime_factors))
rez = []
for i in range(exponents[0]+1):
for j in range(exponents[1]+1):
for k in range(exponents[2]+1):
if 2**i*3**j*5**k <= a:
rez.append((i,j,k))
pprint.pprint(rez)
and you would like to know wether there's a way to make if faster (with less tests). So we're no more on the implementation side, but more on the conception (algorithm) side ?
For example, once the first exponent c[0] has been chosen, the next ones should be selected amongst the one fitting in a//(2**c[a]) as the other answerer proposed i guess
I have some permutations of a list:
>>> import itertools
>>> perms = list(itertools.permutations([0,1,2,3]))
>>> perms
[(0, 1, 2, 3), (0, 1, 3, 2), (0, 2, 1, 3), (0, 2, 3, 1), (0, 3, 1, 2), (0, 3, 2, 1), (1, 0, 2, 3), (1, 0, 3, 2), (1, 2, 0, 3), (1, 2, 3, 0), (1, 3, 0, 2), (1, 3, 2, 0), (2, 0, 1, 3), (2, 0, 3, 1), (2, 1, 0, 3), (2, 1, 3, 0), (2, 3, 0, 1), (2, 3, 1, 0), (3, 0, 1, 2), (3, 0, 2, 1), (3, 1, 0, 2), (3, 1, 2, 0), (3, 2, 0, 1), (3, 2, 1, 0)]
>>> len(perms)
24
What function can I use (without access to the list perm) to get the index of an arbitrary permutation, e.g. (0, 2, 3, 1) -> 3?
(You can assume that permuted elements are always an ascending list of integers, starting at zero.)
Hint: The factorial number system may be involved. https://en.wikipedia.org/wiki/Factorial_number_system
Off the top of my head I came up with the following, didn't test it thoroughly.
from math import factorial
elements = list(range(4))
permutation = (3, 2, 1, 0)
index = 0
nf = factorial(len(elements))
for n in permutation:
nf //= len(elements)
index += elements.index(n) * nf
elements.remove(n)
print(index)
EDIT: replaced nf /= len(elements) with nf //= len(elements)
I suppose this is a challenge, so here is my (recursive) answer:
import math
import itertools
def get_index(l):
# In a real function, there should be more tests to validate that the input is valid, e.g. len(l)>0
# Terminal case
if len(l)==1:
return 0
# Number of possible permutations starting with l[0]
span = math.factorial(len(l)-1)
# Slightly modifying l[1:] to use the function recursively
new_l = [ val if val < l[0] else val-1 for val in l[1:] ]
# Actual solution
return get_index(new_l) + span*l[0]
get_index((0,1,2,3))
# 0
get_index((0,2,3,1))
# 3
get_index((3,2,1,0))
# 23
get_index((4,2,0,1,5,3))
# 529
list(itertools.permutations((0,1,2,3,4,5))).index((4,2,0,1,5,3))
# 529
You need to write your own function. Something like this would work
import math
def perm_loc(P):
N = len(P)
assert set(P) == set(range(N))
def rec(perm):
nums = set(perm)
if not perm:
return 0
else:
sub_res = rec(perm[1:]) # Result for tail of permutation
sub_size = math.factorial(len(nums) - 1) # How many tail permutations exist
sub_index = sorted(nums).index(perm[0]) # Location of first element in permutaiotn
# in the sorted list of number
return sub_index * sub_size + sub_res
return rec(P)
The function that does all the work is rec, with perm_loc just serving as a wrapper around it. Note that this algorithm is based on the nature of the permutation algorithm that itertools.permutation happens to use.
The following code tests the above function. First on your sample, and then on all permutations of range(7):
print perm_loc([0,2,3,1]) # Print the result from the example
import itertools
def test(N):
correct = 0
perms = list(itertools.permutations(range(N)))
for (i, p) in enumerate(perms):
pl = perm_loc(p)
if i == pl:
correct += 1
else:
print ":: Incorrect", p, perms.index(p), perm_loc(N, p)
print ":: Found %d correct results" % correct
test(7) # Test on all permutations of range(7)
from math import factorial
def perm_to_permidx(perm):
# Extract info
n = len(perm)
elements = range(n)
# "Gone"s will be the elements of the given perm
gones = []
# According to each number in perm, we add the repsective offsets
offset = 0
for i, num in enumerate(perm[:-1], start=1):
idx = num - sum(num > gone for gone in gones)
offset += idx * factorial(n - i)
gones.append(num)
return offset
the_perm = (0, 2, 3, 1)
print(perm_to_permidx(the_perm))
# 3
Explanation: All permutations of a given range can be considered as a groups of permutations. So, for example, for the permutations of 0, 1, 2, 3 we first "fix" 0 and permute rest, then fix 1 and permute rest, and so on. Once we fix a number, the rest is again permutations; so we again fix a number at a time from the remaining numbers and permute the rest. This goes on till we are left with one number only. Every level of fixing has a corresponding (n-i)! permutations.
So this code finds the "offsets" for each level of permutation. The offset corresonds to where the given permutation starts when we fix numbers of perm in order. For the given example of (0, 2, 3, 1), we first look at the first number in the given perm which is 0, and figure the offset as 0. Then this goes to gones list (we will see its usage). Then, at the next level of permutation we see 2 as the fixing number. To calculate the offset for this, we need the "order" of this 2 among the remaining three numbers. This is where gones come into play; if an already-fixed and considered number (in this case 0) is less than the current fixer, we subtract 1 to find the new order. Then offset is calculated and accumulated. For the next number 3, the new order is 3 - (1 + 1) = 1 because both previous fixers 0 and 2 are at the "left" of 3.
This goes on till the last number of the given perm since there is no need to look at it; it will have been determined anyway.
I'm trying to write a Python script to determine if a given nonogram is unique. My current script just takes way too long to run so I was wondering if anyone had any ideas.
I understand that the general nonogram problem is NP-hard. However, I know two pieces of information about my given nonograms:
When representing the black/white boxes as 0s and 1s, respectively, I know how many of each I have.
I'm only considering 6x6 nonograms.
I initially used a brute force approach (so 2^36 cases). Knowing (1), however, I was able to narrow it down to n-choose-k (36-choose-number of zeroes) cases. However, when k is near 18, this is still ~2^33 cases. Takes days to run.
Any ideas how I might speed this up? Is it even possible?
Again, I don't care what the solution is -- I already have it. What I'm trying to determine is if that solution is unique.
EDIT:
This isn't exactly the full code but has the general idea:
def unique(nonogram):
found = 0
# create all combinations with the same number of 1s and 0s as incoming nonogram
for entry in itertools.combinations(range(len(nonogram)), nonogram.count(1)):
blank = [0]*len(nonogram) # initialize blank nonogram
for element in entry:
blank[element] = 1 # distribute 1s across nonogram
rows = find_rows(blank) # create row headers (like '2 1')
cols = find_cols(blank)
if rows == nonogram_rows and cols == nonogram_cols:
found += 1 # row and col headers same as original nonogram
if found > 1:
break # obviously not unique
if found == 1:
print('Unique nonogram')
I can't think of a clever way to prove uniqueness other than to solve the problem, but 6x6 is small enough that we can basically do a brute-force solution. To speed things up, instead of looping over every possible nonogram we can loop over all satisfying rows. Something like this (note: untested) should work:
from itertools import product, groupby
from collections import defaultdict
def vec_to_spec(v):
return tuple(len(list(g)) for k,g in groupby(v) if k)
def build_specs(n=6):
specs = defaultdict(list)
for v in product([0,1], repeat=n):
specs[vec_to_spec(v)].append(v)
return specs
def check(rowvecs, row_counts, col_counts):
colvecs = zip(*rowvecs)
row_pass = all(vec_to_spec(r) == tuple(rc) for r,rc in zip(rowvecs, row_counts))
col_pass = all(vec_to_spec(r) == tuple(rc) for r,rc in zip(colvecs, col_counts))
return row_pass and col_pass
def nonosolve(row_counts, col_counts):
specs = build_specs(len(row_counts))
possible_rows = [specs[tuple(r)] for r in row_counts]
sols = []
for poss in product(*possible_rows):
if check(poss, row_counts, col_counts):
sols.append(poss)
return sols
from which we learn that
>>> rows = [[2,2],[4], [1,1,1,], [2], [1,1,1,], [3,1]]
>>> cols = [[1,1,2],[1,1],[1,1],[4,],[2,1,],[3,2]]
>>> nonosolve(rows, cols)
[((1, 1, 0, 0, 1, 1), (0, 0, 1, 1, 1, 1), (1, 0, 0, 1, 0, 1),
(0, 0, 0, 1, 1, 0), (1, 0, 0, 1, 0, 1), (1, 1, 1, 0, 0, 1))]
>>> len(_)
1
is unique, but
>>> rows = [[1,1,1],[1,1,1], [1,1,1,], [1,1,1], [1,1,1], [1,1,1]]
>>> cols = rows
>>> nonosolve(rows, cols)
[((0, 1, 0, 1, 0, 1), (1, 0, 1, 0, 1, 0), (0, 1, 0, 1, 0, 1), (1, 0, 1, 0, 1, 0), (0, 1, 0, 1, 0, 1), (1, 0, 1, 0, 1, 0)),
((1, 0, 1, 0, 1, 0), (0, 1, 0, 1, 0, 1), (1, 0, 1, 0, 1, 0), (0, 1, 0, 1, 0, 1), (1, 0, 1, 0, 1, 0), (0, 1, 0, 1, 0, 1))]
>>> len(_)
2
isn't.
[Note that this isn't a very good solution for the problem in general as it throws away most of the information, but it was straightforward.]
I have a tuple of zeros and ones, for instance:
(1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1)
It turns out:
(1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1) == (1, 0, 1, 1) * 3
I want a function f such that if s is a non-empty tuple of zeros and ones, f(s) is the shortest subtuple r such that s == r * n for some positive integer n.
So for instance,
f( (1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1) ) == (1, 0, 1, 1)
What is a slick way to write the function f in Python?
Edit:
The naive method I am currently using
def f(s):
for i in range(1,len(s)):
if len(s)%i == 0 and s == s[:i] * (len(s)/i):
return s[:i]
I believe I have an O(n) time solution (actually 2n+r, n is length of tuple, r is sub tuplle) which does not use suffix trees, but uses a string matching algorithm (like KMP, which you should find off-the shelf).
We use the following little known theorem:
If x,y are strings over some alphabet,
then xy = yx if and only if x = z^k and y = z^l for some string z and integers k,l.
I now claim that, for the purposes of our problem, this means that all we need to do is determine if the given tuple/list (or string) is a cyclic shift of itself!
To determine if a string is a cyclic shift of itself, we concatenate it with itself (it does not even have to be a real concat, just a virtual one will do) and check for a substring match (with itself).
For a proof of that, suppose the string is a cyclic shift of itself.
The we have that the given string y = uv = vu.
Since uv = vu, we must have that u = z^k and v= z^l and hence y = z^{k+l} from the above theorem. The other direction is easy to prove.
Here is the python code. The method is called powercheck.
def powercheck(lst):
count = 0
position = 0
for pos in KnuthMorrisPratt(double(lst), lst):
count += 1
position = pos
if count == 2:
break
return lst[:position]
def double(lst):
for i in range(1,3):
for elem in lst:
yield elem
def main():
print powercheck([1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1])
if __name__ == "__main__":
main()
And here is the KMP code which I used (due to David Eppstein).
# Knuth-Morris-Pratt string matching
# David Eppstein, UC Irvine, 1 Mar 2002
def KnuthMorrisPratt(text, pattern):
'''Yields all starting positions of copies of the pattern in the text.
Calling conventions are similar to string.find, but its arguments can be
lists or iterators, not just strings, it returns all matches, not just
the first one, and it does not need the whole text in memory at once.
Whenever it yields, it will have read the text exactly up to and including
the match that caused the yield.'''
# allow indexing into pattern and protect against change during yield
pattern = list(pattern)
# build table of shift amounts
shifts = [1] * (len(pattern) + 1)
shift = 1
for pos in range(len(pattern)):
while shift <= pos and pattern[pos] != pattern[pos-shift]:
shift += shifts[pos-shift]
shifts[pos+1] = shift
# do the actual search
startPos = 0
matchLen = 0
for c in text:
while matchLen == len(pattern) or \
matchLen >= 0 and pattern[matchLen] != c:
startPos += shifts[matchLen]
matchLen -= shifts[matchLen]
matchLen += 1
if matchLen == len(pattern):
yield startPos
For your sample this outputs
[1,0,1,1]
as expected.
I compared this against shx2's code(not the numpy one), by generating a random 50 bit string, then replication to make the total length as 1 million. This was the output (the decimal number is the output of time.time())
1362988461.75
(50, [1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1])
1362988465.96
50 [1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1]
1362988487.14
The above method took ~4 seconds, while shx2's method took ~21 seconds!
Here was the timing code. (shx2's method was called powercheck2).
def rand_bitstring(n):
rand = random.SystemRandom()
lst = []
for j in range(1, n+1):
r = rand.randint(1,2)
if r == 2:
lst.append(0)
else:
lst.append(1)
return lst
def main():
lst = rand_bitstring(50)*200000
print time.time()
print powercheck(lst)
print time.time()
powercheck2(lst)
print time.time()
The following solution is O(N^2), but has the advantage of not creating any copies (or slices) of your data, as it is based on iterators.
Depending on the size of your input, the fact you avoid making copies of the data can result in a significant speed-up, but of course, it would not scale as well for huge inputs as algorithms with lower complexity (e.g. O(N*logN)).
[This is the second revision of my solution, the first one is given below. This one is simpler to understand, and is more along the lines of OP's tuple-multiplication, only using iterators.]
from itertools import izip, chain, tee
def iter_eq(seq1, seq2):
""" assumes the sequences have the same len """
return all( v1 == v2 for v1, v2 in izip(seq1, seq2) )
def dup_seq(seq, n):
""" returns an iterator which is seq chained to itself n times """
return chain(*tee(seq, n))
def is_reps(arr, slice_size):
if len(arr) % slice_size != 0:
return False
num_slices = len(arr) / slice_size
return iter_eq(arr, dup_seq(arr[:slice_size], num_slices))
s = (1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1)
for i in range(1,len(s)):
if is_reps(s, i):
print i, s[:i]
break
[My original solution]
from itertools import islice
def is_reps(arr, num_slices):
if len(arr) % num_slices != 0:
return False
slice_size = len(arr) / num_slices
for i in xrange(slice_size):
if len(set( islice(arr, i, None, num_slices) )) > 1:
return False
return True
s = (1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1)
for i in range(1,len(s)):
if is_reps(s, i):
print i, s[:i]
break
You can avoid the call to set() by using something like:
def is_iter_unique(seq):
""" a faster version of testing len(set(seq)) <= 1 """
seen = set()
for x in seq:
seen.add(x)
if len(seen) > 1:
return False
return True
and replacing this line:
if len(set( islice(arr, i, None, num_slices) )) > 1:
with:
if not is_iter_unique(islice(arr, i, None, num_slices)):
Simplifying Knoothe's solution. His algorithm is right, but his implementation is too complex. This implementation is also O(n).
Since your array is only composed of ones and zeros, what I do is use existing str.find implementation (Bayer Moore) to implement Knoothe's idea. It's suprisingly simpler and amazingly faster at runtime.
def f(s):
s2 = ''.join(map(str, s))
return s[:(s2+s2).index(s2, 1)]
Here's another solution (competing with my earlier iterators-based solution), leveraging numpy.
It does make a (single) copy of your data, but taking advantage of the fact your values are 0s and 1s, it is super-fast, thanks to numpy's magics.
import numpy as np
def is_reps(arr, slice_size):
if len(arr) % slice_size != 0:
return False
arr = arr.reshape((-1, slice_size))
return (arr.all(axis=0) | (~arr).all(axis=0)).all()
s = (1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1) * 1000
a = np.array(s, dtype=bool)
for i in range(1,len(s)):
if is_reps(a, i):
print i, s[:i]
break
Just a different approach to the problem
I first determine all the factors of the length and then split the list and check if all the parts are same
>>> def f(s):
def factors(n):
#http://stackoverflow.com/a/6800214/977038
return set(reduce(list.__add__,
([i, n//i] for i in range(2, int(n**0.5) + 1) if n % i == 0)))
_len = len(s)
for fact in reversed(list(factors(_len))):
compare_set = set(izip(*[iter(s)]*fact))
if len(compare_set) == 1:
return compare_set
>>> f(t)
set([(1, 0, 1, 1)])
You can archive it in sublinear time by XOR'ing the rotated binary form for the input array:
get the binary representation of the array, input_binary
loop from i = 1 to len(input_array)/2, and for each loop, rotate the input_binary to the right by i bits, save it as rotated_bin, then compare the XOR of rotated_bin and input_binary.
The first i that yields 0, is the index to which is the desired substring.
Complete code:
def get_substring(arr):
binary = ''.join(map(str, arr)) # join the elements to get the binary form
for i in xrange(1, len(arr) / 2):
# do a i bit rotation shift, get bit string sub_bin
rotated_bin = binary[-i:] + binary[:-i]
if int(rotated_bin) ^ int(binary) == 0:
return arr[0:i]
return None
if __name__ == "__main__":
test = [1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1]
print get_substring(test) # [1,0,1,1]
This one is just a dumb recursive comparison in Haskell. It takes about one second for Knoothe's million long string (f a). Cool problem! I'll think about it some more.
a = concat $ replicate 20000
[1,1,1,0,0,1,0,1,0,0,1,0,0,1,1,1,0,0,
0,0,0,0,1,1,1,1,0,0,0,1,1,0,1,1,1,1,
1,1,1,0,0,1,1,1,0,0,0,0,0,1]
f s =
f' s [] where
f' [] result = []
f' (x:xs) result =
let y = result ++ [x]
in if concat (replicate (div (length s) (length y)) y) == s
then y
else f' xs y
I have been playing with this a while, and just cannot see an obvious solution. I want to remove the recursion from the XinY_Go function.
def XinY_Go(x,y,index,slots):
if (y - index) == 1:
slots[index] = x
print slots
slots[index] = 0
return
for i in range(x+1):
slots[index] = x-i
XinY_Go(x-(x-i), y, index + 1, slots)
def XinY(x,y):
return XinY_Go(x,y,0,[0] * y)
The function is calculating the number of ways to put X marbles in Y slots. Here is some sample output:
>>> xy.XinY(1,2)
[1, 0]
[0, 1]
>>> xy.XinY(2,3)
[2, 0, 0]
[1, 1, 0]
[1, 0, 1]
[0, 2, 0]
[0, 1, 1]
[0, 0, 2]
Everything we think of as recursion can also be thought of as a stack-based problem, where the recursive function just uses the program's call stack rather than creating a separate stack. That means any recursive function can be re-written using a stack instead.
I don't know python well enough to give you an implementation, but that should point you in the right direction. But in a nutshell, push the initial arguments for the function onto the stack and add a loop that runs as long as the size of the stack is greater than zero. Pop once per loop iteration, push every time the function currently calls itself.
A naive implementation of #Joel Coehoorn's suggestion follows:
def XinY_Stack(x, y):
stack = [(x, 0, [0]*y)]
while stack:
x, index, slots = stack.pop()
if (y - index) == 1:
slots[index] = x
print slots
slots[index] = 0
else:
for i in range(x + 1):
slots[index] = x-i
stack.append((i, index + 1, slots[:]))
Example:
>>> XinY_Stack(2, 3)
[0, 0, 2]
[0, 1, 1]
[0, 2, 0]
[1, 0, 1]
[1, 1, 0]
[2, 0, 0]
Based on itertools.product
def XinY_Product(nmarbles, nslots):
return (slots
for slots in product(xrange(nmarbles + 1), repeat=nslots)
if sum(slots) == nmarbles)
Based on nested loops
def XinY_Iter(nmarbles, nslots):
assert 0 < nslots < 22 # 22 -> too many statically nested blocks
if nslots == 1: return iter([nmarbles])
# generate code for iter solution
TAB = " "
loopvars = []
stmt = ["def f(n):\n"]
for i in range(nslots - 1):
var = "m%d" % i
stmt += [TAB * (i + 1), "for %s in xrange(n - (%s)):\n"
% (var, '+'.join(loopvars) or 0)]
loopvars.append(var)
stmt += [TAB * (i + 2), "yield ", ','.join(loopvars),
', n - 1 - (', '+'.join(loopvars), ')\n']
print ''.join(stmt)
# exec the code within empty namespace
ns = {}
exec(''.join(stmt), ns, ns)
return ns['f'](nmarbles + 1)
Example:
>>> list(XinY_Product(2, 3))
[(0, 0, 2), (0, 1, 1), (0, 2, 0), (1, 0, 1), (1, 1, 0), (2, 0, 0)]
>>> list(XinY_Iter(2, 3))
def f(n):
for m0 in xrange(n - (0)):
for m1 in xrange(n - (m0)):
yield m0,m1, n - 1 - (m0+m1)
[(0, 0, 2), (0, 1, 1), (0, 2, 0), (1, 0, 1), (1, 1, 0), (2, 0, 0)]
Look at this code for creating all permutations, I guess I'd be relatively simple to implement something similar for your problem.
How to generate all permutations of a list in python?