I'm trying to solve the general problem of getting the unique combinations from a list in Python
Mathematically from https://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html I can see that the formula for the number of combinations is n!/r!(n-r)! where n is the length of the sequence and r is the number to choose.
As shown by the following python where n is 4 and r is 2:
lst = 'ABCD'
result = list(itertools.combinations(lst, len(lst)/2))
print len(result)
6
The following is a helper function to show the issue I have:
def C(lst):
l = list(itertools.combinations(sorted(lst), len(lst)/2))
s = set(l)
print 'actual', len(l), l
print 'unique', len(s), list(s)
If I run this from iPython I can call it thus:
In [41]: C('ABCD')
actual 6 [('A', 'B'), ('A', 'C'), ('A', 'D'), ('B', 'C'), ('B', 'D'), ('C', 'D')]
unique 6 [('B', 'C'), ('C', 'D'), ('A', 'D'), ('A', 'B'), ('A', 'C'), ('B', 'D')]
In [42]: C('ABAB')
actual 6 [('A', 'A'), ('A', 'B'), ('A', 'B'), ('A', 'B'), ('A', 'B'), ('B', 'B')]
unique 3 [('A', 'B'), ('A', 'A'), ('B', 'B')]
In [43]: C('ABBB')
actual 6 [('A', 'B'), ('A', 'B'), ('A', 'B'), ('B', 'B'), ('B', 'B'), ('B', 'B')]
unique 2 [('A', 'B'), ('B', 'B')]
In [44]: C('AAAA')
actual 6 [('A', 'A'), ('A', 'A'), ('A', 'A'), ('A', 'A'), ('A', 'A'), ('A', 'A')]
unique 1 [('A', 'A')]
What I want to get is the unique count as shown above but doing a combinations and then set doesn't scale.
As when the length of lst which is n gets longer it slows down as the combinations get greater and greater.
Is there a way of using math or Python tricks to to solve the issue of counting the unique combinations ?
Here's some Python code based on the generating function approach outlined in this Math Forum article. For each letter appearing in the input we create a polynomial 1 + x + x^2 + ... + x^k, where k is the number of times that the letter appears. We then multiply those polynomials together: the nth coefficient of the resulting polynomial then tells you how many combinations of length n there are.
We'll represent a polynomial simply as a list of its (integer) coefficients, with the first coefficient representing the constant term, the next coefficient representing the coefficient of x, and so on. We'll need to be able to multiply such polynomials, so here's a function for doing so:
def polymul(p, q):
"""
Multiply two polynomials, represented as lists of coefficients.
"""
r = [0]*(len(p) + len(q) - 1)
for i, c in enumerate(p):
for j, d in enumerate(q):
r[i+j] += c*d
return r
With the above in hand, the following function computes the number of combinations:
from collections import Counter
from functools import reduce
def ncombinations(it, k):
"""
Number of combinations of length *k* of the elements of *it*.
"""
counts = Counter(it).values()
prod = reduce(polymul, [[1]*(count+1) for count in counts], [1])
return prod[k] if k < len(prod) else 0
Testing this on your examples:
>>> ncombinations("abcd", 2)
6
>>> ncombinations("abab", 2)
3
>>> ncombinations("abbb", 2)
2
>>> ncombinations("aaaa", 2)
1
And on some longer examples, demonstrating that this approach is feasible even for long-ish inputs:
>>> ncombinations("abbccc", 3) # the math forum example
6
>>> ncombinations("supercalifragilisticexpialidocious", 10)
334640
>>> from itertools import combinations # double check ...
>>> len(set(combinations(sorted("supercalifragilisticexpialidocious"), 10)))
334640
>>> ncombinations("supercalifragilisticexpialidocious", 20)
1223225
>>> ncombinations("supercalifragilisticexpialidocious", 34)
1
>>> ncombinations("supercalifragilisticexpialidocious", 35)
0
>>> from string import printable
>>> ncombinations(printable, 50) # len(printable)==100
100891344545564193334812497256
>>> from math import factorial
>>> factorial(100)//factorial(50)**2 # double check the result
100891344545564193334812497256
>>> ncombinations("abc"*100, 100)
5151
>>> factorial(102)//factorial(2)//factorial(100) # double check (bars and stars)
5151
Start with a regular recursive definition of combinations() but add a test to only recurse when the lead value at that level hasn't been used before:
def uniq_comb(pool, r):
""" Return an iterator over a all distinct r-length
combinations taken from a pool of values that
may contain duplicates.
Unlike itertools.combinations(), element uniqueness
is determined by value rather than by position.
"""
if r:
seen = set()
for i, item in enumerate(pool):
if item not in seen:
seen.add(item)
for tail in uniq_comb(pool[i+1:], r-1):
yield (item,) + tail
else:
yield ()
if __name__ == '__main__':
from itertools import combinations
pool = 'ABRACADABRA'
for r in range(len(pool) + 1):
assert set(uniq_comb(pool, r)) == set(combinations(pool, r))
assert dict.fromkeys(uniq_comb(pool, r)) == dict.fromkeys(combinations(pool, r))
It seems that this is called a multiset combination. I've faced the same problem and finally came up rewriting a function from sympy (here).
Instead of passing your iterable to something like itertools.combinations(p, r), you pass collections.Counter(p).most_common() to the following function to directly retrieve distinct combinations. It's a lot faster than filtering all combinations and also memory safe!
def counter_combinations(g, n):
if sum(v for k, v in g) < n or not n:
yield []
else:
for i, (k, v) in enumerate(g):
if v >= n:
yield [k]*n
v = n - 1
for v in range(min(n, v), 0, -1):
for j in counter_combinations(g[i + 1:], n - v):
rv = [k]*v + j
if len(rv) == n:
yield rv
Here is an example:
from collections import Counter
p = Counter('abracadabra').most_common()
print(p)
c = [_ for _ in counter_combinations(p, 4)]
print(c)
print(len(c))
Output:
[('a', 5), ('b', 2), ('r', 2), ('c', 1), ('d', 1)]
[['a', 'a', 'a', 'a'], ['a', 'a', 'a', 'b'], ['a', 'a', 'a', 'r'], ['a', 'a', 'a', 'c'], ['a', 'a', 'a', 'd'], ['a', 'a', 'b', 'b'], ['a', 'a', 'b', 'r'], ['a', 'a', 'b', 'c'], ['a', 'a', 'b', 'd'], ['a', 'a', 'r', 'r'], ['a', 'a', 'r', 'c'], ['a', 'a', 'r', 'd'], ['a', 'a', 'c', 'd'], ['a', 'b', 'b', 'r'], ['a', 'b', 'b', 'c'], ['a', 'b', 'b', 'd'], ['a', 'b', 'r', 'r'], ['a', 'b', 'r', 'c'], ['a', 'b', 'r', 'd'], ['a', 'b', 'c', 'd'], ['a', 'r', 'r', 'c'], ['a', 'r', 'r', 'd'], ['a', 'r', 'c', 'd'], ['b', 'b', 'r', 'r'], ['b', 'b', 'r', 'c'], ['b', 'b', 'r', 'd'], ['b', 'b', 'c', 'd'], ['b', 'r', 'r', 'c'], ['b', 'r', 'r', 'd'], ['b', 'r', 'c', 'd'], ['r', 'r', 'c', 'd']]
31
Related
I have those python lists :
x = [('D', 'F'), ('A', 'D'), ('B', 'G'), ('B', 'C'), ('A', 'B')]
priority_list = ['A', 'B', 'C', 'D', 'F', 'G'] # Ordered from highest to lowest priority
How can I, for each tuple in my list, keep the value with the highest priority according to priority_list? The result would be :
['D', 'A', 'B', 'B', 'A']
Another examples:
x = [('B', 'D'), ('E', 'A'), ('B', 'A'), ('D', 'F'), ('E', 'C')]
priority_list = ['A', 'B', 'C', 'D', 'E', 'F']
# Result:
['B', 'A', 'A', 'D', 'C']
x = [('B', 'C'), ('F', 'E'), ('B', 'A'), ('D', 'F'), ('E', 'C')]
priority_list = ['F', 'E', 'D', 'C', 'B', 'A'] # Notice the change in priorities
# Result:
['C', 'F', 'B', 'F', 'E']
Thanks in advance, I might be over complicating this.
You can try
[sorted(i, key=priority_list.index)[0] for i in x]
though it will throw an exception if you find a value not in the priority list.
You can try:
def get_priority_val(data, priority_list):
for single_val in priority_list:
if single_val in data:
return single_val
x = [('D', 'F'), ('A', 'D'), ('B', 'G'), ('B', 'C'), ('A', 'B')]
priority_list = ['A', 'B', 'C', 'D', 'F', 'G']
final_data = []
for data in x:
final_data.append(get_priority_val(data, priority_list))
print(final_data)
Output:
['D', 'A', 'B', 'B', 'A']
you can try using list comprehension:
ans = [d[0] if priority_list.index(d[0]) < priority_list.index(d[1] ) else d[1] for d in x ]
output:
['D', 'A', 'B', 'B', 'A']
You can do it in one line using a list comprehension :
[y[0] if priority_list.index(y[0]) < priority_list.index(y[1]) else y[1] for y in x]
Output :
['D', 'A', 'B', 'B', 'A']
You can create a dict containing priority values and just use min with a custom key
>>> priority_dict = {k:i for i,k in enumerate(priority_list)}
>>> [min(t, key=priority_dict.get) for t in x]
['D', 'A', 'B', 'B', 'A']
This question already has answers here:
How to get the cartesian product of multiple lists
(17 answers)
Closed 3 years ago.
If I have the list my_list = [['a', 'b'], ['c', 'd', 'e'], how can I create a list of all possible tuples where each tuple contains a single element from each sublist? For example:
('a', 'e') is valid
('a', 'b') is invalid
('b', 'c') is valid
('c', 'd') is invalid
Importantly, my_list can contain any number of elements (sublists) and each sublist can be of any length. I tried to get a recursive generator off the ground, but it's not quite there.
I’d like to try and use recursion rather than itertools.
The logic was to iterate through 2 sublists at a time, and store those results as input to the next sublist's expansion.
def foil(lis=l):
if len(lis) == 2:
for x in l[0]:
for y in l[1]:
yield x + y
else:
for p in foil(l[:-1]):
yield p
for i in foil():
print(i)
However, this obviously only works for the len(my_list) == 2. It also needs to work with, say, my_list = [['a'], ['b'], ['c', 'd']] which would return:
('a', 'b', 'c')
('a', 'b', 'd')
Cheers!
Use itertools.product:
import itertools
my_list = [['a', 'b'], ['c', 'd', 'e']]
print(list(itertools.product(*my_list)))
# [('a', 'c'), ('a', 'd'), ('a', 'e'), ('b', 'c'), ('b', 'd'), ('b', 'e')]
my_list = [['a'], ['b'], ['c', 'd']]
print(list(itertools.product(*my_list)))
#[('a', 'b', 'c'), ('a', 'b', 'd')]
I managed to generate a list of all possible combinations of characters 'a', 'b' and 'c' (code below). Now I want to add a fourth character, which can be either 'd' or 'f' but NOT both in the same combination. How could I achieve this ?
items = ['a', 'b', 'c']
from itertools import permutations
for p in permutations(items):
print(p)
('a', 'b', 'c')
('a', 'c', 'b')
('b', 'a', 'c')
('b', 'c', 'a')
('c', 'a', 'b')
('c', 'b', 'a')
Created a new list items2 for d and f. Assuming that OP needs all combinations of [a,b,c,d] and [a,b,c,f]
items1 = ['a', 'b', 'c']
items2 = ['d','f']
from itertools import permutations
for x in items2:
for p in permutations(items1+[x]):
print(p)
A variation on #Van Peer's solution. You can modify the extended list in-place:
from itertools import permutations
items = list('abc_')
for items[3] in 'dg':
for p in permutations(items):
print(p)
itertools.product is suitable for representing these distinct groups in a way which generalizes well. Just let the exclusive elements belong to the same iterable passed to the Cartesian product.
For instance, to get a list with the items you're looking for,
from itertools import chain, permutations, product
list(chain.from_iterable(map(permutations, product(*items, 'df'))))
# [('a', 'b', 'c', 'd'),
# ('a', 'b', 'd', 'c'),
# ('a', 'c', 'b', 'd'),
# ('a', 'c', 'd', 'b'),
# ('a', 'd', 'b', 'c'),
# ('a', 'd', 'c', 'b'),
# ('b', 'a', 'c', 'd'),
# ('b', 'a', 'd', 'c'),
# ('b', 'c', 'a', 'd'),
# ('b', 'c', 'd', 'a'),
# ('b', 'd', 'a', 'c'),
# ('b', 'd', 'c', 'a'),
# ('c', 'a', 'b', 'd'),
# ('c', 'a', 'd', 'b'),
# ...
like this for example
items = ['a', 'b', 'c','d']
from itertools import permutations
for p in permutations(items):
print(p)
items = ['a', 'b', 'c','f']
from itertools import permutations
for p in permutations(items):
print(p)
I have arrays such as these, and each pattern designates a combination shape with each number representing the size of the combination.
pattern 0: [1, 1, 1, 1]
pattern 1: [2, 1, 1]
pattern 2: [3, 1]
pattern 3: [4]
...
I also have a char-valued list like below. len(chars) equals the sum of the upper array's value.
chars = ['A', 'B', 'C', 'D']
I want to find all combinations of chars following a given pattern. For example, for pattern 1, 4C2 * 2C1 * 1C1 is the number of combinations.
[['A', 'B'], ['C'], ['D']]
[['A', 'B'], ['D'], ['C']]
[['A', 'C'], ['B'], ['D']]
[['A', 'C'], ['D'], ['B']]
[['A', 'D'], ['B'], ['C']]
[['A', 'D'], ['C'], ['B']]
...
But I don't know how to create such combination arrays. Of course I know there are a lot of useful functions for combinations in python. But I don't know how to use them to create a combination array of combinations.
EDITED
I'm so sorry my explanation is confusing. I show a simple example.
pattern 0: [1, 1]
pattern 1: [2]
chars = ['A', 'B']
Then, the result should be like below. So first dimension should be permutation, but second dimension should be combination.
pat0: [['A'], ['B']]
pat0: [['B'], ['A']]
pat1: [['A', 'B']] # NOTE: [['B', 'A']] is same in my problem
You can use recursive function that takes the first number in pattern and generates all the combinations of that length from remaining items. Then recurse with remaining pattern & items and generated prefix. Once you have consumed all the numbers in pattern just yield the prefix all the way to caller:
from itertools import combinations
pattern = [2, 1, 1]
chars = ['A', 'B', 'C', 'D']
def patterns(shape, items, prefix=None):
if not shape:
yield prefix
return
prefix = prefix or []
for comb in combinations(items, shape[0]):
child_items = items[:]
for char in comb:
child_items.remove(char)
yield from patterns(shape[1:], child_items, prefix + [comb])
for pat in patterns(pattern, chars):
print(pat)
Output:
[('A', 'B'), ('C',), ('D',)]
[('A', 'B'), ('D',), ('C',)]
[('A', 'C'), ('B',), ('D',)]
[('A', 'C'), ('D',), ('B',)]
[('A', 'D'), ('B',), ('C',)]
[('A', 'D'), ('C',), ('B',)]
[('B', 'C'), ('A',), ('D',)]
[('B', 'C'), ('D',), ('A',)]
[('B', 'D'), ('A',), ('C',)]
[('B', 'D'), ('C',), ('A',)]
[('C', 'D'), ('A',), ('B',)]
[('C', 'D'), ('B',), ('A',)]
Note that above works only with Python 3 since it's using yield from.
Given the iterable [A, B, C] and the function f(x) I want to get the following:
[ A, B, C]
[ A, B, f(C)]
[ A, f(B), C]
[ A, f(B), f(C)]
[f(A), B, C]
[f(A), B, f(C)]
[f(A), f(B), C]
[f(A), f(B), f(C)]
Unfortunately I didn't find anything suitable in the itertools module.
>>> from itertools import product
>>> L = ["A", "B", "C"]
>>> def f(c): return c.lower()
...
>>> fL = [f(x) for x in L]
>>> for i in product(*zip(L, fL)):
... print i
...
('A', 'B', 'C')
('A', 'B', 'c')
('A', 'b', 'C')
('A', 'b', 'c')
('a', 'B', 'C')
('a', 'B', 'c')
('a', 'b', 'C')
('a', 'b', 'c')
Explanation:
Call f for each item in L to generate fL
>>> fL
['a', 'b', 'c']
Use zip to zip the two lists into pairs
>>> zip(L, fL)
[('A', 'a'), ('B', 'b'), ('C', 'c')]
Take the cartesian product of those tuples using itertools.product
product(*zip(L, fL))
is equivalent to
product(*[('A', 'a'), ('B', 'b'), ('C', 'c')])
and that is equivalent to
product(('A', 'a'), ('B', 'b'), ('C', 'c'))
looping over that product, gives exactly the result we need.
You can use itertools.combinations, like this
def f(char):
return char.lower()
iterable = ["A", "B", "C"]
indices = range(len(iterable))
from itertools import combinations
for i in range(len(iterable) + 1):
for items in combinations(indices, i):
print [f(iterable[j]) if j in items else iterable[j] for j in range(len(iterable))]
Output
['A', 'B', 'C']
['a', 'B', 'C']
['A', 'b', 'C']
['A', 'B', 'c']
['a', 'b', 'C']
['a', 'B', 'c']
['A', 'b', 'c']
['a', 'b', 'c']
import itertools
def func_combinations(f, l):
return itertools.product(*zip(l, map(f, l)))
Demo:
>>> for combo in func_combinations(str, range(3)):
... print combo
...
(0, 1, 2)
(0, 1, '2')
(0, '1', 2)
(0, '1', '2')
('0', 1, 2)
('0', 1, '2')
('0', '1', 2)
('0', '1', '2')
This function first computes f once for every element of the input. Then, it uses zip to turn the input and the list of f values into a list of input-output pairs. Finally, it uses itertools.product to produce each possible way to select either input or output.