MRocklin - 3 months ago 21

Python Question

Give an algorithm (or straight Python code) that yields all partitions of a collection of N items into K bins such that each bin has at least one item. I need this in both the case where order matters and where order does not matter.

Example where order matters

`>>> list(partition_n_in_k_bins_ordered((1,2,3,4), 2))`

[([1], [2,3,4]), ([1,2], [3,4]), ([1,2,3], [4])]

>>> list(partition_n_in_k_bins_ordered((1,2,3,4), 3))

[([1], [2], [3,4]), ([1], [2,3], [4]), ([1,2], [3], [4])]

>>> list(partition_n_in_k_bins_ordered((1,2,3,4), 4))

[([1], [2], [3], [4])]

Example where order does not matter

`>>> list(partition_n_in_k_bins_unordered({1,2,3,4}, 2))`

[{{1}, {2,3,4}}, {{2}, {1,3,4}}, {{3}, {1,2,4}}, {{4}, {1,2,3}},

{{1,2}, {3,4}}, {{1,3}, {2,4}}, {{1,4}, {2,3}}]

These functions should produce lazy iterators/generators, not lists. Ideally they would use primitives found in

`itertools`

While I've asked for this in Python I'm also willing to translate a clear algorithm.

Answer

Enrico's algorithm, Knuth's, and only my glue are needed to paste together something that returns the list of lists or set of sets (returned as lists of lists in case elements are not hashable).

```
def kbin(l, k, ordered=True):
"""
Return sequence ``l`` partitioned into ``k`` bins.
Examples
========
The default is to give the items in the same order, but grouped
into k partitions:
>>> for p in kbin(range(5), 2):
... print p
...
[[0], [1, 2, 3, 4]]
[[0, 1], [2, 3, 4]]
[[0, 1, 2], [3, 4]]
[[0, 1, 2, 3], [4]]
Setting ``ordered`` to None means that the order of the elements in
the bins is irrelevant and the order of the bins is irrelevant. Though
they are returned in a canonical order as lists of lists, all lists
can be thought of as sets.
>>> for p in kbin(range(3), 2, ordered=None):
... print p
...
[[0, 1], [2]]
[[0], [1, 2]]
[[0, 2], [1]]
"""
from sympy.utilities.iterables import (
permutations, multiset_partitions, partitions)
def partition(lista, bins):
# EnricoGiampieri's partition generator from
# http://stackoverflow.com/questions/13131491/
# partition-n-items-into-k-bins-in-python-lazily
if len(lista) == 1 or bins == 1:
yield [lista]
elif len(lista) > 1 and bins > 1:
for i in range(1, len(lista)):
for part in partition(lista[i:], bins - 1):
if len([lista[:i]] + part) == bins:
yield [lista[:i]] + part
if ordered:
for p in partition(l, k):
yield p
else:
for p in multiset_partitions(l, k):
yield p
```