Generate filtered binary cartesian products

Question

Problem statement

I'm looking for an efficient way to generate full binary cartesian products (tables with all combinations of True and False with a certain number of columns), filtered by certain exclusive conditions. For example, for three columns/bits n=3 we would get the full table

df_combs = pd.DataFrame(itertools.product(*([[True, False]] * n)))
       0      1      2
0   True   True   True
1   True   True  False
2   True  False   True
3   True  False  False
...

This is supposed to be filtered by dictionaries defining mutually exclusive combinations as follows:

mutually_excl = [{0: False, 1: False, 2: True},
                 {0: True, 2: True}]

Where the keys denote the columns in the table above. The example would be read as:

• If 0 is False and 1 is False, 2 can't be True
• If 0 is True, 2 can't be True

Based on these filters, the expected output is:

       0      1      2
1   True   True  False
3   True  False  False
4  False   True   True
5  False   True  False
7  False  False  False

In my use case, the filtered table is multiple orders of magnitude smaller than the full cartesian product (e.g. some 1000 instead of 2**24 (16777216)).

Below are my three current solutions, each with their own pros and cons, discussed at the very end.

---

import random
import pandas as pd
import itertools
import wrapt
import time
import operator
import functools

def get_mutually_excl(n, nfilt):  # generate random example filter
    ''' Example: `get_mutually_excl(9, 2)` creates a list of two filters with
    maximum index `n=9` and each filter length between 2 and `int(n/3)`:
    `[{1: True, 2: False}, {3: False, 2: True, 6: False}]` '''
    random.seed(2)
    return [{random.choice(range(n)): random.choice([True, False])
                           for _ in range(random.randint(2, int(n/3)))}
                           for _ in range(nfilt)]

@wrapt.decorator
def timediff(f, _, args, kwargs):
    t = time.perf_counter()
    res = f(*args)
    return res, time.perf_counter() - t

---

Solution 1: Filter first, then merge.

Expand each single filter entry (e.g. {0: True, 2: True}) into a sub-table with columns corresponding to the indices in this filter entry ([0, 2]). Remove single filtered row from this sub-table ([True, True]). Merge with full table to get the complete list of filtered combinations.

@timediff
def make_df_comb_filt_merge(n, nfilt):

    mutually_excl = get_mutually_excl(n, nfilt)

    # determine missing (unfiltered) columns
    cols_missing = set(range(n)) - set(itertools.chain.from_iterable(mutually_excl))

    # complete dataframe of unfiltered columns with column "temp" for full outer merge
    df_comb = pd.DataFrame(itertools.product(*([[True, False]] * len(cols_missing))),
                            columns=cols_missing).assign(temp=1)

    for filt in mutually_excl:  # loop through individual filters

        # get columns and bool values of this filters as two tuples with same order
        list_col, list_bool = zip(*filt.items())

        # construct dataframe
        df = pd.DataFrame(itertools.product(*([[True, False]] * len(list_col))),
                                columns=list_col)

        # filter remove a *single* row (by definition)
        df = df.loc[df.apply(tuple, axis=1) != list_bool]

        # determine which rows to merge on
        merge_cols = list(set(df.columns) & set(df_comb.columns))
        if not merge_cols:
            merge_cols = ['temp']
            df['temp'] = 1

        # merge with full dataframe
        df_comb = pd.merge(df_comb, df, on=merge_cols)

    df_comb.drop('temp', axis=1, inplace=True)
    df_comb = df_comb[range(n)]
    df_comb = df_comb.sort_values(df_comb.columns.tolist(), ascending=False)

    return df_comb.reset_index(drop=True)

---

Solution 2: Full expansion, then filter

Generate DataFrame for full cartesian product: The entire thing ends up in memory. Loop through filters and create a mask for each. Apply each mask to table.

@timediff
def make_df_comb_exp_filt(n, nfilt):

    mutually_excl = get_mutually_excl(n, nfilt)

    # expand all bool combinations into dataframe
    df_comb = pd.DataFrame(itertools.product(*([[True, False]] * n)),
                           dtype=bool)

    for filt in mutually_excl:

        # generate total filter mask for given excluded combination
        mask = pd.Series(True, index=df_comb.index)
        for col, bool_act in filt.items():
            mask = mask & (df_comb[col] == bool_act)

        # filter dataframe
        df_comb = df_comb.loc[~mask]

    return df_comb.reset_index(drop=True)

---

Solution 3: Filter iterator

Keep the full cartesian product an iterator. Loop while checking for each row whether it is excluded by any of the filters.

@timediff
def make_df_iter_filt(n, nfilt):

    mutually_excl = get_mutually_excl(n, nfilt)

    # switch to [[(1, 13), (True, False)], [(4, 9), (False, True)], ...]
    mutually_excl_index = [list(zip(*comb.items()))
                                for comb in mutually_excl]

    # create iterator
    combs_iter = itertools.product(*([[True, False]] * n))

    @functools.lru_cache(maxsize=1024, typed=True)  # small benefit
    def get_getter(list_):
        # Used to access combs_iter row values as indexed by the filter
        return operator.itemgetter(*list_)

    def check_comb(comb_inp, comb_check):
        return get_getter(comb_check[0])(comb_inp) == comb_check[1]

    # loop through the iterator
    # drop row if any of the filter matches
    df_comb = pd.DataFrame([comb_inp for comb_inp in combs_iter
                       if not any(check_comb(comb_inp, comb_check)
                                  for comb_check in mutually_excl_index)])

    return df_comb.reset_index(drop=True)

---

Run examples

dict_time = dict.fromkeys(itertools.product(range(16, 23, 2), range(3, 20)))

for n, nfilt in dict_time:
    dict_time[(n, nfilt)] = {'exp_filt': make_df_comb_exp_filt(n, nfilt)[1],
                             'filt_merge': make_df_comb_filt_merge(n, nfilt)[1],
                             'iter_filt': make_df_iter_filt(n, nfilt)[1]}

---

Analysis

import seaborn as sns
import matplotlib.pyplot as plt

df_time = pd.DataFrame.from_dict(dict_time, orient='index',
                                 ).rename_axis(["n", "nfilt"]
                                 ).stack().reset_index().rename(columns={'level_2': 'solution', 0: 'time'})

g = sns.FacetGrid(df_time.query('n in %s' % str([16,18,20,22])),
                  col="n",  hue="solution", sharey=False)
g = (g.map(plt.plot, "nfilt", "time", marker="o").add_legend())

Solution 3: The iterator based approach (comb_iterator) has dismal running times, but no significant use of memory. I feel there is room for improvement, though the inevitable loop likely imposes hard bounds in terms of running time.

Solution 2: Expanding the full cartesian product into a DataFrame (exp_filt) causes significant spikes in memory, which I would like to avoid. Running times are ok though.

Solution 1: Merging DataFrames created from the individual filters (filt_merge) feels like a good solution for my practical application (note the reduction in running time for greater numbers of filters, which is a result of the smaller cols_missing table). Still, this approach is not entirely satisfying: If a single filter includes all columns, the whole cartesian product (2**n) would end up in memory, making this solution worse than comb_iterator.

Question: Any other ideas? A crazy smart numpy two-liner? Could the iterator based approach be improved somehow?

Answer 1

Try timing the following:

def in_filter(arr, arr_filt, n):
    return ((arr[:, None] >> (n-1-arr_filt[:, 0])) & 1 == arr_filt[:, 1]).all(axis=1)

def bits_to_boolean(arr, n):
    return ((arr[:, None] >> np.arange(n, dtype=arr.dtype)[::-1]) & 1).astype(bool)

@timediff
def recursive_filter(n, nfilt, dtype='uint32'):
    filts = get_mutually_excl(n, nfilt)
    out = np.arange(2**n, dtype=dtype)
    for filt in filts:
        arr_filt = np.array(list(filt.items()))
        out = out[~in_filter(out, arr_filt, n)]
    return bits_to_boolean(out, n)[::-1]

It treats the Cartesian binary products as the bits encoded in the range of integers 0..<2**n and uses vectorised functions to recursively remove numbers that have bit sequences that match the given filters.

The memory efficiency is better than allocating all [True, False] Cartesian products as each Boolean would be stored with at least 8 bits each (using 7 bits more than required), but it will use more memory than an iterator-based approach. If you require a solution for large n, you could break down this task by allocating and operating on one sub-range at a time. I did have this in my first implementation, but it didn't offer much benefit for n<=22 and it required the calculation of the size of the output array, which gets complicated when there are overlapping filters.

Answer 2

Based on @ayhan's comment I implemented an or-tools SAT-based solution. While the idea is great, this really struggles for larger number of binary variables. I suspect this is similar to large IP problems, which ain't no walk in the park either. However, the strong dependency on filter numbers could make this a valid option for certain parameter configurations. But as a general solution I wouldn't use it.

from ortools.sat.python import cp_model

class VarArraySolutionCollector(cp_model.CpSolverSolutionCallback):

    def __init__(self, variables):
        cp_model.CpSolverSolutionCallback.__init__(self)
        self.__variables = variables
        self.solution_list = []

    def on_solution_callback(self):
        self.solution_list.append([self.Value(v) for v in self.__variables])


@timediff
def make_df_comb_sat(n, nfilt):

    mutually_excl = get_mutually_excl(n, nfilt)

    model = cp_model.CpModel()

    make_var_name = 'x{:02d}'.format
    vrs = dict.fromkeys(map(make_var_name, range(n)))
    for var_name in vrs:
        vrs[var_name] = model.NewBoolVar(var_name)

    for filt in mutually_excl:
        list_expr = [vrs[make_var_name(iv)]
                     if not bool_ else getattr(vrs[make_var_name(iv)], 'Not')()
                     for iv, bool_ in filt.items()]
        model.AddBoolOr(list_expr)

    solver = cp_model.CpSolver()
    solution_printer = VarArraySolutionCollector(vrs.values())
    solver.SearchForAllSolutions(model, solution_printer)

    df_comb = pd.DataFrame(solution_printer.solution_list).astype(bool)
    df_comb = df_comb.sort_values(df_comb.columns.tolist(), ascending=False)
    df_comb = df_comb.reset_index(drop=True)

    return df_comb