Qfyilyi

The contents of this post were originally meant to be a part of
Pandas Merging 101,
but due to the nature and size of the content required to fully do
justice to this topic, it has been moved to its own QnA.

Given two simple DataFrames;

left = pd.DataFrame({'col1' : ['A', 'B', 'C'], 'col2' : [1, 2, 3]})

right = pd.DataFrame({'col1' : ['X', 'Y', 'Z'], 'col2' : [20, 30, 50]})



left



  col1  col2

0    A     1

1    B     2

2    C     3



right



  col1  col2

0    X    20

1    Y    30

2    Z    50

The cross product of these frames can be computed, and will look something like:

A       1      X      20

A       1      Y      30

A       1      Z      50

B       2      X      20

B       2      Y      30

B       2      Z      50

C       3      X      20

C       3      Y      30

C       3      Z      50

What is the most performant method of computing this result?

Let's start by establishing a benchmark. The easiest method for solving this is using a temporary "key" column:

def cartesian_product_basic(left, right):

    return (

       left.assign(key=1).merge(right.assign(key=1), on='key').drop('key', 1))



cartesian_product_basic(left, right)



  col1_x  col2_x col1_y  col2_y

0      A       1      X      20

1      A       1      Y      30

2      A       1      Z      50

3      B       2      X      20

4      B       2      Y      30

5      B       2      Z      50

6      C       3      X      20

7      C       3      Y      30

8      C       3      Z      50

How this works is that both DataFrames are assigned a temporary "key" column with the same value (say, 1). merge then performs a many-to-many JOIN on "key".

While the many-to-many JOIN trick works for reasonably sized DataFrames, you will see relatively lower performance on larger data.

A faster implementation will require NumPy. Here are some famous NumPy implementations of 1D cartesian product. We can build some of the performant solutions to get our desired output. My favourite, however, is @senderle's first implementation.

def cartesian_product(*arrays):

    la = len(arrays)

    dtype = np.result_type(*arrays)

    arr = np.empty([len(a) for a in arrays] + [la], dtype=dtype)

    for i, a in enumerate(np.ix_(*arrays)):

        arr[...,i] = a

    return arr.reshape(-1, la)  

Generalizing: CROSS JOIN on Unique or Non-Unique Indexed DataFrames

This trick will work on any kind of DataFrame. We compute the cartesian product of the DataFrames' numeric indices using the aforementioned cartesian_product, use this to reindex the DataFrames, and

def cartesian_product_generalized(left, right):

    la, lb = len(left), len(right)

    idx = cartesian_product(np.ogrid[:la], np.ogrid[:lb])

    return pd.DataFrame(

        np.column_stack([left.values[idx[:,0]], right.values[idx[:,1]]]))



cartesian_product_generalized(left, right)



   0  1  2   3

0  A  1  X  20

1  A  1  Y  30

2  A  1  Z  50

3  B  2  X  20

4  B  2  Y  30

5  B  2  Z  50

6  C  3  X  20

7  C  3  Y  30

8  C  3  Z  50



np.array_equal(cartesian_product_generalized(left, right),

               cartesian_product_basic(left, right))

True

And, along similar lines,

left2 = left.copy()

left2.index = ['s1', 's2', 's1']



right2 = right.copy()

right2.index = ['x', 'y', 'y']





left2

   col1  col2

s1    A     1

s2    B     2

s1    C     3



right2

  col1  col2

x    X    20

y    Y    30

y    Z    50



np.array_equal(cartesian_product_generalized(left, right),

               cartesian_product_basic(left2, right2))

True

This solution can generalise to multiple DataFrames. For example,

def cartesian_product_multi(*dfs):

    idx = cartesian_product(*[np.ogrid[:len(df)] for df in dfs])

    return pd.DataFrame(

        np.column_stack([df.values[idx[:,i]] for i,df in enumerate(dfs)]))



cartesian_product_multi(*[left, right, left]).head()



   0  1  2   3  4  5

0  A  1  X  20  A  1

1  A  1  X  20  B  2

2  A  1  X  20  C  3

3  A  1  X  20  D  4

4  A  1  Y  30  A  1

Further Simplification

A simpler solution not involving @senderle's cartesian_product is possible when dealing with just two DataFrames. Using np.broadcast_arrays, we can achieve almost the same level of performance.

def cartesian_product_simplified(left, right):

    la, lb = len(left), len(right)

    ia2, ib2 = np.broadcast_arrays(*np.ogrid[:la,:lb])



    return pd.DataFrame(

        np.column_stack([left.values[ia2.ravel()], right.values[ib2.ravel()]]))



np.array_equal(cartesian_product_simplified(left, right),

               cartesian_product_basic(left2, right2))

True

Performance Comparison

Benchmarking these solutions on some contrived DataFrames with unique indices, we have

Do note that timings may vary based on your setup, data, and choice of cartesian_product helper function as applicable.

Performance Benchmarking Code

This is the timing script. All functions called here are defined above.

from timeit import timeit

import pandas as pd

import matplotlib.pyplot as plt



res = pd.DataFrame(

       index=['cartesian_product_basic', 'cartesian_product_generalized', 

              'cartesian_product_multi', 'cartesian_product_simplified'],

       columns=[1, 10, 50, 100, 200, 300, 400, 500, 600, 800, 1000, 2000],

       dtype=float

)



for f in res.index: 

    for c in res.columns:

        # print(f,c)

        left2 = pd.concat([left] * c, ignore_index=True)

        right2 = pd.concat([right] * c, ignore_index=True)

        stmt = '{}(left2, right2)'.format(f)

        setp = 'from __main__ import left2, right2, {}'.format(f)

        res.at[f, c] = timeit(stmt, setp, number=5)



ax = res.div(res.min()).T.plot(loglog=True) 

ax.set_xlabel("N"); 

ax.set_ylabel("time (relative)");



plt.show()

Using itertools product and recreate the value in dataframe

import itertools

l=list(itertools.product(left.values.tolist(),right.values.tolist()))

pd.DataFrame(list(map(lambda x : sum(x,),l)))

   0  1  2   3

0  A  1  X  20

1  A  1  Y  30

2  A  1  Z  50

3  B  2  X  20

4  B  2  Y  30

5  B  2  Z  50

6  C  3  X  20

7  C  3  Y  30

8  C  3  Z  50