Project Euler #18 in Python: Maximum sum down a triangle of numbers

Question

I've solved Project Euler #18 in Python but think my code and algorithm can be improved. Here is the problem:

By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.

3
7 4
2 4 6
8 5 9 3

That is, 3 + 7 + 4 + 9 = 23.

Find the maximum total from top to bottom of the triangle below:

75
95 64
17 47 82
18 35 87 10
20 04 82 47 65
19 01 23 75 03 34
88 02 77 73 07 63 67
99 65 04 28 06 16 70 92
41 41 26 56 83 40 80 70 33
41 48 72 33 47 32 37 16 94 29
53 71 44 65 25 43 91 52 97 51 14
70 11 33 28 77 73 17 78 39 68 17 57
91 71 52 38 17 14 91 43 58 50 27 29 48
63 66 04 68 89 53 67 30 73 16 69 87 40 31
04 62 98 27 23 09 70 98 73 93 38 53 60 04 23

NOTE: As there are only 16384 routes, it is possible to solve this problem by trying every route. However, Problem 67, is the same challenge with a triangle containing one-hundred rows; it cannot be solved by brute force, and requires a clever method!

And I want with an bottom-up calculation to find the best path down. Here is my code:

import time

num_pyr = '''75
95 64
17 47 82
18 35 87 10
20 04 82 47 65
19 01 23 75 03 34
88 02 77 73 07 63 67
99 65 04 28 06 16 70 92
41 41 26 56 83 40 80 70 33
41 48 72 33 47 32 37 16 94 29
53 71 44 65 25 43 91 52 97 51 14
70 11 33 28 77 73 17 78 39 68 17 57
91 71 52 38 17 14 91 43 58 50 27 29 48
63 66 04 68 89 53 67 30 73 16 69 87 40 31
04 62 98 27 23 09 70 98 73 93 38 53 60 04 23'''


def break_pyramid(num_pyr):
    '''
    Breaks the number string into an array
    '''
    pyramid = []
    rows = num_pyr.split('\n')
    for row in rows:
        pyramid.append([int(i) for i in row.split(' ')])
    return pyramid

def add_from_under(upper_row, lower_row):
    '''
    Takes the maximum of the legs under upper row to each index to add up from bottom
    Parameters
    ----------
    upper_row : len n
    lower_row : len n + 1

    Returns
    -------
    new_upper_row
    '''
    greater_vals = [max(lower_row[i:i+2]) for i in range(len(upper_row))]
    return [x + y for x, y in zip(upper_row, greater_vals)]


def main():
    pyramid = break_pyramid(num_pyr)[::-1]
    for i in range(len(pyramid)-1):
        new_upper = add_from_under(pyramid[1], pyramid[0])
        pyramid[1] = new_upper
        del pyramid[0]
    return pyramid[0][0]


if __name__ == '__main__':
    t1 = time.time()
    best_path = main()
    op_time = 1000000 * (time.time() - t1)  # us
    print("Calculated {} in {:.1f} us".format(best_path, op_time))

I think improvements can be made in my main() since I am using del. Should del be avoided for any reason?

Answer 1

In break_pyramid you use list comprehension to split a row into numbers – why not use list comprehension as well to split the input into rows?

def break_pyramid(num_pyr):
    '''
    Breaks the number string into an array
    '''
    pyramid = [[int(col) for col in row.split(' ')] for row in num_pyr.split('\n')]
    return pyramid

---

In add_from_under, this

greater_vals = [max(lower_row[i:i+2]) for i in range(len(upper_row))]

creates many intermediate slices. The alternative

greater_vals = [max(l, r) for l, r in zip(lower_row, lower_row[1:])]

seems to be slightly faster. You can even combine both zip operations into a single one:

def add_from_under(upper_row, lower_row):
    return [upper + max(lower_left, lower_right)
         for upper, lower_left, lower_right in zip(upper_row, lower_row, lower_row[1:])]

so that no intermediate list is created.

---

The main loop applies add_from_under cumulatively to the pyramid rows (in reverse order). That is a "reduce" operation and can be done with functools.reduce:

from functools import reduce

def main():
    pyramid = break_pyramid(num_pyr)
    final_row = reduce(add_from_under, reversed(pyramid))
    return final_row[0]

if we exchange the parameter order in add_from_under to

def add_from_under(lower_row, upper_row):
    return [upper + max(lower_left, lower_right)
         for upper, lower_left, lower_right in zip(upper_row, lower_row, lower_row[1:])]

Now the pyramid variable is just iterated over, but not modified anymore.

Answer 2

Since you are not using i anywhere in the loop, the better approach is to loop for row in pyramid. Now you only need to track your new_upper:

    new_upper = pyramid[0]
    for row in pyramid[1:]:
        new_upper = add_from_under(new_upper, row)
    return new_upper[0]

Other than that, the algorithm is correct, the code is clean and well-structured.