Given a string of numbers and a number of multiplication operators, what is the highest number one can calculate?

Question

This was an interview question I had and I was embarrassingly pretty stumped by it. Wanted to know if anyone could think up an answer to it and provide the big O notation for it

Answer 1

I am assuming here that the required number m of multiplication operators is given as part of the problem, along with the string s of digits.

You can solve this problem using the tabular method (aka "dynamic programming") with O(m |s|²) multiplications of numbers that are O(|s|) digits long. The optimal computational complexity of multiplication is not known, but with the schoolbook multiplication algorithm this is O(m |s|⁴) overall.

(The idea is to compute the answer for each subproblem consisting of a tail of the string and a number m′ ≤ m. There are O(m |s|) such subproblems and solving each one involves O(|s|) multiplications of numbers that are O(|s|) digits long.)

In Python, you could program it like this, using the @memoized decorator from the Python decorator library:

@memoized
def max_product(s, m):
    """Return the maximum product of digits from the string s using m
    multiplication operators.

    """
    if m == 0:
        return int(s)
    return max(int(s[:i]) * max_product(s[i:], m - 1)
               for i in range(1, len(s) - m + 1))

If you're used to the bottom-up form of dynamic programming where you build up a table, this top-down form might look strange, but in fact the @memoized decorator maintains the table in the cache property of the function:

>>> max_product('56789', 1)
51102
>>> max_product.cache
{('89', 0): 89, ('9', 0): 9, ('6789', 0): 6789, ('56789', 1): 51102, ('789', 0): 789}