Wrong answer in SPOJ `CUBERT` [closed]

问题:

I am getting a Wrong Answer for my solution to this problem on SPOJ.

The problem asks to calculate the cube root of an integer(which can be upto 150 digits long), and output the answer truncated upto 10 decimal places.
It also asks to calculate the sum of all the digits in the answer modulo 10 as a 'checksum' value.

Here is the exact problem statement:

Your task is to calculate the cube root of a given positive integer. We can not remember why exactly we need this, but it has something in common with a princess, a young peasant, kissing and half of a kingdom (a huge one, we can assure you).

Write a program to solve this crucial task.

Input

The input starts with a line containing a single integer t <= 20, the number of test cases. t test cases follow.

The next lines consist of large positive integers of up to 150 decimal digits. Each number is on its own separate line of the input file. The input file may contain empty lines. Numbers can be preceded or followed by whitespaces but no line exceeds 255 characters.

Output

For each number in the input file your program should output a line consisting of two values separated by single space. The second value is the cube root of the given number, truncated (not rounded!) after the 10th decimal place. First value is a checksum of all printed digits of the cube root, calculated as the sum of the printed digits modulo 10.

Example

Input:
5
1

8

1000

2 33076161

Output:
1 1.0000000000
2 2.0000000000
1 10.0000000000
0 1.2599210498
6 321.0000000000

Here is my solution:

from math import pow


def foo(num):
    num_cube_root = pow(num, 1.0 / 3)
    # First round upto 11 decimal places
    num_cube_root = "%.11f" % (num_cube_root)
    # Then remove the last decimal digit
    # to achieve a truncation of 10 decimal places
    num_cube_root = str(num_cube_root)[0:-1]

    num_cube_root_sum = 0
    for digit in num_cube_root:
        if digit != '.':
            num_cube_root_sum += int(digit)
    num_cube_root_sum %= 10

    return (num_cube_root_sum, num_cube_root)


def main():
    # Number of test cases
    t = int(input())
    while t:
        t -= 1
        num = input().strip()
        # If line empty, ignore
        if not num:
            t += 1
            continue

        num = int(num)
        ans = foo(num)
        print(str(ans[0]) + " " + ans[1])


if __name__ == '__main__':
    main()

It is working perfectly for the sample cases: Live demo.

Can anyone tell what is the problem with this solution?

回答1:

Your solution has two problems, both related to the use of floating-point arithmetic. The first issue is that Python floats only carry roughly 16 significant decimal digits of precision, so as soon as your answer requires more than 16 significant digits or so (so more than 6 digits before the point, and 10 digits after), you've very little hope of getting the correct trailing digits. The second issue is more subtle, and affects even small values of n. That's that your approach of rounding to 11 decimal digits and then dropping the last digit suffers from potential errors due to double rounding. For an example, take n = 33. The cube root of n, to 20 decimal places or so, is:

3.20753432999582648755...

When that's rounded to 11 places after the point, you end up with

3.20753433000

and now dropping the last digit gives 3.2075343300, which isn't what you wanted. The problem is that that round to 11 decimal places can end up affecting digits to the left of the 11th place digit.

So what can you do to fix this? Well, you can avoid floating-point altogether and reduce this to a pure integer problem. We need the cube root of some integer n to 10 decimal places (rounding the last place down). That's equivalent to computing the cube root of 10**30 * n to the nearest integer, again rounding down, then dividing the result by 10**10. So the essential task here is to compute the floor of the cube root of any given integer n. I was unable to find any existing Stack Overflow answers about computing integer cube roots (still less in Python), so I thought it worth showing how to do so in detail.

Computing cube roots of integers turns out to be quite easy (with the help of a tiny bit of mathematics). There are various possible approaches, but one approach that's both efficient and easy to implement is to use a pure-integer version of the Newton-Raphson method. Over the real numbers, Newton's method for solving the equation x**3 = n takes an approximation x to the cube root of n, and iterates to return an improved approximation. The required iteration is:

x_next = (2*x + n/x**2)/3

In the real case, you'd repeat the iteration until you reached some desired tolerance. It turns out that over the integers, essentially the same iteration works, and with the right exit condition it will give us exactly the correct answer (no tolerance required). The iteration in the integer case is:

a_next = (2*a + n//a**2)//3

(Note the uses of the floor division operator // in place of the usual true division operator / above.) Mathematically, a_next is exactly the floor of (2*a + n/a**2)/3.

Here's some code based on this iteration:

def icbrt_v1(n, initial_guess=None):
    """
    Given a positive integer n, find the floor of the cube root of n.

    Args:
        n : positive integer
        initial_guess : positive integer, optional. If given, this is an
            initial guess for the floor of the cube root. It must be greater
            than or equal to floor(cube_root(n)).

    Returns:
        The floor of the cube root of n, as an integer.
    """
    a = initial_guess if initial_guess is not None else n
    while True:
        d = n//a**2
        if a <= d:
            return a
        a = (2*a + d)//3

And some example uses:

>>> icbrt_v1(100)
4
>>> icbrt_v1(1000000000)
1000
>>> large_int = 31415926535897932384626433
>>> icbrt_v1(large_int**3)
31415926535897932384626433
>>> icbrt_v1(large_int**3-1)
31415926535897932384626432

There are a couple of annoyances and inefficiencies in icbrt_v1 that we'll fix shortly. But first, a brief explanation of why the above code works. Note that we start with an initial guess that's assumed to be greater than or equal to the floor of the cube root. We'll show that this property is a loop invariant: every time we reach the top of the while loop, a is at least floor(cbrt(n)). Furthermore, each iteration produces a value of a strictly smaller than the old one, so our iteration is guaranteed to eventually converge to floor(cbrt(n)). To prove these facts, note that as we enter the while loop, there are two possibilities:

Case 1. a is strictly greater than the cube root of n. Then a > n//a**2, and the code proceeds to the next iteration. Write a_next = (2*a + n//a**2)//3, then we have:

• a_next >= floor(cbrt(n)). This follows from the fact that (2*a + n/a**2)/3 is at least the cube root of n, which in turn follows from the AM-GM inequality applied to a, a and n/a**2: the geometric mean of these three quantities is exactly the cube root of n, so the arithmetic mean must be at least the cube root of n. So our loop invariant is preserved for the next iteration.

• a_next < a: since we're assuming that a is larger than the cube root, n/a**2 < a, and it follows that (2a + n/a**2) / 3 is smaller than a, and hence that floor((2a + n/a**2) / 3) < a. This guarantees that we make progress towards the solution at each iteration.

Case 2. a is less than or equal to the cube root of n. Then a <= floor(cbrt(n)), but from the loop invariant established above we also know that a >= floor(cbrt(n)). So we're done: a is the value we're after. And the while loop exits at this point, since a <= n // a**2.

There are a couple of issues with the code above. First, starting with an initial guess of n is inefficient: the code will spend its first few iterations (roughly) dividing the current value of a by 3 each time until it gets into the neighborhood of the solution. A better choice for the initial guess (and one that's easily computable in Python) is to use the first power of two that exceeds the cube root of n.

initial_guess = 1 << -(-n.bit_length() // 3)

Even better, if n is small enough to avoid overflow, is to use floating-point arithmetic to provide the initial guess, with something like:

initial_guess = int(round(n ** (1/3.)))

But this brings us to our second issue: the correctness of our algorithm requires that the initial guess is no smaller than the actual integer cube root, and as n gets large we can't guarantee that for the float-based initial_guess above (though for small enough n, we can). Luckily, there's a very simple fix: for any positive integer a, if we perform a single iteration we always end up with a value that's at least floor(cbrt(a)) (using the same AM-GM argument that we used above). So all we have to do is perform at least one iteration before we start testing for convergence.

With that in mind, here's a more efficient version of the above code:

def icbrt(n):
    """
    Given a positive integer n, find the floor of the cube root of n.

    Args:
        n : positive integer

    Returns:
        The floor of the cube root of n, as an integer.
    """
    if n.bit_length() < 1024:  # float(n) safe from overflow
        a = int(round(n**(1/3.)))
        a = (2*a + n//a**2)//3  # Ensure a >= floor(cbrt(n)).
    else:
        a = 1 << -(-n.bit_length()//3)

    while True:
        d = n//a**2
        if a <= d:
            return a
        a = (2*a + d)//3

And with icbrt in hand, it's easy to put everything together to compute cube roots to ten decimal places. Here, for simplicity, I output the result as a string, but you could just as easily construct a Decimal instance.

def cbrt_to_ten_places(n):
    """
    Compute the cube root of `n`, truncated to ten decimal places.

    Returns the answer as a string.
    """
    a = icbrt(n * 10**30)
    q, r = divmod(a, 10**10)
    return "{}.{:010d}".format(q, r)

Example outputs:

>>> cbrt_to_ten_places(2)
'1.2599210498'
>>> cbrt_to_ten_places(8)
'2.0000000000'
>>> cbrt_to_ten_places(31415926535897932384626433)
'315536756.9301821867'
>>> cbrt_to_ten_places(31415926535897932384626433**3)
'31415926535897932384626433.0000000000'

回答2:

You may try to use the decimal module with a sufficiently large precision value.

EDIT: Thanks to @DSM, I realised that decimal module will not produce very exact cube roots. I suggest that you check whether all digits are 9s and round it to a integer if that is a case.

Also, I now perform the 1/3 division with Decimals as well, because passing the result of 1/3 to Decimal constructor leads to reduced precision.

import decimal

def cbrt(n):
    nd = decimal.Decimal(n)
    with decimal.localcontext() as ctx:
        ctx.prec = 50
        i = nd ** (decimal.Decimal(1) / decimal.Decimal(3))
    return i

ret = str(cbrt(1233412412430519230351035712112421123121111))
print(ret)
left, right = ret.split('.')
print(left + '.' + ''.join(right[:10]))

Output:

107243119477324.80328931501744819161741924145124146
107243119477324.8032893150

Output of cbrt(10) is:

9.9999999999999999999999999999999999999999999999998