The Power of Math

Author: Szymon Lipiński
Published at: 2016-10-13

In a couple of previous posts I was considering which is better in Python: map or list comprehension. The final idea was that none of them is as fast as a C++ program, even using exactly the same algorithm.

What about changing an algorithm a little bit?

The Idea

I was thinking about speeding up the Python code. Then I thought about the simple equation for calculating nth Fibonacci number. It is very simple, the algorithm complexity is not that bad, it is something like a^n + b^n. This is much faster for bigger n then the recursive/iterative version.

If I was going to sum all the numbers from 1 to n, then I wouldn’t write this:

def sum_numbers(n):
    return sum([x for x in range(1, n+1)])

I’d rather use the quite simple equation for summing it:

sum(n) = (1 + n) * n / 2

Then the program looks like:

def sum_numbers(n):
    return (1 + n) * n / 2

The time difference is huge: n=10*1000*1000: first program: 5s, second program: 15ms. So that’s just 333 times faster.

The Solution

So what about the equation for calculating the sum of all squares of even numbers from 1 to 1,000,000?. It is very simple:

sum(n) = (4 + 6 + 2n)/3

So let’s implement it and check the time:

def sum_numbers(n):
    return (4 * n**3 + 6 * n**2 + 2*n) / 3   

This runs in 10ms. Much better compared to the previous solution, which took 70s.

The Final Conclusion

Sometimes the best solution is brute force. Sometimes just good old math.

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