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Find the square root of an integer with trigonometry and Lagrange’s four-square theorem


My friend was solving the following problem during a interview for Citigroup’s IT department.

Problem1

Find the square root of a integer n, without using the built in sqrt function. (The range of the result was not specified, I assume it’s double)

This is a common interview question.

There are many ways to do it. I want to come up with a way no one else would think of, something that could amaze the interviewer. I mean, she might interviewed enough people to get bored with the standard answers.

I present the following highly inefficient but somewhat creative solution. The code is here.

How does it work?

We know is an integer. By Lagrange’s four-square theorem, for integer . . Thus is the magnitude of the vector . can be calculated by brute force search(therefore runs in time).

Note a simple improvement of the naive algorithm can reduce the computation time to by doing a binary search for the last square.

A much smarter randomized algorithm by Michael O. Rabin and Jeffrey Shallit have a running time of .

A recursive algorithm using the following relation can find the magnitude of any vector(assume ) It’s easy to see, this breaks a -dimension vector into orthogonal vectors of -dimensions and -dimension. We get a right triangle. Trigonometry comes in handy.

Posted by Chao Xu on .
Tags: math.