$\star$ We are given $n$ points in the unit circle, $p_i = (x_i, y_i)$, such that $0 < x_i^2 + y_i^2 \le 1$ for $i = 1, 2, \ldots, n$. Suppose that the points are uniformly distributed; that is, the probability of finding a point in any region of the circle is proportional to the area of that region. Design an algorithm with an average-case running time of $\Theta(n)$ to sort the $n$ points by their distances $d_i = \sqrt{x_i^2 + y_i^2}$ from the origin. (

Hint:Design the bucket sizes in`BUCKET-SORT`

to reflect the uniform distribution of the points in the unit circle.)

The unit circle has area $\pi 1^2 = \pi$. We need to split it in $n$ discs, each having area $\pi / n$. The radius of such a disc is $\pi(b^2 - a^2)$, where $b$ is the radius of the outer edge and $a$ is the radius of the inner edge of the disc.

Let the points $a_0, a_1, a_2, \ldots a_n$ divide the circle in n dics, the $i$th disc having radiuses $a_i$ and $a_{i-1}$. We know that $a_0 = 0$ and $a_n = 1$. For any two discs we have:

$$ \pi^2 \pi(a_i^2 - a_{i-1}^2) = \pi(a_j^2 - a_{j-1}^2) = \frac{\pi}{n} \\ \Downarrow \\ a_i^2 - a_{i-1}^2 = a_j^2 - a_{j-1}^2 = \frac{1}{n} \\ \Downarrow \\ a_i^2 = \frac{1}{n} + a_{i-1}^2 $$

We get the following recurrence:

$$ \begin{align} a_0 &= 0 \\ a_i &= \sqrt{1/n + a_{i-1}^2} \end{align} $$

If we check some small values, we see the following pattern:

$$ \pi n^2 $$

$$ \begin{align} a_0 & = 0 = \frac{1}{\sqrt n} \\ a_1 & = \sqrt{\frac{1}{n} + \frac{1}{n}} = \frac{\sqrt 2}{\sqrt n} \\ a_2 & = \sqrt{\frac{1}{n} + \frac{2}{n}} = \frac{\sqrt 2}{\sqrt n} \\ & \ldots \\ a_i & = \frac{\sqrt i}{\sqrt n} \end{align} $$

The last step is easy to prove by induction. If we assume it for $i \le k$, then:

$$ a_{k+1} = \sqrt{\frac{1}{n} + a_k^2} = \sqrt{\frac{1}{n} + \frac{k}{n}} = \frac{\sqrt{k+1}}{\sqrt n} $$

Thus, we create buckets for the following intervals:

$$ \bigg[0, \frac{1}{\sqrt n} \bigg), \bigg[\frac{1}{\sqrt n}, \frac{\sqrt 2}{\sqrt n}\bigg) \cdots \bigg[\frac{\sqrt{n-1}}{\sqrt n}, 1 \bigg] $$

And distribute elements according to their distance. To calculate the bucket $k$ for a distance $d$ in constant time, just take:

$$ k = \begin{cases} \lfloor d^2n \rfloor + 1 & \text{if } d < 1, \\ n & \text{id } d = 1 \end{cases} $$

...for 1-based buckets.