We are given $n$ points in the unit circle, $P_i=(x_i,y_i)$, such that $0<x_i^2+y_i^2<1$ for $i=1,2,….,n$.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.)
