Maximum and minimum depth of quicksort

Question

This was a problem of CLR (Introduction to Algorithms) The question goes as follow:

Suppose that the splits at every level of quicksort are in the proportion 1 - α to α, where 0 < α ≤ 1/2 is a constant. Show that the minimum depth of a leaf in the recursion tree is approximately - lg n/ lg α and the maximum depth is approximately -lg n/ lg(1 - α). (Don't worry about integer round-off.)http://integrator-crimea.com/ddu0043.html

I'm not getting how to reach this solution. as per the link they show that for a ratio of 1:9 the max depth is log n/log(10/9) and minimum log n/log(10). Then how can the above formula be proved. Please help me as to where am I going wrong as I'm new to Algorithms and Data Structures course.

Answer 1

First, let us consider this simple problem. Assume you a number n and a fraction (between 0 and 1) p. How many times do you need to multiply n with p so that resulting number is less than or equal to 1?

n*p^k <= 1
log(n)+k*log(p) <= 0
log(n) <= -k*log(p)
k => -log(n)/log(p)

Now, let us consider your problem. Assume you send the shorter of the two segments to the left child and longer to the right child. For the left-most chain, the length is given by substituting \alpha as p in the above equation. For the right most chain, the length is calculated by substituting 1-\alpha as p. Which is why you have those numbers as answers.