Time complexity of Prune and Search

Question

I read a lot of stuff about Prune and Search algorithm and I even asked some of it for confirmation.

This is a great source. However, some things are hard for me to understand. Like the time complexity of Prune and Search:

Can someone provide a brief explanation for this?

This is the time complexity of a particular prune and search type algorithm. You should describe the algorithm and what you don't understand in this computation. — hivert, Jul 16 '13 at 09:44
@DavidEisenstat Yes, the one that finds the median. It's kinda confusing. — KC-Chan, Jul 16 '13 at 15:22

score 0 · Answer 1 · answered Jul 16 '13 at 12:27

They're solving the recurrence T(n) <= T(n/5) + T(3n/4) + Cn (C is the big-O constant) via some bizarre method that I don't recognize. Modulo the missing base case and floor and ceiling operators, we can solve it via Akra–Bazzi or the substitution method (this answer).

The inductive hypothesis is that T(n') <= 20Cn' for all n' < n. Then

T(n) <= T(n/5) + T(3n/4) + Cn
     <= 20C(n/5) + 20C(3n/4) + Cn
      = 20C(4n/20) + 20C(15n/20) + Cn
      = 4Cn + 15Cn + Cn
      = 20Cn.

Time complexity of Prune and Search

1 Answers1