GATE CSE
First time here? Checkout the FAQ!
x
+1 vote
199 views
I've been struggling to come to exact solution for this. Master's theorem is not applicable and likely way to get to answer is Recursion tree. Which is giving me Theta(n) as an answer.

Steps : =>

1)  T(n) = 2T(n/4) + √3

2) Emerging Pattern => (1/2^k) * n

3) Considering when we break down to T(1) : :   (1/2^k) * n  =1

4) k = log (n)

5) Each level has 2^i nodes :: 2^(log (n)) :=> n

6) If you sum up cost , n + n/2 + n/4 + ..... = n/2

Which is incorrect ,

Answer given is ( √n log n ) , would appreciate if someone could shed light how so ?
asked in Algorithms by Junior (769 points)   | 199 views
why masters theorem not applicable
Case (III) Regularity condition fails.
It is not case 3.  It is Case 1     $\sqrt {3}$ =$O(n^{1/2 - e})$
My bad , it is case (I) , but then  √3 =O(n1/2−e) does not always hold , so is it still valid ?

 

And , if you apply considering it does hold , answer given is ( √n log n ) and master's theorem would gives us √n .

2 Answers

+3 votes
Adding cost of each level;

$\begin{align*} T(n) &= 2^0.\sqrt{3} + 2^1.\sqrt{3} + 2^2.\sqrt{3} + .... + 2^{\log_4 n}.\sqrt{3} \\ &= \sqrt{3}.\left [ \frac{2^{\log_4 {n}+1}-1}{2-1} \right ] \\ &= \sqrt{3}.O(n^{\frac{1}{2}}) \\ &= O(\sqrt{n}) \end{align*}$
answered by Veteran (46.9k points)  

This is right , I was summing up thinking it's √3n ; anyways I'm not able to figure out  if √n * log n is wrong answer then ? 

+3 votes

In these type of questions where master method doesnt apply you can do the following

$T(n) = 2T(n/4) + \sqrt{3}$

Reframe it like this, by putting i in the specified positions

$T(n) = 2^{i}T(n/4^{i}) + i\sqrt{3}$

Now for n/4i = 1, put i = log4n

So the equation becomes

$T(n) = 2^{log_{4}n} + log_{4}n\sqrt{3}$

Or

$T(n) = n^{log_{4}2} + log_{4}n\sqrt{3}$

$T(n) = \sqrt{n} + log_{4}n\sqrt{3}$

And therefore

$T(n) = O(\sqrt{n})$

answered by Junior (783 points)  

Related questions

+2 votes
2 answers
1
0 votes
1 answer
2
asked in Algorithms by iarnav Active (1.1k points)   | 80 views


Top Users May 2017
  1. akash.dinkar12

    3154 Points

  2. pawan kumarln

    1630 Points

  3. sh!va

    1590 Points

  4. Arjun

    1350 Points

  5. Devshree Dubey

    1246 Points

  6. Angkit

    1044 Points

  7. Debashish Deka

    1022 Points

  8. Bikram

    972 Points

  9. LeenSharma

    836 Points

  10. Prashant.

    692 Points

Monthly Topper: Rs. 500 gift card
Top Users 2017 May 22 - 28
  1. pawan kumarln

    256 Points

  2. Ahwan

    236 Points

  3. jjayantamahata

    114 Points

  4. joshi_nitish

    114 Points

  5. Arnab Bhadra

    94 Points


22,731 questions
29,061 answers
65,101 comments
27,627 users