# Can I simplify the recurrence T(n) = 2T((n+1)/2) + c by ignoring the “+1” part?

I have written a recurrence relation to describe a recursive algorithm finding the maximum element in an array. The algorthim has an overlap, meaning both of the subarrays that are recurred on contain the middle element of the array.

$T(n) = 2T((n+1)/2) + c$

However, I want to simplify this recurrence relation more.

Since you can often times omit floors and ceilings in recurrence relations, can I omit the $+ 1$?

If so, then my relation is now: $T(n) = 2T(n/2) + c$

Would this alter my big-Theta time complexity? Why or why not?

• What have you tried and where did you get stuck? Hint: solve the simplified recurrence and prove the answer correct for the original one. See also our reference question. Oh, nitpick: "time complexity" is not a useful term here. You are solving a recurrence, period. – Raphael Jan 26 '16 at 11:19
• I just used mathematical induction to prove that my recurrence relation is indeed $T(n) = \Theta(n)$. However, I'm just wondering if there's an easier way to justify me omitting the +1 without showing the inductive proof and just the Master Theorem proof. – leviless Jan 26 '16 at 17:36
• There may be ways to generalize the proof you just performed to arbitrary $a$, $b$, $c$, and added constant. – Raphael Jan 26 '16 at 23:52
• It's probably safe to go the extra mile and write the inductive proofs. I've generalized my inductive proofs with constants which should be enough. Thanks! – leviless Jan 27 '16 at 2:13

This probably does not change the asymptotic order.

From $T'(n) = 2 T'(n/2) + c$, you can obtain $T'(n) = \Theta(n)$.

Then you can try to prove that your original $T(n) = \Theta(n)$ by mathematical induction.

We can rewrite the original recurrence as

$$T(n-1+1)=2T(\frac{n-1}2+1)+c$$ or, setting $m:=n-1$ and $S(m):=T(m+1)$,

$$S(m)=2S(\frac m2)+c,$$ which is of the simplified type.

After solving for $S(m)$,

$$T(n)=S(n-1)$$ and the solutions are virtually identical.