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Given $i < n/2$ and denoting $[x]$ to be an integer part of $x$ (floor$(x)$) and $(a \operatorname{rem} b)$ to be a reminder when $a$ is divided by $b$.

$$ T(n,i) = \left(\left\lfloor\frac{n-i}{i}\right\rfloor \cdot i\right) + T\left(i + (n \operatorname{rem} i), (n \operatorname{rem} i)\right) $$ then is $T(n,i) = O(n)$?

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What is $T(n, 0)$? –  user2468 Aug 30 '12 at 21:14
    
T(n,0) = 0 and T(n,1) = n –  rTeja Aug 30 '12 at 21:17
    
I'm not sure if the following makes any sense or even correct. Anyways it might help you towards a correct answer. Fix an $i < n/2$. Then we have $\lfloor \frac{n-i}{i} \rfloor < n-1$, and $n \bmod i < i-1$. Hence $T(n, i) <$ $(n-1) +$ $T(2i-1, i-1)$. If $i$ is fixed to a constant, then $T(n, i) < n-1,$ hence $T(n, i)$ $\in$ $\mathcal{O}(n)$. Continuing on this gibberish, if $i$ is not constant: we actually have $T(n, i)$ $<$ $(n-1)$ $+ (n-2)$ $+ \cdots +$ $(n-i)$ $\in$ $\mathcal{O}(ni - i^2)$ $\in$ $\mathcal{O}(n^2)$. –  user2468 Aug 30 '12 at 22:10
    
Proceeding on the lines of analysis for Euclid's GCD algorithm we can actually say T(n,i) = O(nLogn). I want to know if we can better analyse the sum and actually say it is O(n) or not. –  rTeja Aug 31 '12 at 3:33
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