this is for holding javascript data
Xavier Holt edited begin_claim_The_algorithm_runs__.tex
about 8 years ago
Commit id: 4687a9471a721f98de4da6adbd6bb7d835c20200
deletions | additions
diff --git a/begin_claim_The_algorithm_runs__.tex b/begin_claim_The_algorithm_runs__.tex
index c268f11..ec542cb 100644
--- a/begin_claim_The_algorithm_runs__.tex
+++ b/begin_claim_The_algorithm_runs__.tex
...
&=2\left(2R(n/4) + O(\log n/2) + O(n/2)\right) + O(\log n) + O(n)\\
&= iO(n) + \sum_{j=0}^{i-1} 2^j O(\log \frac{n}{2^i}) + 2^iR(\frac{n}{2^i})\\
&= O(n\log n) + \sum_{j=0}^{\log n-1} 2^j O(\log \frac{n}{2^i}) + 2^{\log n}R(1)\\
&= O(n\log n) + n^{\log 2}O(1) + \sum_{j=0}^{\log n-1} 2^j (\log n -
\log{2^j}) \\ \log{2^j})\\
&= O(n\log n) + O(n) + \left(\log n\sum_{j=0}^{\log n-1} 2^j - \sum_{j=0}^{\log n-1} 2^j\log{2^j} \right)\\
&= O(n\log n) + \left(\log n \times 2^{\log n} - C \right) &\text{with $C\geq0$}\\
&= O(n\log n) + \left(n\log n - C \right) &\text{inner bracket $\geq
0$} \\ 0$}\\
\end{align*}
\end{proof}