this is for holding javascript data
Xavier Holt edited begin_claim_The_algorithm_runs__.tex
about 8 years ago
Commit id: bc2f345519775c0d5388e9d4ff3ea593ddfc8f7a
deletions | additions
diff --git a/begin_claim_The_algorithm_runs__.tex b/begin_claim_The_algorithm_runs__.tex
index 3a355b0..1dfc0fa 100644
--- a/begin_claim_The_algorithm_runs__.tex
+++ b/begin_claim_The_algorithm_runs__.tex
...
&= 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})\\
&= 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 \tag{with $C\geq0$}\\
&= O(n\log n) + \left(n\log n - C \right)
&&\tag{inner \tag{inner bracket $\geq 0$}\\
&\leq O(n\log n) + O(n\log n)\\
&\in O(n\log n)
\end{align*}