Accuracy finished.

diff --git a/opt.tex b/opt.tex
index e058beb86f433fb3905c2504e025d1f538ea9eec..ce1ecd6ea364dd267fda19863038584d48c2b0a4 100644 (file)
--- a/opt.tex
+++ b/opt.tex
@@ -284,7 +284,7 @@ has disappeared since $\<rank>(v)>r$ and therefore the desired bound is at least
 We will now sum the sizes of the lists over all vertices containing corrupted items.
 
 \lemma
-After~$n$ items have been inserted, the heap contains at most~$n/2^{r-3}$ corrupted
+After~$n$ items have been inserted, the heap contains at most~$n/2^{r-2}$ corrupted
 items at any given time.
 
 \proof
@@ -302,20 +302,22 @@ tree of every queue in the heap. The actual trees can be much sparser, but the
 above claim guarantees that the total size of the master trees is bounded by the
 number of insertions properly.
 
-So let us have a~complete tree of~rank~$k$. It has exactly $2^{k-i}$ vertices
+So let us consider a~complete tree of~rank~$k$. It has exactly $2^{k-i}$ vertices
 of rank~$i$ and each such vertex contains a~list of at most~$2^{\lceil i/2\rceil - r/2}$
 items by the previous lemma. Summing over all ranks greater than~$r$, we get that
 the total number of corrupted items in this tree is at most:
 $$
 \sum_{i=r+1}^k 2^{k-i}\cdot 2^{\lceil i/2\rceil - r/2}
-= 2^{k-r/2} \cdot \sum_i 2^{\lceil i/2\rceil - i}
-\le 2^{k-r/2} \cdot \sum_i 2^{-i/2}
+= 2^{k-r/2} \cdot \sum_{i=r+1}^k 2^{\lceil i/2\rceil - i}
+\le 2^{k-r/2} \cdot \sum_{i=r+1}^k 2^{-i/2}
+\le 2^{k-r} \cdot \sum_{i=0}^\infty 2^{-i/2}.
 $$
-
-\FIXME{Finish the proof and update the claim of the lemma.}
-
-
+The sum of a~geometric series with quotient $2^{-1/2}$ is less than four, so the
+last formula is less than $2^{k-r+2}$. Since the tree contains $n_k=2^k$ black vertices,
+this makes less than $n_k/2^{k-2}$ corrupted items as we asserted.
 \qed
 
+\paran{Analysis of time complexity}
+
 
 \endpart