Bug fixes to fully dynamic algorithms.

diff --git a/dyn.tex b/dyn.tex
index 33c4b44ea4cdf74aa53aed77d129e79f8b71a11d..302425656cd02273ce3b65ac7c7edca6ba1df913 100644 (file)
--- a/dyn.tex
+++ b/dyn.tex
@@ -503,14 +503,14 @@ This does not hurt the complexity of insertions and deletions, but allows for fa
 \qed
 
 \rem
-An~$\Omega(\log n/\log\log n)$ lower bound for amortized complexity of the dynamic connectivity
+An~$\Omega(\log n/\log\log n)$ lower bound for the amortized complexity of the dynamic connectivity
 problem has been proven by Henzinger and Fredman \cite{henzinger:lowerbounds} in the cell
 probe model with $\O(\log n)$-bit words. Thorup has answered by a~faster algorithm
 \cite{thorup:nearopt} that achieves $\O(\log n\log^3\log n)$ time per update and
 $\O(\log n/\log^{(3)} n)$ per query on a~RAM with $\O(\log n)$-bit words. (He claims
 that the algorithm runs on a~Pointer Machine, but it uses arithmetic operations,
 so it does not fit the definition of the PM we use. The algorithm only does not
-need indexing of arrays.) So far, it is not known how to extend this algorithm
+need direct indexing of arrays.) So far, it is not known how to extend this algorithm
 to fit our needs, so we omit the details.
 
 %--------------------------------------------------------------------------------
@@ -595,18 +595,19 @@ as~candidates for the replacement edge, because the algorithm always picks the l
 candidate first. Such edges therefore have been already moved a~level up.
 
 The case of edges of~$C$ that do not touch~$T_1$ is easy to handle: Such edges do not exist.
-If they did, at least two edges of~$C$ would have to connect~$T_1$ with the other trees of level~$\ell$,
-so one of them that is lighter than~$e$ would be selected as the replacement edge before~$e$ could be considered.
+If they did, at least one more edge of~$C$ besides~$e$ would have to connect~$T_1$ with the other
+trees of level~$\ell$. We already know that this could not be a~tree edge. If it were a~non-tree
+edge, it could not have level greater than~$\ell$ by~I1 nor smaller than~$\ell$ by~I3. Therefore
+it would be a~level~$\ell$ edge lighter than~$e$, and as such it would have been selected as the
+replacement edge before $e$~was.
 \qed
 
-\FIXME{The previous paragraph is incomplete, it does not take tree edges into account.}
-
 We can conclude:
 
 \thmn{Decremental MSF, Holm et al.~\cite{holm:polylog}}
 When we start with a~graph on $n$~vertices with~$m$ edges and we perform a~sequence of
 edge deletions, the MSF can be initialized in time $\O((m+n)\cdot\log^2 n)$ and then
-updated in time $\O(log^2 n)$ amortized per operation.
+updated in time $\O(\log^2 n)$ amortized per operation.
 
 \paran{Fully dynamic MSF}%
 The decremental MSF algorithm can be turned to a~fully dynamic one by a~blackbox