Most of the early algorithms for dynamic connectivity also imply $\O(n^\varepsilon)$
algorithms for dynamic maintenance of the MSF. Henzinger and King \cite{henzinger:twoec,henzinger:randdyn}
have generalized their randomized connectivity algorithm to maintain the MSF in $\O(\log^5 n)$ time per
-operation, or $\O(k\log^3 n)$ if only~$k$ different values of edge weights are allowed. They have solved
+operation, or $\O(k\log^3 n)$ if only $k$ different values of edge weights are allowed. They have solved
the decremental version of the problem first (which starts with a~given graph and only edge deletions
are allowed) and then presented a~general reduction from the fully dynamic MSF to its decremental version.
We will describe the algorithm of Holm, de Lichtenberg and Thorup \cite{holm:polylog}, who have followed
are at level~0. Sole deletions of edges (both tree and non-tree) cannot violate I3, so we need
to check only the replaces, in particular the place when an~edge~$e$ gets its level increased.
-For the violation to happen, $e$~must be the heaviest on some cycle~$C$, so by the Cycle
-rule, $e$~must be non-tree. The increase of $\ell(e)$ must therefore take place when~$e$ is
-considered as a~replacement edge incident with some tree~$T_1$ at level $\ell=\ell(e)$.
-We will pause the computation just before this increase and we will prove that
-all other edges of~$C$ already are at levels greater than~$\ell$.
+For the violation to happen for the first time, $e$~must be the heaviest on
+some cycle~$C$, so by the Cycle rule, $e$~must be non-tree. The increase of
+$\ell(e)$ must therefore take place when~$e$ is considered as a~replacement
+edge incident with some tree~$T_1$ at level $\ell=\ell(e)$. We will pause the
+computation just before this increase and we will prove that all other edges
+of~$C$ already are at levels greater than~$\ell$, so the violation cannot occur.
-Let us first show that for edges of~$C$ incident with~$T_1$. All edges of~$T_1$ itself
+Let us first show this for edges of~$C$ incident with~$T_1$. All edges of~$T_1$ itself
already are at the higher levels as they were moved there at the very beginning of the
-search for the replacement edge. As the algorithm always picks the lightest candidate
-for the replacement edge available, all non-tree edges incident with~$T_1$ that are lighter
-than~$e$ were already considered and thus also moved one level up. This includes all
-other edges of~$C$ that are incident with~$T_1$.
+search for the replacement edge. The other tree edges incident with~$T_1$ would have
+lower levels, which is impossible since the invariant would be already violated.
+Non-tree edges of~$C$ incident with~$T_1$ are lighter than~$e$, so they were already considered
+as~candidates for the replacement edge, because the algorithm always picks the lightest
+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 be non-tree edges connecting~$T_1$
-with the other trees at 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 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.
\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}}
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