More bits of ET trees.

diff --git a/dyn.tex b/dyn.tex
index 666da7df7c67e001fce201f8cb8d58c436b7b02a..59362a671f62e0446e2d50c2f9c533937cce1bfc 100644 (file)
--- a/dyn.tex
+++ b/dyn.tex
@@ -214,17 +214,32 @@ order vertices on an~Eulerian tour in the tree with all edges doubled. Let us ob
 to the ET sequence when we modify the tree.
 
 When we \em{delete} an~edge $uv$ from the tree~$T$ (let $u$~be the parent of~$v$), the sequence
-$\Eul(T) = Au_iv_1Bv_du_{i+1}C$ splits to two ET sequences $AuC$ and $v_1Bv_d$, Here $v_1$~is
-the first occurrence of~$v$ in $\Eul(T)$ and $v_d$~is the last one. If there was only a~single
-occurrence, $v$~was a~leaf and thus the second sequence will consists of $v$~alone. If $v_1$~or~$v_d$
-was the designated occurrence of~$v$, we redirect it to the new occurrence.
+$\Eul(T) = AuvBvuC$ (with no~$u$ nor~$v$ in~$B$) splits to two ET sequences $AuC$ and $vBv$.
+If there was only a~single occurrence of~$v$, it corresponded to a~leaf and thus the second
+sequence should consist of $v$~alone.
 
-\em{Changing the root} of the tree~$T$ from~$v$ to~$w$ changes $\Eul(T) = vAw_1Bw_dCv$
-to $w_1Bw_dCvAw_{d+1}$.
+\em{Changing the root} of the tree~$T$ from~$v$ to~$w$ changes $\Eul(T) = vAwBwCv$ (no~$w$ in~$B$)
+to $wBwCvAw$. If $w$~was a~leaf, the sequence changes from $vAwCv$ to $wCvAw$. If $vw$ was the only
+edge of~$T$, the sequence $vw$ becomes $wv$.
 
-\em{Joining} the roots of two trees by a~new edge, their ET sequences
-$vAv$ and~$wBw$ combine to $vAvwBwv$.
+\em{Joining} the roots of two trees by a~new edge makes their ET sequences $vAv$ and~$wBw$
+combine to $vAvwBwv$. Again, we have to handle the cases when $v$ or~$w$ has degree~1 separately:
+$v$~and~$wBw$ combine to $vwBwv$, and $v$~with~$w$ makes $vwv$.
 
+If any of the occurrences that we have removed from the sequence was designated, there is always
+a~new occurrence of the same vertex that can stand in its place and inherit the auxiliary data.
+
+The ET trees will represent the ET sequences by $(a,b)$-trees with the parameter~$a$ set upon
+initialization of the structure and with $b=2a$. We know from the standard theorems of $(a,b)$-trees
+(see for example \cite{clrs}) that the depth of a~tree with $n$~leaves is always $\O(\log_a n)$
+and that all basic operations including insertion, deletion, search, splitting and joining the trees
+run in time $\O(a\log_a n)$ in the worst case.
+
+We will use the ET trees to maintain a~spanning forest of the current graph. The auxiliary data of
+each vertex will hold a~list of edges incident with the given vertex, which do not lie in the
+forest. Such edges are usually called the \df{non-tree edges.}
+
+\defnn{Eulerial Tour trees}