From: Martin Mares <mj@ucw.cz>
Date: Wed, 9 Apr 2008 18:27:07 +0000 (+0200)
Subject: First bits of ET trees.
X-Git-Tag: printed~107
X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=8d8b17f0affcab162e84075e5f8b43f07daf7b96;p=saga.git

First bits of ET trees.
---

diff --git a/biblio.bib b/biblio.bib
index 2c1aedd..f5d6355 100644
--- a/biblio.bib
+++ b/biblio.bib
@@ -1296,3 +1296,13 @@
   publisher = {ACM},
   address = {New York, NY, USA},
 }
+
+@article{ henzinger:randdyn,
+  title={{Randomized Fully Dynamic Graph Algorithms with Polylogarithmic Time per Operation}},
+  author={Henzinger, M.R. and King, V.},
+  journal={Journal of the ACM},
+  volume={46},
+  number={4},
+  pages={502--516},
+  year={1999}
+}
diff --git a/dyn.tex b/dyn.tex
index efa1f24..666da7d 100644
--- a/dyn.tex
+++ b/dyn.tex
@@ -180,12 +180,52 @@ need to change anything) and so is deletion of bridges (we just remove the bridg
 from the Link-Cut tree, knowing that there is no edge to replace it). The hard part
 is the search for replacement edges after an~edge of the MSF is deleted.
 
-As this very problem has to be solved by algorithms for fully dynamic connectivity,
+This very problem also has to be solved by algorithms for fully dynamic connectivity,
 we will take a~look on them first.
 
 %--------------------------------------------------------------------------------
 
 \section{Eulerian Tour trees}
 
+An~important stop on the road to fully dynamic algorithms has the name \df{Eulerian Tour trees} or
+simply \df{ET-trees}. It is a~representation of forests introduced by Henzinger and King
+\cite{henzinger:randdyn} in their randomized dynamic algorithms. It is similar to the one by Sleator
+and Tarjan, but without orientation and without the path operations. Their advantage lies in
+simplicity and also in their ease of extension to contain various auxiliary data attached to
+vertices and edges of the original tree.
+
+\defn\id{eulseq}%
+The \df{Eulerian Tour sequence} $\Eul(T)$ of a~rooted tree~$T$ is the sequence of vertices of~$T$ as visited
+by the depth-first traversal of~$T$. More precisely, it is generated by the following algorithm $\<ET>(v)$
+when it is invoked on the root of the tree:
+\algo
+\:Record~$v$ in the sequence.
+\:For each son~$w$ of~$v$:
+\::Call $\<ET>(w)$ recursively.
+\::Record~$w$.
+\endalgo
+\>One of the occurrences of each vertex is defined as its \df{designated occurrence} and it will
+be used to store auxiliary data associated with the vertex.
+
+\obs
+The ET sequence contains a~vertex of degree~$d$ exactly $d$~times except for the root which
+occurs $d+1$ times. The whole sequence therefore contains $2n-1$ elements. It indeed describes the
+order vertices on an~Eulerian tour in the tree with all edges doubled. Let us observe what happens
+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.
+
+\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{Joining} the roots of two trees by a~new edge, their ET sequences
+$vAv$ and~$wBw$ combine to $vAvwBwv$.
+
+
+
 
 \endpart
diff --git a/macros.tex b/macros.tex
index b350a78..6d063a7 100644
--- a/macros.tex
+++ b/macros.tex
@@ -54,6 +54,7 @@
 \def\E{{\bb E}}
 \def\crpt{\mathbin{\Uparrow}}
 \def\C{{\cal C}}
+\def\Eul{{\cal E}}
 
 \def\brk{\hfil\break}
 
diff --git a/notation.tex b/notation.tex
index 8693a5d..daba11c 100644
--- a/notation.tex
+++ b/notation.tex
@@ -92,6 +92,7 @@
 \n{$a(x,n)$}{The inverse of the $x$-th row of the Ackermann's function \[ackerinv]}
 \n{$a(n)$}{The diagonal inverse of the Ackermann's function \[ackerinv]}
 \n{$\alpha(m,n)$}{$\alpha(m,n) := \min\{ x\ge 1 \mid A(x,4\lceil m/n\rceil) > \log n \}$ \[ackerinv]}
+\n{$\Eul(T)$}{The Eulerian tour sequence for a~tree~$T$ \[eulseq]}
 }
 
 %--------------------------------------------------------------------------------