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\lecture{6 --- February 17, 2005}{Spring 2005}{Prof.\ Erik Demaine}{Jelani Nelson}

\section{Overview}

In the last lecture we discussed the link-cut tree: a dynamic tree that
achieves $O(\lg n)$ amortized time per operation.  In this lecture, leave
the BST model and enter the pointer-machine model.  In this model, we are
allowed to have nodes which each hold $O(1)$ pointers and $O(1)$ fields
(e.g. integers).  Our machine is also allowed to keep track of $O(1)$
``fingers'' to nodes at a time.  The operations we are allowed to do are to
copy a finger, follow a pointer to a node, and to retrieve or change the
contents of a node's fields.  So, the BST model allows for a subset of the
data structures allowable in the pointer-machine model.  We will focus on
data structures in this model that solve the dynamic connectivity problem.

\section{Euler-tour trees}

The Euler-tour tree data structure is due to Henzinger and King in
\cite{henzinger}.  An Euler-tour of a tree is a path along the tree that
begins at the root and ends at the root, traversing each edge exactly twice
--- once to enter the subtree at the other endpoint and once to leave it.
You can think of an Euler-tour as just being a depth first traversal where
we return to the root at the end.

We will represent the Euler-tour of the tree as a balanced binary search
tree with one node for each time a node in the represented tree was
visited, and each node in the tree is keyed by its time of visit.  Each
node in the represented tree will also hold pointers to the nodes
representing the first and last time it was visited.

While the link-cut trees we discussed last lecture are good for maintaining
aggregates on paths of a tree (making it a good choice data structure in
network flow algorithms), Euler-tour trees are better at keeping aggregate
information on subtrees.  We will use this feature of Euler-tour trees
toward the end of the lecture notes.

We want Euler-tour trees to be able to perform the following three
operations:

\begin{itemize}
\item[-] \textsc{{\textbf Cut}}\textbf{(v)}\\
      What this entails in the BST is cutting out the interval between the
      first and last visits to $v$.  So, we split $v$'s BST before the last visit to $v$ and after the last
      visit.  We then concatenate the first and last trees.

\item[-] \textsc{{\textbf Link}}\textbf{(v,w)}\\
      We split $w$'s BST after the first visit to $w$ then put in $v$'s BST
      using two concatenates.
\item[-] \textsc{{\textbf Findroot}}\textbf{(v)}\\
      Our items may be in different represented trees, so we may wish to find the root
      of $v$'s Euler-tour tree.  This can be achieved by just following
      parent pointers in the Euler-tour tree to the root.
\end{itemize}

We know all the above operations can be achieved in $O(\lg n)$ using any
common balanced BST data structure.

\section{Dynamic Graph Problems}
We wish to maintain an undirected graph subject to vertex insertion and
deletion, and edge insertion and deletion.  Deleting a vertex also deletes
all incident edges (or alternatively the vertex must have no incident
edges).  Since a data structure that is never observed can perform any
operation in zero time, we will also say that the graph can be queried in
some way.

\subsection{Connectivity}
The query type we focus on in this lecture is connectivity.  Given a graph
$G$, we wish to be able to handle queries of the form \textsc{\textbf
Connected}\textbf{(v,w)}, which returns true if and only if there is a path
from $v$ to $w$ in $G$.  Also, we would like to handle queries of the form \textsc{\textbf
Connected}\textbf{($G$)}, which returns whether or not $G$ is connected.
We have already seen how to achieve $O(\lg n)$ time per operation when $G$
is a tree --- use the link-cut tree data-stracture and compare \textsc{\textbf
Findroot}\textbf{(v)} with \textsc{\textbf
Findroot}\textbf{(w)}.  There have been several other results for graphs
that are not forests:

\begin{itemize}
\item[-] $O(\lg n)$ update and query for plane graphs \cite{eppstein}.
\item[-] $O(\lg n(\lg\lg n)^3)$ update\\ $O(\lg n / \lg\lg\lg n)$ query
\cite{thorupstoc}
\item[-] $O(\lg^2 n)$ update\\$O(\lg n / \lg\lg n)$ query \cite{hlt}
\item[-] $O(\lg n\cdot x)$ update requires $\Omega(\lg n / \lg x)$ query for
$x > 1$ \cite{patdem}.
\item[-] \textbf{OPEN: } Is $o(\lg n)$ update and $O(\poly(\lg n))$
query achievable?
\item[-] \textbf{OPEN: } Is $O(\lg n)$ update and query for general graphs achievable?

\end{itemize}

For the first result, a {\em plane} graph is similar to a {\em planar}
graph, but the planar embedding is fixed (we are not allowed to change the
embedding as we add edges to the graph).  Also, all the bounds above are
amortized.  There has only been one significant worst-case bound proven,
done in \cite{egin}.  They showed that $O(\sqrt{n})$ update and $O(1)$
query is achievable.  Note that the bounds in \cite{thorupstoc,hlt,egin}
are in a sense optimal in that they achieve the trade-off bounds shown in
\cite{patdem}.

There are some modifications we can play in the types of operations we
allow to be done to our graph.  For {\em incremental} dynamic graphs, we
are never allowed to delete edges or vertices.  For this problem, we know
that we can achieve $O(\alpha(n))$ update and query by using the union-find
data structure ($\alpha(n)$ being the inverse Ackermann function).  It is
also known that achieving $\Theta(x)$ update requires $\Theta(\lg n / \lg
x)$ for $x > 1$.  This has also been achieved.  For {\em decremental}
dynamic graphs we allow deletions but no insertions.  In \cite{thorupjacm}
it was shown how to achieve $O(m\lg n + n\poly(\log n))$ for all updates,
and $O(1)$ query (where $m$ is the number of edges).

\subsection{Other Dynamic Graph Problems}

\subsubsection{Minimum Spanning Forest}
Here we would like an MST for each connected component of the graph, each
represented as a dynamic tree.  An amortized solution to this was described
in \cite{hlt} that achieves $O(\lg^4 n)$ update.  A worst case bound of
$O(\sqrt{n})$ for general graphs was achieved in \cite{egin}, and $O(\lg n)$ for
plane graphs was achieved in \cite{eitjwy}.  The problem of determining
graph bipartiteness can also be
solved by reduction to the minimum spanning forest problem.

\subsubsection{k-connectivity (vertex or edge)}
A pair of vertices $(v,w)$ is {\em k-connected} if there are $k$
vertex-disjoint (or edge-disjoint) paths from $v$ to $w$.  A graph is
k-connected if each vertex pair in the graph is k-connected.  To
determine whether the whole graph is k-connected can be solved as a
max-flow problem.  An $O(\sqrt{n}\poly(\lg n))$ algorithm for
$O(\poly(\lg n))$-edge-connectivity was shown in \cite{thorup2001}.
There have been many results for k-connectivity between a single pair
of vertices:

\begin{itemize}
\item[-] $O(\poly(\lg n))$ for $k=2$ \cite{hlt}
\item[-] $O(\lg^2 n)$ for planar, decremental graphs \cite{italiano}
\end{itemize}

The above are amortized bounds.  There have also been worst case bounds
shown in \cite{egin}:

\begin{itemize}
\item[-] $O(\sqrt{n})$ for 2-edge-connectivity
\item[-] $O(n)$ for 2-vertex-connectivity and 3-vertex-connectivity
\item[-] $O(n^{2/3})$ for 3-edge-connectivity
\item[-] $O(n\alpha(n))$ for $k=4$
\item[-] $O(n\lg n)$ for $O(1)$-edge-connectivity
\item[-] \textbf{OPEN: } Is $O(\poly(\lg n))$ achievable for $k=O(1)$?
Or perhaps $k=\poly(\lg n)$?
\end{itemize}

\subsubsection{Planarity Testing}
For this type of query, we would like to ask if inserting some edge
$e=(v,w)$ into our graph violates planarity.  It was shown in \cite{gis} that we can do
this in $O(n^{2/3})$.  In \cite{ipr} it is shown how to achieve $O(\lg^2
n)$ worst case for a fixed embedding.  La Poutr\'{e} showed an $O(\alpha(m,n)\cdot m+n)$
algorithm for a total of $m$ operations with an incremental graph in
\cite{poutre}.

Planarity is just one type of minor-closed property, i.e. by Kuratowski's
theorem we know that a graph is planar if and only if it does not contain a
subgraph isomorphic to $K_5$ or to $K_{3,3}$.  In general, it is an open
problem to test whether or not a graph has a fixed minor.  A minor of $G$
is a graph that can be obtained by a sequence of edge deletions and contractions.
A property is minor-closed if $G$ having the property implies that any
minor of $G$ must also have the property.  It was shown in \cite{robertson} that $G$ has a certain minor-closed property if
and only if it excludes a finite set of minors.

\section{Dynamic Connectivity}
Now we will focus on the explanation of the dynamic connectivity algorithm
described in \cite{hlt}.  The high level idea is that we will store a
spanning forest implemented using the Euler-tour tree.  We will have $\lg
n$ levels of spanning forests of subgraphs.

The {\em level} of an edge is some integer between $0$ and $\lg n$ that is
non-increasing over time.  $G_i$ is the subgraph consisting of edges that
are at level $i$ or less.  The $F_i$ will also be a set of spanning forests
of the $G_i$.  We will keep two invariants during the execution
of the algorithm:

\paragraph{Invariant 1: }
Every connected component of $G_i$ has at most $2^i$ vertices.

\paragraph{Invariant 2: }  $F_i = F_{\lg n}$, and $F_{\lg n}$ is a minimum
spanning forest of $G_{\lg n}$, using edge levels as weights.

We will also keep an adjacency matrix for each $G_i$.  There are three
operations we would like our data structure to support:

\begin{itemize}
\item[-] \textsc{{\textbf Insert}}\textbf{(e=(v,w))}\\
      We will set the level of $e$ to $\lg n$ and change the adjacency
      lists of $v$ and $w$.  Also, if $v$ and $w$ are in separate connected
      components of $F_{\lg n}$ then we add $e$ to $F_{\lg n}$ (we can tell
      this by calling \textsc{Findroot} on $v$ and $w$.

\item[-] \textsc{{\textbf Delete}}\textbf{(e=(v,w))}\\
      First we remove $e$ from the adjacency lists of $v$ and $w$.  Then,
      we do the following:
      \begin{itemize}
          \item[-] if $e$ is in $F_{\lg n}$
          \begin{itemize}
              \item[-] delete $e$ from $F_i$ for $i\geq$level$(e)$
              \item[-] look for a replacement edge to reconnect $v$ and $w$
              \begin{itemize}
                  \item[-] the replacement edge cannot be at a level less
                  than level$(e)$ by invariant 2 (each $F_i$ is a minimum
                  spanning forest).
                  \item[-] We will start searching for a replacement edge
                  at level$(e)$ to preserve the second invariant.
              \end{itemize}
              \item[-] for $i=$level$(e)$,...,$\lg n$:
              \begin{itemize}
                  \item[-] Let $T_v$ be the tree containing $v$ and $T_w$
                  the tree containing $w$.
                  \item[-] Relabel $v$ and $w$ so that $|T_v|\leq |T_w|$.
                  \item[-] By invariant 1, we know that $|T_v| + |T_w|\leq
                  2^i\Rightarrow |T_v|\leq 2^{i-1}$ (so we can afford to
                  push all the edges of $T_v$ down to level $i-1$, but we
                  don't do that).
                  \item[-] for each edge $e'=(x,y)$ with $x$ in $T_v$ and
                  level$(e')=i$
                  %\begin{itemize}
                      \item[] \ \ \ \ - if $y$ is in $T_w$: add $(x,y)$ to
                      $F_i,F_{i+1},...,F_{\lg n}$ and stop. %H4X0rZ
                      \item[] \ \ \ \ - else set level$(e')$ to $i-1$ and delete
                      $e'$ from $F_i$.
                  %\end{itemize}
              \end{itemize}
          \end{itemize}
      \end{itemize}
      
\item[-] \textsc{{\textbf Connected}}\textbf{(v,w)}\\
      We can modify the branching factor of nodes in $F_{\lg n}$ to
      $\Theta(\lg n)$ using a balanced binary search tree.  This gives us $O(\lg n / \lg\lg n)$
      query.  Also, using a balanced BST gives us $O(\lg^2n / \lg\lg n)$ update
      time for our other two operations.
\end{itemize}

We have to augment our Euler-tour trees to do delete so that we can test if
$|T_v| \leq |T_w|$ in $O(1)$ time (this part is standard and easy).  We
also would like to know for each node $v$ in the minimum spanning forest
$F_i$ whether or not $v$'s subtree contains any nodes incident to level-$i$
edges.  We can find the next level-$i$ edge incident to a node in $T_v$ in
$O(\lg n)$ time using successor, jumping over empty subtrees.  There is
also an $O(\lg n)$ charge to each edge's decrease in level.

The total cost of delete is then $O(\lg^2 n)$ since each inserted edge is
charged $O(\lg n)$ times.  The remaining part of delete also takes $O(\lg^2
n)$ time since we have to delete the edge from at most $O(\lg n)$ levels,
and deleting from each level takes $O(\lg n)$.

%\bibliography{mybib}
\bibliographystyle{alpha}

\begin{thebibliography}{77}

\bibitem{henzinger}
Monika Rauch Henzinger, Valerie King: Randomized dynamic graph algorithms
with polylogarithmic time per operation. STOC 1995: 519-527

\bibitem{eppstein}
David Eppstein, Giuseppe F. Italiano, Roberto Tamassia, Robert Endre 
Tarjan, Jeffery Westbrook, Moti Yung: Maintenance of a Minimum Spanning Forest in a Dynamic Plane Graph. J. Algorithms 13(1): 33-54 (1992)

\bibitem{thorupstoc}
Mikkel Thorup: Near-optimal fully-dynamic graph connectivity. STOC 2000: 343-350

\bibitem{hlt}
Jacob Holm, Kristian de Lichtenberg, Mikkel Thorup: Poly-logarithmic
deterministic fully-dynamic algorithms for connectivity, minimum spanning
tree, 2-edge, and biconnectivity. J. ACM 48(4): 723-760 (2001)

\bibitem{patdem}
Mihai Patrascu, Erik D. Demaine: Lower bounds for dynamic
connectivity. STOC 2004: 546-553

\bibitem{egin}
David Eppstein, Zvi Galil, Giuseppe F. Italiano, Amnon Nissenzweig:
Sparsification - a technique for speeding up dynamic graph
algorithms. J. ACM 44(5): 669-696 (1997)

\bibitem{thorupjacm}
Mikkel Thorup: Decremental Dynamic Connectivity. J. Algorithms 33(2):
229-243 (1999)

\bibitem{eitjwy}
David Eppstein, Giuseppe F. Italiano, Roberto Tamassia, Robert Endre
Tarjan, Jeffery Westbrook, Moti Yung: Maintenance of a Minimum Spanning
Forest in a Dynamic Plane Graph. J. Algorithms 13(1): 33-54 (1992)

\bibitem{thorup2001}
Mikkel Thorup: Fully-dynamic min-cut. STOC 2001: 224-230

\bibitem{italiano}
Dora Giammarresi, Giuseppe F. Italiano: Decremental 2- and 3-Connectivity
on Planar Graphs. Algorithmica 16(3): 263-287 (1996)

\bibitem{gis}
Zvi Galil, Giuseppe F. Italiano, Neil Sarnak: Fully Dynamic Planarity
Testing with Applications. J. ACM 46(1): 28-91 (1999)

\bibitem{ipr}
Giuseppe F. Italiano, Johannes A. La Poutr\'{e}, Monika Rauch: Fully Dynamic
Planarity Testing in Planar Embedded Graphs (Extended Abstract). ESA 1993:
212-223

\bibitem{poutre}
Johannes A. La Poutr\'{e}: Alpha-algorithms for incremental planarity
testing (preliminary version). STOC 1994: 706-715

\bibitem{robertson}
Neil Robertson, Paul D. Seymour: Graph Minors. XX. Wagner's conjecture. J. Comb. Theory, Ser. B 92(2): 325-357 (2004)

\end{thebibliography}

\end{document}