\ifx\endpart\undefined
\input macros.tex
\fi

\chapter{Dynamic Spanning Trees}\id{dynchap}%

\section{Dynamic graph algorithms}

In many applications, we often need to solve a~certain graph problem for a~sequence of graphs that
differ only a~little, so recomputing the solution for every graph from scratch would be a~waste of
time. In such cases, we usually turn our attention to \df{dynamic graph algorithms.} A~dynamic
algorithm is in fact a~data structure that remembers a~graph. It offers operations that modify the
structure of the graph and also operations that query the result of the problem for the current
state of the graph. A~typical example of a~problem of this kind is dynamic maintenance of connected
components:

\problemn{Dynamic connectivity}
Maintain an~undirected graph under a~sequence of the following operations:
\itemize\ibull
\:$\<Init>(n)$ --- Create a~graph with $n$~isolated vertices $\{1,\ldots,n\}$.\foot{%
The structure could support dynamic addition and removal of vertices, too,
but this is easy to add and infrequently used, so we will rather keep the set
of vertices fixed for clarity.}
\:$\<Insert>(G,u,v)$ --- Insert an~edge $uv$ to~$G$ and return its unique
identifier. This assumes that the edge did not exist yet.
\:$\<Delete>(G,e)$ --- Delete an~edge specified by its identifier from~$G$.
\:$\<Connected>(G,u,v)$ --- Test if vertices $u$ and~$v$ are in the same connected component of~$G$.
\endlist

\para
We have already encountered a~special case of dynamic connectivity when implementing the
Kruskal's algorithm in Section \ref{classalg}. At that time, we did not need to delete
any edges from the graph, which makes the problem substantially easier. This special
case is customarily called an~\df{incremental} or \df{semidynamic} graph algorithm.
We mentioned the Disjoint Set Union data structure of Tarjan and van Leeuwen (Theorem \ref{dfu})
which can be used for that: Connected components are represented by equivalence classes.
Queries on connectedness translate to \<Find>, edge insertions to \<Find>
followed by \<Union> if the new edge joins two different components. This way,
a~sequence of $m$~operations starting with an~empty graph on $n$~vertices is
processed in time $\O(n+m\timesalpha(m,n))$ and this holds even for the Pointer
Machine. Fredman and Saks \cite{fredman:cellprobe} have proven a~matching lower
bound in the cell-probe model which is stronger than RAM with $\O(\log n)$-bit
words.

In this chapter, we will focus on the dynamic version of the minimum spanning forest.
This problem seems to be intimately related to the dynamic connectivity. Indeed, all known
algorithms for dynamic connectivity maintain some sort of a~spanning forest. For example, in the
incremental algorithm we have just mentioned, this forest is formed by the edges that have
triggered the \<Union>s. This suggests that a~dynamic MSF algorithm could be obtained by modifying
the mechanism to keep the forest minimum. This will indeed turn out to be true, although we cannot
be sure that it will lead to the most efficient solution possible --- as of now, the known lower
bounds are very far.

Incremental MST will be easy to achieve even in the few pages of this section, but making it fully
dynamic will require more effort, so we will review some of the required building blocks before
going into that.

We however have to answer one important question first: What should be the output of
our MSF data structure? Adding an~operation that would return the MSF of the current
graph is of course possible, but somewhat impractical as this operation has to
spend $\Omega(n)$ time on the mere writing of its output. A~better way seems to
be making the \<Insert> and \<Delete> operations report the list of modifications
of the MSF implied by the change in the graph.

Let us see what happens when we \<Insert> an~edge~$e$ to a~graph~$G$ with its minimum spanning
forest~$F$, obtaining a~new graph~$G'$ with its MSF~$F'$. If $e$~connects two components of~$G$ (and
therefore also of~$F$), we have to add~$e$ to~$F$. Otherwise, one of the following cases happens:
Either $e$~is $F$-heavy and so the forest~$F$ is also the MSF of the new graph. Or it is $F$-light
and we have to modify~$F$ by exchanging the heaviest edge~$f$ of the path $F[e]$ with~$e$.

Correctness of the former case follows immediately from the Minimality Theorem (\ref{mstthm}),
because any $F'$-light would be also $F$-light, which is impossible as $F$~was
minimum. In the latter case, the edge~$f$ is not contained in~$F'$ because it is the heaviest
on the cycle $F[e]+e$ (by the Red lemma, \ref{redlemma}). We can now use the Blue lemma
(\ref{bluelemma}) to prove that it should be replaced with~$e$. Consider the tree~$T$
of~$F$ that contains both endpoints of the edge~$e$. When we remove~$f$ from~$F$, this tree falls
apart to two components $T_1$ and~$T_2$. The edge~$f$ was the lightest in the cut~$\delta_G(T_1)$
and $e$~is lighter than~$f$, so $e$~is the lightest in~$\delta_{G'}(T_1)$ and hence $e\in F'$.

A~\<Delete> of an~edge that is not contained in~$F$ does not change~$F$. When we delete
an~MSF edge, we have to reconnect~$F$ by choosing the lightest edge of the cut separating
the new components (again the Blue lemma in action). If there is no such
replacement edge, we have deleted a~bridge, so the MSF has to remain
disconnected.

The idea of reporting differences in the MSF indeed works very well. We can summarize
what we have shown in the following lemma and use it to define the dynamic MSF.

\lemma
An~\<Insert> or \<Delete> of an~edge in~$G$ causes at most one edge addition, edge
removal or edge exchange in $\msf(G)$.

\problemn{Dynamic minimum spanning forest}
Maintain an~undirected graph with distinct weights on edges (drawn from a~totally ordered set)
and its minimum spanning forest under a~sequence of the following operations:
\itemize\ibull
\:$\<Init>(n)$ --- Create a~graph with $n$~isolated vertices $\{1,\ldots,n\}$.
\:$\<Insert>(G,u,v)$ --- Insert an~edge $uv$ to~$G$. Return its unique
	identifier and the list of additions and deletions of edges in $\msf(G)$.
\:$\<Delete>(G,e)$ --- Delete an~edge specified by its identifier from~$G$.
	Return the list of additions and deletions of edges in $\msf(G)$.
\endlist

\paran{Incremental MSF}%
To obtain an~incremental MSF algorithm, we need to keep the forest in a~data structure that
supports search for the heaviest edge on the path connecting a~given pair
of vertices. This can be handled efficiently by the Link-Cut trees of Sleator and Tarjan:

\thmn{Link-Cut Trees, Sleator and Tarjan \cite{sleator:trees}}\id{sletar}%
There is a~data structure that represents a~forest of rooted trees on~$n$ vertices.
Each edge of the forest has a~weight drawn from a~totally ordered set. The structure
supports the following operations in time $\O(\log n)$ amortized:\foot{%
The Link-Cut trees can offer a~plethora of other operations, but we do not mention them
as they are not needed in our application.}
\itemize\ibull
\:$\<Parent>(v)$ --- Return the parent of~$v$ in its tree or \<null> if $v$~is a~root.
\:$\<Root>(v)$ --- Return the root of the tree containing~$v$.
\:$\<Weight>(v)$ --- Return the weight of the edge $(\<Parent(v)>,v)$.
\:$\<PathMax>(v)$ --- Return the vertex~$w$ closest to $\<Root>(v)$ such that the edge
	$(\<Parent>(w),w)$ is the heaviest of those on the path from the root to~$v$.
	If more edges have the maximum weight, break the tie arbitrarily.
	If there is no such edge ($v$~is the root itself), return \<null>.
\:$\<Link>(u,v,x)$ --- Connect the trees containing $u$ and~$v$ by an~edge $(u,v)$ of
	weight~$x$. Assumes that $v~$is a tree root and $u$~lies in a~different tree.
\:$\<Cut>(v)$ --- Split the tree containing the non-root vertex $v$ to two trees by
	removing the edge $(\<Parent>(v),v)$. Returns the weight of this edge.
\:$\<Evert>(v)$ --- Modify the orientations of edges to make~$v$ the root of its tree.
\endlist

%% \>Additionally, all edges on the path from~$v$ to $\<Root>(v)$ can be enumerated in
%% time $\O(\ell + \log n)$, where $\ell$~is the length of that path. This operation
%% (and also the path itself) is called $\<Path>(v)$.
%% 
%% \>If the weights are real numbers (or in general an~arbitrary group), the $\O(\log n)$
%% operations also include:
%% 
%% \itemize\ibull
%% \:$\<PathWeight>(v)$ --- Return the sum of the weights on $\<Path>(v)$.
%% \:$\<PathUpdate>(v,x)$ --- Add~$x$ to the weights of all edges on $\<Path>(v)$.
%% \endlist

\proof
See \cite{sleator:trees}.
\qed

Once we have this structure, we can turn our ideas on updating of the MSF to
an~incremental algorithm:

\algn{\<Insert> in an~incremental MSF}
\algo
\algin A~graph~$G$ with its MSF $F$ represented as a~Link-Cut forest, an~edge~$uv$
with weight~$w$ to be inserted.
\:$\<Evert>(u)$. \cmt{$u$~is now the root of its tree.}
\:If $\<Root>(v) \ne u$: \cmt{$u$~and~$v$ lie in different trees.}
\::$\<Link>(v,u,w)$. \cmt{Connect the trees.}
\::Return ``$uv$ added''.
\:Otherwise: \cmt{both are in the same tree}
\::$y\=\<PathMax>(v)$.
\::$x\=\<Parent>(y)$.  \cmt{Edge~$xy$ is the heaviest on $F[uv]$.}
\::If $\<Weight>(y) > w$: \cmt{We have to exchange~$xy$ with~$uv$.}
\:::$\<Cut>(y)$, $\<Evert>(v)$, $\<Link>(u,v,w)$.
\:::Return ``$uv$~added, $xy$~removed''.
\::Otherwise return ``no changes''.
\algout The list of changes in~$F$.
\endalgo

\thmn{Incremental MSF}
When only edge insertions are allowed, the dynamic MSF can be maintained in time $\O(\log n)$
amortized per operation.

\proof
Every \<Insert> performs $\O(1)$ operations on the Link-Cut forest, which take
$\O(\log n)$ each by Theorem \ref{sletar}.
\qed

\rem
We can easily extend the semidynamic MSF algorithm to allow an~operation commonly called
\<Backtrack> --- removal of the most recently inserted edge. It is sufficient to keep the
history of all MSF changes in a~stack and reverse the most recent change upon backtrack.

What are the obstacles to making the structure fully dynamic?
Deletion of edges that do not belong to the MSF is trivial (we do not
need to change anything) and so is deletion of bridges (we just remove the bridge
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 belonging to the MSF is deleted.

This very problem also has to be solved by algorithms for fully dynamic connectivity,
we will take a~look at 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 Link-Cut
trees, but it is much simpler and instead of path operations it offers efficient operations on
subtrees. It is also possible to attach auxiliary data to vertices and edges of the original tree.

\defn\id{eulseq}%
Let~$T$ be a~rooted tree. We will call a~sequence of vertices of~$T$ its \df{Eulerian Tour sequence (ET-sequence)}
if it lists the vertices visited by the depth-first traversal of~$T$.
More precisely, it can be generated by the following procedure $\<ET>(v)$
when it is invoked on the root of the tree:
\algo
\:Record~$v$ in the sequence.
\:For each son~$w$ of~$v$:
\::Call $\<ET>(w)$.
\::Record~$w$.
\endalgo
\>A~single tree can have multiple ET-sequences, corresponding to different orders in which the
sons can be enumerated in step~2.

In every ET-tree, one of the occurrences of each vertex is defined as its \df{active occurrence} and
it will be used to store auxiliary data associated with that vertex.

\obs
An~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 an~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
$AuvBvuC$ (with no~$u$ nor~$v$ in~$B$) splits to two sequences $AuC$ and $vBv$.
If there was only a~single occurrence of~$v$, then $v$~was a~leaf and thus the sequence
transforms from $AuvuC$ to $AuC$ and $v$~alone.

\em{Changing the root} of the tree~$T$ from~$v$ to~$w$ changes its ET-sequence from $vAwBwCv$ 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 $wv$. Note that this works regardless of the possible presence of~$w$ inside~$B$.

\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 $vwv$.

\float{\valign{\vfil#\vfil\cr
\hbox{\epsfbox{pic/ettree.eps}}\cr
\noalign{\qquad\quad}
  \halign{#\hfil\cr
    $T_1: 0121034546474308980$,~~$T_2: aba$. \cr
    $T_1-34: 01210308980, 4546474$. \cr
    $T_1\hbox{~rooted at~3}: 3454647430898012103$. \cr
    $T_1+0a+T_2$: $0121034546474308980aba0$. \cr
  }\cr
}}{Trees and their ET-sequences}

If any of the occurrences that we have removed from the sequence was active, 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 store the ET-sequences as $(a,b)$-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 $\O(b\log_a n)$ in the worst case.

We will use the ET-trees to maintain a~spanning forest of the dynamic graph. The auxiliary data of
each vertex will hold a~list of edges incident with the given vertex, that do not lie in the
forest. Such edges are usually called the \df{non-tree edges.}

\defn
\df{Eulerian Tour trees (ET-trees)} are a~data structure that represents a~forest of trees and a~set of non-tree
edges associated with the vertices of the forest. To avoid confusion, we will distinguish between
\df{original} vertices and edges (of the given trees) and the vertices and edges of the
data structure. The structure consists of:
\itemize\ibull
\:A~collection of $(a,b)$-trees of some fixed parameters $a$ and~$b$.
	Each such tree corresponds to one of the original trees~$T$. Its
	leaves (in the usual tree order) correspond to the elements
	of an~ET-sequence for~$T$. Each two consecutive leaves $u$ and~$v$ are separated
	by a~unique key stored in an~internal vertex of the $(a,b)$-tree. This key is used to represent
	the original edge~$uv$. Each original edge is therefore kept in both its orientations.
\:Mappings \<act>, \<edge> and \<twin>:
	\itemize\icirc
		\:$\<act>(v)$ maps each original vertex to the leaf containing its active occurrence;
		\:$\<edge>(e)$ of an~original edge~$e$ is one of the internal keys representing~it;
		\:$\<twin>(k)$ pairs an~internal key~$k$ with the other internal key of the same original edge.
	\endlist
\:A~list of non-tree edges placed in each leaf. The lists are allowed to be non-empty only
	in the leaves that represent active occurrences of original vertices.
\:Boolean \df{markers} in the internal vertices that signal presence of a~non-tree
	edge anywhere in the subtree rooted at the internal vertex.
\:Counters $\<leaves>(v)$ that contain the number of leaves in the subtree rooted at~$v$.
\endlist

\defn
The ET-trees support the following operations on the original trees:
\itemize\ibull
\:\<Create> --- Create a~single-vertex tree.
\:$\<Link>(u,v)$ --- Join two different trees by an~edge~$uv$ and return a~unique identifier
	of this edge.
\:$\<Cut>(e)$ --- Split a~tree by removing the edge~$e$ given by its identifier.
\:$\<Connected>(u,v)$ --- Test if the vertices $u$ and~$v$ lie in the same tree.
\:$\<Size>(v)$ --- Return the number of vertices in the tree containing the vertex~$v$.
\:$\<InsertNontree>(v,e)$ --- Add a~non-tree edge~$e$ to the list at~$v$ and return a~unique
	identifier of this edge.
\:$\<DeleteNontree>(e)$ --- Delete a~non-tree edge~$e$ given by its identifier.
\:$\<ScanNontree>(v)$ --- Return a~list of non-tree edges associated with the vertices
	of the $v$'s tree.
\endlist

\impl
We will implement the operations on the ET-trees by translating the intended changes of the
ET-sequences to operations on the $(a,b)$-trees. The role of identifiers of the original vertices
and edges will be of course played by pointers to the respective leaves and internal keys of
the $(a,b)$-trees.

\<Cut> of an~edge splits the $(a,b)$-tree at both internal keys representing the given edge
and joins them back in the different order.

\<Link> of two trees can be accomplished by making both vertices the roots of their trees first
and joining the roots by an~edge afterwards. Re-rooting involves splits and joins of $(a,b)$-trees.
As we can split at any occurrence of the new root vertex, we will use the active occurrence
which we remember. Linking of the roots is translated to joining of the $(a,b)$-trees.

\<Connected> follows parent pointers from both $u$ and~$v$ to the roots of their trees.
Then it checks if the roots are equal.

\<Size> finds the root and returns its counter.

\<InsertNontree> finds the leaf $\<act>(v)$ containing the list of $v$'s non-tree edges
and inserts the new edge there. The returned identifier will consist from the pointer to
the edge and the vertex in whose list it is stored. Then we have to recalculate the markers
on the path from $\<act>(v)$ to the root. \<DeleteNontree> is analogous.

Whenever any other operation changes a~vertex of the tree, it will also update its marker and
counter and, if necessary, the markers and counters on the path to the root.

\<ScanNontree> traverses the tree recursively from the root, but it does not enter the
subtrees whose roots are not marked.

Analysis of time complexity of the operations is now straightforward:

\thmn{Eulerian Tour trees, Henzinger and Rauch \cite{henzinger:randdyn}}\id{etthm}%
The ET-trees perform the operations \<Link> and \<Cut> in time $\O(a\log_a n)$, \<Create>
in $\O(1)$, \<Connected>, \<Size>, \<InsertNontree>, and \<DeleteNontree> in $\O(\log_a n)$, and
\<ScanNontree> in $\O(a\log_a n)$ per edge reported. Here $n$~is the number of vertices
in the original forest and $a\ge 2$ is an~arbitrary constant.

\proof
We set $b=2a$. Our implementation performs $\O(1)$ operations on the $(a,b)$-trees
per operation on the ET-tree, plus $\O(1)$ other work. We apply the standard theorems
on the complexity of $(a,b)$-trees \cite{clrs}.
\qed

\examplen{Connectivity acceleration}\id{accel}%
In most cases, the ET-trees are used with $a$~constant, but sometimes choosing~$a$ as a~function
of~$n$ can also have its beauty. Suppose that there is a~data structure which maintains an~arbitrary
spanning forest of a~dynamic graph. Suppose also that the structure works in time $\O(\log^k n)$
per operation and that it reports $\O(1)$ changes in the spanning forest for every change
in the graph. If we keep the spanning forest in ET-trees with $a=\log n$, the updates of the
data structure cost an~additional $\O(\log^2 n / \log\log n)$, but connectivity queries accelerate to $\O(\log
n/\log\log n)$.

\paran{ET-trees with weights}
In some cases, we will also need a~representation of weighted graphs and enumerate the non-tree
edges in order of their increasing weights (in fact, it will be sufficient to find the
lightest one, remove it and iterate). This can be handled by a~minute modification of the
ET-trees.

The tree edges will remember their weight in the corresponding internal keys of the ET-tree.
We replace each list of non-tree edges by an~$(a,b)$-tree keeping the edges sorted by weight.
We also store the minimum element of that tree separately, so that it can be accessed in constant
time. The boolean \em{marker} will then become the minimum weight of a~non-tree edge attached to the
particular subtree, which can be recalculated as easy as the markers can. Searching for the
lightest non-tree edge then just follows the modified markers.

The time complexities of all operations therefore remain the same, with a~possible
exception of the operations on non-tree edges, to which we have added the burden of
updating the new $(a,b)$-trees. This however consists of $\O(1)$ updates per a~single
call to \<InsertNontree> or \<DeleteNontree>, which takes $\O(a\log_a n)$ time only.
We can therefore conclude:

\corn{Weighted ET-trees}\id{wtet}%
The time bounds in Theorem \ref{etthm} hold for the weighted ET-trees, too.


%--------------------------------------------------------------------------------

\section{Dynamic connectivity}

The fully dynamic connectivity problem has a~long and rich history. In the 1980's, Frederickson \cite{frederickson:dynamic}
has used his topological trees to construct a~dynamic connectivity algorithm of complexity $\O(\sqrt m)$ per update and
$\O(1)$ per query. Eppstein et al.~\cite{eppstein:sparsify} have introduced a~sparsification technique which can bring the
updates down to $\O(\sqrt n)$. Later, several different algorithms with complexity on the order of $n^\varepsilon$
were presented by Henzinger and King \cite{henzinger:mst} and also by Mare\v{s} \cite{mares:dga}.
A~polylogarithmic time bound was first reached by the randomized algorithm of Henzinger and King \cite{henzinger:randdyn}.
The best result known as of now is the $\O(\log^2 n)$ time deterministic algorithm by Holm,
de~Lichtenberg and Thorup \cite{holm:polylog}, which will we describe in this section.

The algorithm will maintain a~spanning forest~$F$ of the current graph~$G$, represented by an~ET-tree
which will be used to answer connectivity queries. The edges of~$G\setminus F$ will be stored as~non-tree
edges in the ET-tree. Hence, an~insertion of an~edge to~$G$ either adds it to~$F$ or inserts it as non-tree.
Deletions of non-tree edges are also easy, but when a~tree edge is deleted, we have to search for its
replacement among the non-tree edges.

To govern the search in an~efficient way, we will associate each edge~$e$ with a~level $\ell(e) \le
L = \lfloor\log_2 n\rfloor$. For each level~$i$, we will use~$F_i$ to denote the subforest
of~$F$ containing edges of level at least~$i$. Therefore $F=F_0 \supseteq F_1 \supseteq \ldots \supseteq F_L$.
We will maintain the following \em{invariants:}

{\narrower
\def\iinv{{\bo I\the\itemcount~}}
\numlist\iinv
\:$F$~is the maximum spanning forest of~$G$ with respect to the levels. (In other words,
if $uv$ is a~non-tree edge, then $u$ and~$v$ are connected in~$F_{\ell(uv)}$.)
\:For each~$i$, the components of~$F_i$ have at most $\lfloor n/2^i \rfloor$ vertices each.
(This implies that it does not make sense to define~$F_i$ for $i>L$, because it would be empty
anyway.)
\endlist
}

At the beginning, the graph contains no edges, so both invariants are trivially
satisfied. Newly inserted edges can enter level~0, which cannot break I1 nor~I2.

When we delete a~tree edge at level~$\ell$, we split a~tree~$T$ of~$F_\ell$ to two
trees $T_1$ and~$T_2$. Without loss of generality, let us assume that $T_1$ is the
smaller one. We will try to find the replacement edge of the highest possible
level that connects the spanning tree back. From I1, we know that such an~edge cannot belong to
a~level greater than~$\ell$, so we start looking for it at level~$\ell$. According
to~I2, the tree~$T$ had at most $\lfloor n/2^\ell\rfloor$ vertices, so $T_1$ has
at most $\lfloor n/2^{\ell+1} \rfloor$ of them. Thus we can increase the levels
of all edges of~$T_1$ without violating either invariant.

We now start enumerating the non-tree edges incident with~$T_1$. Each such edge
is either local to~$T_1$ or it joins $T_1$ with~$T_2$. We will therefore check each edge
whether its other endpoint lies in~$T_2$ and if it does, we have found the replacement
edge, so we insert it to~$F_\ell$ and stop. Otherwise we move the edge one level up. (This
will be the grist for the mill of our amortization argument: We can charge most of the work at level
increases and we know that the level of each edge can reach at most~$L$.)

If the non-tree edges at level~$\ell$ are exhausted, we try the same in the next
lower level and so on. If there is no replacement edge at level~0, the tree~$T$
remains disconnected.

\impl
For each level, we will use a~separate ET-tree ${\cal E}_\ell$ with~$a$ set to~2,
which will represent the forest~$F_i$ and the non-tree edges at that particular level.
Besides operations on the non-tree edges, we also need to find the tree edges of level~$\ell$
when we want to bring them one level up. This can be accomplished either by modifying the ET-trees
to attach two lists of edges attached to vertices instead of one, or by using a~second ET-tree.

\algn{Insertion of an~edge}
\algo
\algin An~edge $uv$ to insert.
\:$\ell(uv) \= 0$.
\:Ask the ET-tree ${\cal E}_0$ if $u$ and~$v$ are in the same component. If they are:
\::Add $uv$ to the list of non-tree edges in ${\cal E}_0$ at both $u$ and~$v$.
\:Otherwise:
\::Add $uv$ to~$F_0$.
\endalgo

\algn{Deletion of an~edge}
\algo
\algin An~edge $uv$ to delete.
\:$\ell \= \ell(uv)$.
\:If $uv$ is a~non-tree edge:
\::Remove $uv$ from the lists of non-tree edges at both $u$ and~$v$ in~${\cal E}_{\ell}$.
\:Otherwise:
\::Remove $uv$ from~$F_\ell$ and hence also from $F_0,\ldots,F_{\ell-1}$.
\::Call $\<Replace>(uv,\ell)$ to get the replacement edge~$f$.
\::Insert $f$ to~$F_0,\ldots,F_{\ell(f)}$.
\endalgo

\algn{$\<Replace>(uv,i)$ -- Search for replacement for edge~$uv$ at level~$i$}
\algo
\algin An~edge~$uv$ to replace and a~level~$i$ such that there is no replacement
at levels greater than~$i$.
\:Let $T_1$ and~$T_2$ be the trees in~$F_i$ containing $u$ and~$v$ respectively.
\:If $n(T_1) > n(T_2)$, swap $T_1$ with~$T_2$.
\:Find all level~$i$ edges in~$T_1$ using ${\cal E}_i$ and move them to level~$i+1$.
\:Enumerate non-tree edges incident with vertices of~$T_1$ and stored in ${\cal E}_i$.
  For each edge~$xy$, $x\in T_1$, do:
\::If $y\in T_2$, remove~$xy$ from~${\cal E}_i$ and return it to the caller.
\::Otherwise increase $\ell(xy)$ by one.
  \hfil\break
  This includes deleting~$xy$ from~${\cal E}_i$ and inserting it to~${\cal E}_{i+1}$.
\:If $i>0$, call $\<Replace>(xy,i-1)$.
\:Otherwise return \<null>.
\algout The replacement edge.
\endalgo

\>As promised, time complexity will be analysed by amortization on the levels.

\thmn{Fully dynamic connectivity, Holm et al.~\cite{holm:polylog}}\id{dyncon}%
Dynamic connectivity can be maintained in time $\O(\log^2 n)$ amortized per
\<Insert> and \<Delete> and in time $\O(\log n/\log\log n)$ per \<Connected>
in the worst case.

\proof
The direct cost of an~\<Insert> is $\O(\log n)$ for the operations on the ET-trees
(by Theorem \ref{etthm}). We will also have the insertion pre-pay all level increases of the new
edge. Since the levels never decrease, each edge can be brought a~level up at most
$L=\lfloor\log n\rfloor$ times. Every increase costs $\O(\log n)$ on the ET-tree
operations, so we pay $\O(\log^2 n)$ for all of them.

A~\<Delete> costs $\O(\log^2 n)$ directly, as we might have to update all~$L$
ET-trees. Additionally, we call \<Replace> up to $L$ times. The initialization of
\<Replace> costs $\O(\log n)$ per call, the rest is paid for by the edge level
increases.

To bring the complexity of the operation \<Connected> from $\O(\log n)$ down to $\O(\log n/\log\log n)$,
we apply the trick from Example \ref{accel} and store~$F_0$ in a~ET-tree with $a=\max(\lfloor\log n\rfloor,2)$.
This does not hurt the complexity of insertions and deletions, but allows for faster queries.
\qed

\rem\id{dclower}%
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 direct indexing of arrays.) So far, it is not known how to extend this algorithm
to fit our needs, so we omit the details.

%--------------------------------------------------------------------------------

\section{Dynamic spanning forests}\id{dynmstsect}%

Let us turn our attention back to the dynamic MSF now.
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
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
the same path. They have modified their dynamic connectivity algorithm to solve the decremental MSF
in $\O(\log^2 n)$ and obtained the fully dynamic MSF working in $\O(\log^4 n)$ per operation.

\paran{Decremental MSF}%
Turning the algorithm from the previous section to the decremental MSF requires only two
changes: First, we have to start with the forest~$F$ equal to the MSF of the initial
graph. As we require to pay $\O(\log^2 n)$ for every insertion, we can use almost arbitrary
MSF algorithm to find~$F$. Second, when we search for an~replacement edge, we need to pick
the lightest possible choice. We will therefore use the weighted version of the ET-trees (Corollary \ref{wtet})
and scan the lightest non-tree edge incident with the examined tree first. We must ensure
that the lower levels cannot contain a~lighter replacement edge, but fortunately the
light edges tend to ``bubble up'' in the hierarchy of levels. This can be formalized as
the following invariant:

{\narrower
\def\iinv{{\bo I\the\itemcount~}}
\numlist\iinv
\itemcount=2
\:On every cycle, the heaviest edge has the smallest level.
\endlist
}

\>This immediately implies that we always select the right replacement edge:

\lemma\id{msfrepl}%
Let $F$~be the minimum spanning forest and $e$ any its edge. Then among all replacement
edges for~$e$, the lightest one is at the maximum level.

\proof
Let us consider any two edges $f_1$ and~$f_2$ replacing~$e$. By minimality of~$F$ and the Cycle
rule (Lemma \ref{redlemma}), each $f_i$ is the heaviest edge on the cycle~$C_i = F[f_i] + f_i$.
In a~moment, we will show that the symmetric difference~$C$ of these two cycles is again a~cycle.
This implies that if $f_1$ is heavier than~$f_2$, then $f_1$~is the heaviest edge on~$C$, so
$\ell(f_1) \le \ell(f_2)$ by I3. Therefore the lightest of all replacement edges must have
the maximum level.

Why is~$C$ a~cycle:
Let $F^a$ and~$F^b$ be the trees of~$F-e$ in which the endpoints of~$e$ lie, and for
every edge~$g$ going between $F^a$ and~$F^b$ let $g^a$ and~$g^b$ be its respective endpoints.
We know that $C_i$ consists of the path $F[f_i^a,e^a]$, the edge~$e$, the path $F[e^b,f_i^b]$,
and the edge~$f_i$. Thus~$C$ must contain the paths $F[f_1^a,f_2^a]$ and $F[f_1^b,f_2^b]$ and
the edges $f_1$ and~$f_2$, which together form a~simple cycle.
\qed

We now have to make sure that the additional invariant is indeed observed:

\lemma\id{ithree}%
After every operation, the invariant I3 is satisfied.

\proof
When the structure is freshly initialized, I3 is obviously satisfied, as all edges
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 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 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. 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 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

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.

\paran{Fully dynamic MSF}%
The decremental MSF algorithm can be turned to a~fully dynamic one by a~blackbox
reduction whose properties are summarized in the following theorem:

\thmn{MSF dynamization, Holm et al.~\cite{holm:polylog}}
Suppose that we have a~decremental MSF algorithm with the following properties:
\numlist\ndotted
\:For any $a$,~$b$, it can be initialized on a~graph with~$a$ vertices and~$b$ edges.
\:Then it executes an~arbitrary sequence of deletions in time $\O(b\cdot t(a,b))$, where~$t$ is a~non-decreasing function.
\endlist
\>Then there exists a~fully dynamic MSF algorithm for a~graph on $n$~vertices, starting
with no edges, that performs $m$~insertions and deletions in amortized time:
$$
\O\left( \log^3 n + \sum_{i=1}^{\log m} \sum_{j=1}^i \; t(\min(n,2^j), 2^j) \right) \hbox{\quad per operation.}
$$

\proofsketch
The reduction is very technical, but its essence is the following: We maintain a~logarithmic number
of decremental structures $A_0,\ldots,A_{\lfloor\log n\rfloor}$ of exponentially increasing sizes. Every non-tree
edge is contained in exactly one~$A_i$, tree edges can belong to multiple structures.

When an~edge is inserted, we union it with some of the $A_i$'s, build a~new decremental structure
and amortize the cost of the build over the insertions. Deletes of non-tree edges are trivial.
Delete of a~non-tree edge is performed on all $A_i$'s containing it and the replacement edge is
sought among the replacement edges found in these structures. The unused replacement edges then have
to be reinserted back to the structure.

The original reduction of Henzinger et al.~\cite{henzinger:twoec} handles these reinserts by a~mechanism of batch insertions
supported by their decremental structure, which is not available in our case. Holm et al.~have
replaced it by a~system of auxiliary edges inserted at various places in the structure.
We refer to the article \cite{holm:polylog} for details.
\qed

\corn{Fully dynamic MSF}\id{dynmsfcorr}%
There is a~fully dynamic MSF algorithm that works in time $\O(\log^4 n)$ amortized
per operation for graphs on $n$~vertices.

\proof
Apply the reduction from the previous theorem to the decremental algorithm we have
developed. This results in an~algorithm of amortized complexity $\O(\log^4\max(m,n))$ where~$m$
is the number of operations performed. This could exceed $\O(\log^4 n)$ if
$m$~is very large, but we can rebuild the whole structure after $n^2$~operations,
which brings $\log m$ down to $\O(\log n)$. The $\O(n^2\log^4 n)$ cost of the
rebuild then incurs only $\O(\log^4 n)$ additional cost on each operation.
\qed

\rem
The limitation of MSF structures based on the Holm's algorithm for connectivity
to only edge deletions seems to be unavoidable. The invariant I3 could be easily
broken for many cycles at once whenever a~very light non-tree edge is inserted.
We could try increasing the level of the newly inserted edge, but we would quite
likely hit I1 before we managed to skip the levels of all the heaviest edges on the
particular cycles.

On the other hand, if we decided to drop I3, we would encounter different problems. The ET-trees can
bring the lightest non-tree incident with the current tree~$T_1$, but the lightest replacement edge
could also be located in the super-trees of~$T_1$ at the lower levels, which are too large to scan
and both I1 and I2 prevent us from charging the time on increasing levels there.

An~interesting special case in which insertions are possible is when all non-tree
edges have the same weight. This leads to the following algorithm for dynamic MSF
on~graphs with a~small set of allowed edge weights. It is based on an~idea similar
to the $\O(k\log^3 n)$ algorithm of Henzinger and King \cite{henzinger:randdyn},
but adapted to use the better results on dynamic connectivity we have at hand.

\paran{Dynamic MSF with limited edge weights}%
Let us assume for a~while that our graph has edges of only two different weights (let us say
1~and~2). We will forget our rule that all edge weights are distinct for a~moment and we recall
the observation in \ref{multiweight} that the basic structural properties of
the MST's from Section \ref{mstbasics} still hold.

We split the graph~$G$ to two subgraphs~$G_1$ and~$G_2$ according to the edge
weights. We use one instance~$\C_1$ of the dynamic connectivity algorithm maintaining
an~arbitrary spanning forest~$F_1$ of~$G_1$, which is obviously minimum. Then we add
another instance~$\C_2$ to maintain a~spanning forest~$F_2$ of the graph $G_2\cup F_1$
such that all edges of~$F_1$ are forced to be in~$F_2$. Obviously, $F_2$~is the
MSF of the whole graph~$G$ --- if any edge of~$F_1$ were not contained in~$\msf(G)$,
we could use the standard exchange argument to create an~even lighter spanning tree.\foot{This
is of course the Blue lemma in action, but we have to be careful as we did not have proven it
for graphs with multiple edges of the same weight.}

When a~weight~2 edge is inserted to~$G$, we insert it to~$\C_2$ and it either enters~$F_2$
or becomes a~non-tree edge. Similarly, deletion of a~weight~2 edge is a~pure deletion in~$\C_2$,
because such edges can be replaced only by other weight~2 edges.

Insertion of edges of weight~1 needs more attention: We insert the edge to~$\C_1$. If~$F_1$
stays unchanged, we are done. If the new edge enters~$F_1$, we use Sleator-Tarjan trees
kept for~$F_2$ to check if the new edge covers some tree edge of weight~2. If this is not
the case, we insert the new edge to~$\C_2$ and hence also to~$F_2$ and we are done.
Otherwise we exchange one of the covered weight~2 edges~$f$ for~$e$ in~$\C_2$. We note
that~$e$ can inherit the level of~$f$ and $f$~can become a~non-tree edge without
changing its level. This adjustment can be performed in time $\O(\log^2 n)$, including
paying for the future level increases of the new edge.

Deletion of weight~1 edges is once more straightforward. We delete the edge from~$\C_1$.
If it has no replacement, we delete it from~$\C_2$ as well. If it has a~replacement,
we delete the edge from~$\C_2$ and insert the replacement on its place as described
above. We observe than this pair of operations causes an~insertion, deletion or
a~replacement in~$\C_2$.

This way, we can handle every insertion and deletion in time $\O(\log^2 n)$ amortized.
This construction can be iterated in an~obvious way: if we have $k$~distinct edge weights,
we build $k$~connectivity structures $\C_1,\ldots,\C_k$. The structure~$\C_i$ contains edges of
weight~$i$ together with the MSF edges from~$\C_{i-1}$. Bounding the time complexity is then easy:

\thmn{MSF with limited edge weights}
There is a~fully dynamic MSF algorithm that works in time $\O(k\cdot\log^2 n)$ amortized
per operation for graphs on $n$~vertices with only $k$~distinct edge weights allowed.

\proof
A~change in the graph~$G$ involving an~edge of weight~$w$ causes a~change in~$\C_w$,
which can propagate to~$\C_{w+1}$ and so on, possibly up to~$\C_k$. In each~$\C_i$,
we spend time $\O(\log^2 n)$ by updating the connectivity structure according to
Theorem \ref{dyncon} and $\O(\log n)$ on operations with the Sleator-Tarjan trees
by Theorem \ref{sletar}.
\qed

%--------------------------------------------------------------------------------

\section{Almost minimum trees}\id{kbestsect}%

In some situations, finding the single minimum spanning tree is not enough and we are interested
in the $K$~lightest spanning trees, usually for some small value of~$K$. Katoh, Ibaraki
and Mine \cite{katoh:kmin} have given an~algorithm of time complexity $\O(m\log\beta(m,n) + Km)$,
building on the MST algorithm of Gabow et al.~\cite{gabow:mst}.
Subsequently, Eppstein \cite{eppstein:ksmallest} has discovered an~elegant preprocessing step which allows to reduce
the running time to $\O(m\log\beta(m,n) + \min(K^2,Km))$ by eliminating edges
which are either present in all $K$ trees or in none of them.
We will show a~variant of their algorithm based on the MST verification
procedure of Section~\ref{verifysect}.

In this section, we will require the edge weights to be real numbers (or integers), because
comparisons are certainly not enough to determine the second best spanning tree. We will
assume that our computation model is able to add, subtract and compare the edge weights
in constant time.

Let us focus on finding the second best spanning tree first.

\paran{Second best spanning tree}%
Suppose that we have a~weighted graph~$G$ and a~sequence $T_1,\ldots,T_z$ of all its spanning
trees. Also suppose that the weights of these spanning trees are distinct and that the sequence
is ordered by weight, i.e., $w(T_1) < \ldots < w(T_z)$ and $T_1 = \mst(G)$. Let us observe
that each tree is similar to at least one of its predecessors:

\lemma\id{kbl}%
For each $i>1$ there exists $j<i$ such that $T_i$ and~$T_j$ differ by a~single edge exchange.

\proof
We know from the Monotone exchange lemma (\ref{monoxchg}) that $T_1$ can be transformed
to~$T_i$ by a~sequence of edge exchanges which never decreases tree weight. The last
exchange in this sequence therefore obtains~$T_i$ from a~tree of the desired properties.
\qed

\para
This lemma implies that the second best spanning tree~$T_2$ differs from~$T_1$ by a~single
edge exchange and it remains to find which exchange it is. Let us consider the exchange
of an~edge $f\in E\setminus T_1$ with an~edge $e\in T_1[f]$. We get a~tree $T_1-e+f$
of weight $w(T_1)-w(e)+w(f)$. To obtain~$T_2$, we have to find~$e$ and~$f$ such that the
difference $w(f)-w(e)$ is the minimum possible. Thus for every~$f$, the edge $e$~must be always
the heaviest on the path $T_1[f]$. We can now use the algorithm from Corollary \ref{rampeaks}
to find the heaviest edges (peaks) of all such paths and examine all possible choices of~$f$
in linear time. This implies the following:

\lemma
Given~$G$ and~$T_1$, we can find~$T_2$ in time $\O(m)$.

\paran{Third best spanning tree}%
When we know~$T_1$ and~$T_2$, how to get~$T_3$? According to Lemma \ref{kbl}, $T_3$~can be
obtained by a~single exchange from either~$T_1$ or~$T_2$. Therefore we need to find the
best exchange for~$T_2$ and the second best exchange for~$T_1$ and use the better of them.
The latter is not easy to find directly, so we will make a~minor side step.

We know that $T_2=T_1-e+f$ for some edges $e$ and~$f$. We define two auxiliary graphs:
$G_1 := G\sgc e$ ($G$~with the edge~$e$ contracted) and $G_2 := G-e$. The tree~$T_1\sgc e$ is
obviously the MST of~$G_1$ (by the Contraction lemma) and $T_2$ is the MST of~$G_2$ (all
$T_2$-light edges in~$G_2$ would be $T_1$-light in~$G$).

\obs
The tree $T_3$~can be obtained by a~single edge exchange in either $(G_1,T_1\sgc e)$ or $(G_2,T_2)$:

\itemize\ibull
\:If $T_3 = T_1-e'+f'$ for $e'\ne e$, then $T_3\sgc e = (T_1\sgc e)-e'+f'$ in~$G_1$.
\:If $T_3 = T_1-e+f'$, then $T_3 = T_2 - f + f'$ in~$G_2$.
\:If $T_3 = T_2-e'+f'$, then this exchange is found in~$G_2$.
\endlist

\>Conversely, a~single exchange in $(G_1,T_1\sgc e)$ or in $(G_2,T_2)$ corresponds
to an~exchange in either~$(G,T_1)$ or $(G,T_2)$.
Even stronger, a~spanning tree~$T$ of~$G$ either contains~$e$ and then $T\sgc
e$ is a~spanning tree of~$G_1$, or $T$~doesn't contain~$e$ and so it is
a~spanning tree of~$G_2$.

Thus we can run the previous algorithm for finding the best edge exchange
on both~$G_1$ and~$G_2$ and find~$T_3$ again in time $\O(m)$.

\paran{Further spanning trees}%
The construction of auxiliary graphs can be iterated to obtain $T_1,\ldots,T_K$
for an~arbitrary~$K$. We will build a~\df{meta-tree} of auxilary graphs. Each node of this meta-tree
is assigned a~minor of~$G$ and its minimum spanning tree. The root node contains~$(G,T_1)$,
its sons have $(G_1,T_1\sgc e)$ and $(G_2,T_2)$. When $T_3$ is obtained by an~exchange
in one of these sons, we attach two new leaves to that son and we assign the two auxiliary
graphs derived by contracting or deleting the exchanged edge. Then we find the best
edge exchanges among the new sons and repeat the process. By the above observation,
each spanning tree of~$G$ is generated exactly once. Lemma \ref{kbl} guarantees that
the trees are enumerated in the increasing order.

Recalculating the best exchanges in all leaves of the meta-tree after generating each~$T_i$
is of course not necessary, because most leaves stay unchanged. We will rather remember
the best exchange for each leaf and keep their values in a~heap. In every step, we will
delete the minimum from the heap and use the exchange in the particular leaf to generate
a~new tree. Then we will create the new leaves, calculate their best exchanges and insert
them into the heap. The algorithm is now straightforward and so will be its analysis:

\algn{Finding $K$ best spanning trees}\id{kbestalg}%
\algo
\algin A~weighted graph~$G$, its MST~$T_1$ and an~integer $K>0$.
\:$R\=$ a~meta tree whose vertices carry triples $(G',T',F')$. Initially
  it contains just a~root with $(G,T_1,\emptyset)$.
  \hfil\break
  \cmt{$G'$ is a~minor of~$G$, $T'$~is its MST, and~$F'$ is a~set of edges of~$G$
  that are contracted in~$G'$.}
\:$H\=$ a~heap of quadruples $(\delta,r,e,f)$ ordered on~$\delta$, initially empty.
  \hfil\break
  \cmt{Each quadruple describes an~exchange of~$e$ for~$f$ in a~leaf~$r$ of~$R$ and $\delta=w(f)-w(e)$
  is the weight gain of this exchange.}
\:Find the best edge exchange in~$(G,T_1)$ and insert it to~$H$.
\:$i\= 1$.
\:While $i<K$:
\::Delete the minimum quadruple $(\delta,r,e,f)$ from~$H$.
\::$(G',T',F') \=$ the triple carried by the leaf~$r$.
\::$i\=i+1$.
\::$T_i\=(T'-e+f) \cup F'$. \cmt{The next spanning tree}
\::$r_1\=$ a~new leaf carrying $(G'\sgc e,T'\sgc e,F'+e)$.
\::$r_2\=$ a~new leaf carrying $(G'-e,T_i,F')$.
\::Attach~$r_1$ and~$r_2$ as sons of~$r$.
\::Find the best edge exchanges in~$r_1$ and~$r_2$ and insert them to~$H$.
\algout The spanning trees $T_2,\ldots,T_K$.
\endalgo

\lemma\id{kbestl}%
Given~$G$ and~$T_1$, we can find $T_2,\ldots,T_K$ in time $\O(Km + K\log K)$.

\proof
Generating each~$T_i$ requires finding the best exchange for two graphs and $\O(1)$
operations on the heap. The former takes $\O(m)$ according to Corollary \ref{rampeaks},
and each heap operation takes $\O(\log K)$.
\qed

\paran{Arbitrary weights}%
While the assumption that the weights of all spanning trees are distinct has helped us
in thinking about the problem, we should not forget that it is somewhat unrealistic.
We could refine the proof of our algorithm and demonstrate that the algorithm indeed works
without this assumption, but we will rather show that the ties can be broken easily.

Let~$\delta$ be the minimum positive difference of weights of spanning trees
of~$G$ and $e_1,\ldots,e_m$ be the edges of~$G$. We observe that it suffices to
increase $w(e_i)$ by~$\delta_i = \delta/2^{i+1}$. The cost of every spanning tree
has increased by at most $\sum_i\delta_i < \delta/2$, so if $T$~was lighter
than~$T'$, it still is. On the other hand, the no two trees share the same
weight difference, so all tree weights are now distinct.

The exact value of~$\delta$ is not easy to calculate, but examination of the algorithm
reveals that it is not needed at all. The only place where the edge weights are examined
is when we search for the best exchange. In this case, we compare the differences of
pairs of edge weights with each other. Each such difference is therefore adjusted
by $\delta\cdot(2^{-i}-2^{-j})$ for some $i,j>1$, which again does not influence comparison
of originally distinct differences. If the differences were the same, it is sufficient
to look at their values of~$i$ and~$j$, i.e., at the identifiers of the edges.

\paran{Invariant edges}%
Our algorithm can be further improved for small values of~$K$ (which seems to be the common
case in most applications) by the reduction of Eppstein \cite{eppstein:ksmallest}.
We will observe that there are many edges of~$T_1$
which are guaranteed to be contained in $T_2,\ldots,T_K$ as well, and likewise there are
many edges of $G\setminus T_1$ which are certainly not present in those spanning trees.
The idea is the following (we again assume that the tree weights are distinct):

\defn
For an~edge $e\in T_1$, we define its \df{gain} $g(e)$ as the minimum weight gained by an~edge exchange
replacing~$e$. Similarly, we define $G(f)$ for $f\not\in T_1$. Put formally:
$$\eqalign{
g(e) &:= \min\{ w(f)-w(e) \mid f\in E, e\in T[f] \} \cr
G(f) &:= \min\{ w(f)-w(e) \mid e\in E, e\in T[f] \}.\cr
}$$

\lemma\id{gaina}%
When $t_1,\ldots,t_{n-1}$ are the edges of~$T_1$ in order of increasing gain,
the edges $t_K,\ldots,t_n$ are present in all trees $T_2,\ldots,T_K$.

\proof
The best exchanges in~$T_1$ involving $t_1,\ldots,t_{K-1}$ produce~$K-1$ spanning trees
of increasing weights. Any exchange involving $t_K,\ldots,t_n$ produces a~tree
which is heavier or equal than those. (We are ascertained by the Monotone exchange lemma
that the gain of such exchanges cannot be reverted by any later exchanges.)
\qed

\lemma\id{gainb}%
When $q_1,\ldots,q_{m-n+1}$ are the edges of $G\setminus T_1$ in order of increasing gain,
the edges $q_K,\ldots,q_{m-n+1}$ are not present in any of $T_2,\ldots,T_K$.

\proof
Similar to the previous lemma.
\qed

\para
It is therefore sufficient to find $T_2,\ldots,T_K$ in the graph obtained from~$G$ by
contracting the edges $t_K,\ldots,t_n$ and deleting $q_K,\ldots,q_{m-n+1}$. This graph
has only $\O(K)$ vertices and $\O(K)$ edges. The only remaining question is how to
calculate the gains. For edges outside~$T_1$, it again suffices to find the peaks of the
covered paths. The gains of MST edges require a~different algorithm, but Tarjan \cite{tarjan:applpc}
has shown that they can be obtained in time $\O(m\timesalpha(m,n))$.

When we put the results of this section together, we obtain:

\thmn{Finding $K$ best spanning trees}\id{kbestthm}%
For a~given graph~$G$ with real edge weights, the $K$~best spanning trees can be found
in time $\O(m\timesalpha(m,n) + \min(K^2,Km + K\log K))$.

\proof
First we find the MST of~$G$ in time $\O(m\timesalpha(m,n))$ using the Pettie's Optimal
MST algorithm (Theorem \ref{optthm}). Then we calculate the gains of MST edges by the
Tarjan's algorithm from \cite{tarjan:applpc}, again in $\O(m\timesalpha(m,n))$, and
the gains of the other edges using our MST verification algorithm (Corollary \ref{rampeaks})
in $\O(m)$. We Lemma \ref{gaina} to identify edges that are required, and Lemma \ref{gainb}
to find edges that are useless. We contract the former edges, remove the latter ones
and run Algorithm \ref{kbestalg} to find the trees. By Lemma \ref{kbestl}, it runs in
the desired time.

If~$K\ge m$, this reduction does not pay off, so we run Algorithm \ref{kbestalg}
directly on the input graph.
\qed

\paran{Improvements}%
It is an~interesting open question whether the algorithms of Section \ref{verifysect} can
be modified to calculate all gains. The main procedure can, but it requires to reduce
the input to a~balance tree first and here the Bor\o{u}vka trees fail. The Buchsbaum's
Pointer-Machine algorithm (\ref{pmverify}) is more promising.

\paran{Large~$K$}%
When $K$~is large, re-running the verification algorithm for every change of the graph
is too costly. Frederickson \cite{frederickson:ambivalent} has shown how to find the best
swaps dynamically, reducing the overall time complexity of Algorithm \ref{kbestalg}
to $\O(Km^{1/2})$ and improving Theorem \ref{kbestthm} to $\O(m\timesalpha(m,n)
+ \min( K^{3/2}, Km^{1/2} ))$. It is open if the dynamic data structures of this
chapter could be modified to bring the complexity of finding the next tree down
to polylogarithmic.

\paran{Multiple minimum trees}%
Another nice application of Theorem \ref{kbestthm} is finding all minimum spanning
trees in a~graph that does not have distinct edge weights.

\endpart
