\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 from scratch for
-every graph 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 and 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:
+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:
\:$\<Insert>(G,u,v)$ --- 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 $u$ and~$v$ are in the same connected component of~$G$.
+\:$\<Connected>(G,u,v)$ --- Test if vertices $u$ and~$v$ are in the same connected component of~$G$.
\endlist
\para
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 as an~equivalence classes.
+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
bound in the cell-probe model which is stronger than RAM with $\O(\log n)$-bit
words.
-The edges that have triggered the \<Union>s form a~spanning forest of the current graph.
-So far, all known algorithms for dynamic connectivity maintain some sort of a~spanning
-tree. This suggests that a~dynamic MST algorithm could be obtained by modifying the
-dynamic connectivity algorithms. This will indeed turn out to be true. Incremental MST
-is 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.
+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
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 Theorem on Minimality by order
-(\ref{mstthm}), because all $F'$-light would be also $F$-light, which is impossible as $F$~was
+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 edge of the cut~$\delta_G(T_1)$
+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
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 offer many other operations, but we do not mention them
+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.
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 $u~$is a tree root and $v$~lies in a~different tree.
+ 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 the edges in the tree containing~$v$
- to make~$v$ the tree's root.
+\:$\<Evert>(v)$ --- 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
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.}
-\::$\<Link>(u,v,w)$. \cmt{Connect the 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~$uv$.}
-\:::$\<Cut>(y)$, $\<Evert>(v)$, $\<Link>(v,y,w)$.
+\:::$\<Cut>(y)$, $\<Evert>(v)$, $\<Link>(u,v,w)$.
\:::Return ``$uv$~added, $xy$~removed''.
\::Otherwise return ``no changes''.
\algout The list of changes in~$F$.
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 of the MSF is deleted.
+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 on them first.
+we will take a~look at them first.
%--------------------------------------------------------------------------------
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 it is much simpler and instead of path operations it offers efficient operations on
+\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}%
-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 $\<ET>(v)$
+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 $\<ET>(w)$ recursively.
+\::Call $\<ET>(w)$.
\::Record~$w$.
\endalgo
-\>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 the vertex.
+\>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
-The ET-sequence contains a~vertex of degree~$d$ exactly $d$~times except for the root which
+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 the ET-sequence when we modify the tree.
+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
-$\Eul(T) = AuvBvuC$ (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.
+$AuvBvuC$ (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 $\Eul(T)$ from $vAwBwCv$ to $wBwCvAw$.
+\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 presence of~$w$ inside~$B$.
+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:
\hbox{\epsfbox{pic/ettree.eps}}\cr
\noalign{\qquad\quad}
\halign{#\hfil\cr
- $\Eul(T_1) = 0121034546474308980$,~~$\Eul(T_2) = aba$. \cr
- $\Eul(T_1-34) = 01210308980, 4546474$. \cr
- $\Eul(T_1\hbox{~rooted at~3}) = 3454647430898012103$. \cr
- $\Eul(T_1+0a+T_2)$: $0121034546474308980aba0$. \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 represent the ET-sequences by $(a,b)$-trees with the parameter~$a$ set upon
+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 $\O(a\log_a n)$ in the worst case.
+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 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
+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} are a~data structure that represents a~forest of trees and a~set of non-tree
+\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 original trees) and the vertices and edges of the
+\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 with some fixed parameters $a$ and~$b$.
+\: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 the ET-sequence $\Eul(T)$. Each two consecutive leaves $u$ and~$v$ are separated
+ 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.
-\:A~mapping $\<act>(v)$ that maps each original vertex to the leaf containing its active occurrence.
-\:A~mapping $\<edge>(e)$ that maps an~original edge~$e$ to one of the internal keys representing it.
-\:A~mapping $\<twin>(k)$ that maps an~internal key~$k$ to the other internal key of the same
- original edge.
+\: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 that internal vertex.
+ 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
-\>The structure supports the following operations on the original trees:
+
+\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.
-\:$\<Root>(v)$ --- Return the root of the ET-tree containing the vertex~$v$. This can be used
- to test whether two vertices lie in the same tree. However, the root is not guaranteed
- to stay the same when the tree is modified by a~\<Link> or \<Cut>.
+\:$\<Connected>(u,v)$ --- 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 non-tree edges stored in the same tree as~$v$.
+\:$\<ScanNontree>(v)$ --- Return a~list of non-tree edges associated with the vertices
+ of the $v$'s tree.
\endlist
-\>If the non-tree edges have weights,
\impl
We will implement the operations on the ET-trees by translating the intended changes of the
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 again involves splits and joins of $(a,b)$-trees.
+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. Joining of the roots translated to joining of the $(a,b)$-trees.
+which we remember. Linking of the roots is translated to joining of the $(a,b)$-trees.
-\<Root> just follows parent pointers in the $(a,b)$-tree and it walks the path from the leaf
-to the root.
+\<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.
\<ScanNontree> traverses the tree recursively from the root, but it does not enter the
subtrees whose roots are not marked.
-Analysis of the time complexity is now straightforward:
+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 $\O(1)$, \<Root>, \<InsertNontree>, and \<DeleteNontree> in $\O(\log_a n)$, and
+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 operations. We apply the standard theorems
+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
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~extra $\O(\log^2 n / \log\log n)$, but queries accelerate to $\O(\log
+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 to represent weighted graphs and enumerate the non-tree
+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 the tree separately, so that it can be accessed in constant
+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.
\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
+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}.
{\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~$F_{\ell(uw)}$.
-\:For each~$i$, the components of~$F_i$ have at most $\lfloor n/2^i \rfloor$ vertices.
-This implies that all~$F_i$ for $i>L$ are empty.
+\:$F$~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
-satistifed. Newly inserted edges can enter level~0, which cannot break I1 nor~I2.
+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 them back. From I1, we know that such an~edge cannot belong to
-level greater than~$\ell$, so we start looking for it at level~$\ell$. According
+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~$T_1$. For each such edge,
-we test whether its other endpoint lies in~$T_2$. If it does, we have found the replacement
-edge and we insert it to~$F_\ell$. Otherwise we move the edge one level up. (This will
-be the gist of our amortization argument: We can charge most of the work on level
+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 on level~0, the tree~$T$
+lower level and so on. If there is no replacement edge at level~0, the tree~$T$
remains disconnected.
\impl
-We will use a~single ET-tree with~$a$ set to~2 for each level. For the $i$-th level, the ET-tree
-${\cal E}_i$ will represent the forest~$F_i$ and the non-tree edges of
-level~$i$ will be attached to its vertices.
+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
\: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~$F_\ell$.
-\::Call $\<Replace>(uv,\ell)$.
+\::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 an~replacement edge for~$uv$ at level~$i$}
+\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~$T_2$.
-\:Move all level~$i$ edges in~$T_1$ to level~$i+1$ and insert them to~${\cal E}_{i+1}$.
+\: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 $y\in T_2$, add the edge $xy$ to~$F_i$ and return.
-\::Otherwise increase $\ell(xy)$ by one. This includes deleting~$xy$ from~${\cal E}_i$
- and inserting it to~${\cal E}_{i+1}$.
+\::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
-\>Analysis of the time complexity is straightforward:
+\>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 for both
+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 have it pre-pay all level increases of the new
+(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.
\<Replace> costs $\O(\log n)$ per call, the rest is paid for by the edge level
increases.
-To bring the complexity of \<Connected> from $\O(\log n)$ down to $\O(\log n/\log\log n)$,
+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
+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 MSF}
+\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
+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
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 decremental MSF requires only two
+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 try the lightest non-tree edge incident with the examined tree first. We must ensure
+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:
\endlist
}
-\>This easily implies that we select the right replacement edge:
+\>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 on the maximum level.
+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 (\ref{cyclerule}), each $f_i$ is the heaviest edge on the cycle~$C_i = F[f_i] + f_i$.
+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 by I3 $f_1$~is the heaviest edge on~$C$, so
-$\ell(f_1) \le \ell(f_2)$. Therefore the lightest of all replacement edges must have
+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~$g$ between $F^a$ and~$F^b$ let $g^a$ and~$g^b$ be its respective endpoints.
+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 observed:
+We now have to make sure that the additional invariant is indeed observed:
\lemma\id{ithree}%
-After every operation, I3 is satisfied.
+After every operation, the invariant I3 is satisfied.
\proof
-When the structure is freshly initialized, I3 is obviously satisfied, because all edges
+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 when an~edge~$e$ either gets its level increased
-or becomes a~tree edge.
-
-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 happen 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~$\ell$.
-
-Let us first show that for edges of~$C$ incident with~$T_1$. All edges of~$T_1$ itself
-already are on the higher levels as they were moved there at the very beginning of the
-search for the replacement edge. As the algorithm always tries the lightest candidate
-for the replacement edge first, all non-tree edges incident with~$T_1$ which 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~$T_1$.
-
-The case of edges 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.
+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 with $n$~vertices and~$m\ge n$ edges and we perform
-$d$~edge deletions, the MSF can be maintained in time $\O((m+d)\cdot\log^2 n)$.
+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 by the following theorem:
+reduction whose properties are summarized in the following theorem:
-\thmn{MSF dynamization, Henzinger et al.~\cite{henzinger:twoec}, Holm et al.~\cite{holm:polylog}}
-Suppose that we have a~decremental MSF algorithm that for any $a$,~$b$ can be initialized
-on a~graph with~$a$ vertices and~$b$ edges and then it executes an~arbitrary sequence
-of deletions in time $\O(b\cdot t(a,b))$, where~$t$ is a~non-decreasing function.
-Then there exists a~fully dynamic MSF algorithm for a~graph on $n$~vertices, starting
+\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 $A_0,\ldots,A_{\log n}$ of exponentially increasing sizes. Every non-tree
+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
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.~handles the reinserts by a~mechanism of batch insertions
+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}
-There is a~fully dynamic MSF algorithm that works in amortized time $\O(\log^4 n)$
+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 $\O(\log^4 m)$ where~$m$
-is the number of operations performed. This could be more than $\O(\log^4 n)$ if
+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.
\rem
The limitation of MSF structures based on the Holm's algorithm for connectivity
-to edge deletions only seems to be unavoidable. The invariant I3 could be easily
+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
-possibly hit I1 before we skipped the levels of all the heavies edges on the
+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.
-We have enough time to scan all non-tree edges incident to the current tree~$T_1$
---- we can charge it on the level increases of its tree edges and if we use the
-degree reduction from Lemma \ref{degred}, there are at most two non-tree edges
-per vertex. (The reduction can be used dynamically as it always translates a~single
-change of the original graph to $\O(1)$ changes of the reduced graph.) The lightest
-replacement edge however could also be located in the super-trees of~$T_1$ on the
-lower levels, which are too large to scan and both I1 and I2 prevent us from
-charging the time on increasing levels there.
+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.
+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 drop our rule that all edge weights are distinct for a~moment and we recall that
-the basic structural properties of the MST's from Section \ref{mstbasics} still hold.
+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 to maintain
+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
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~$i$ and the MSF edges from~$\C_{i-1}$. Bounding the time complexity is then easy:
+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
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}.
+Theorem \ref{dyncon} and $\O(\log n)$ on operations with the Sleator-Tarjan trees
+by Theorem \ref{sletar}.
\qed
projects / saga.git / blobdiff