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\chapter{Introduction}
\bigskip

This thesis tells the story of two well-established problems of algorithmic
graph theory: the minimum spanning trees and ranks of permutations. At distance,
both problems seem to be simple, boring and already solved, because we have poly\-nom\-ial-time
algorithms for them since ages. But when we come closer and seek algorithms that
are really efficient, the problems twirl and twist and withstand many a~brave
attempt at the optimum solution. They also reveal a~vast and diverse landscape
of a~deep and beautiful theory. Still closer, this landscape turns out to be interwoven
with the intricate details of various models of computation and even of arithmetics
itself.

We have tried to cover all known important results on both problems and unite them
in a~single coherent theory. At many places, we have attempted to contribute my own
little stones to this mosaic: several new results, simplifications of existing
ones, and last, but not least filling in important details where the original
authors have missed some.

When compared with the earlier surveys on the minimum spanning trees, most
notably Graham and Hell \cite{graham:msthistory} and Eisner \cite{eisner:tutorial},
this work adds many of the recent advances, the dynamic algorithms and
also the relationship with computational models. No previous work covering
the ranking problems in their entirety is known.

We~have tried to stick to the usual notation except where it was too inconvenient.
Most symbols are defined at the place where they are used for the first time.
To avoid piling up too many symbols at places that speak about a~single fixed graph,
this graph is always called~$G$, its set of vertices and edges are denoted by $V$
and~$E$ respectively, and we~also use~$n$ for the number of its vertices and $m$~for
the number of edges. At places where there could be a~danger of confusion, more explicit notation
is used instead.

\chapter{Minimum Spanning Trees}

\section{The Problem}

The problem of finding a minimum spanning tree of a weighted graph is one of the
best studied problems in the area of combinatorial optimization since its birth.
Its colorful history (see \cite{graham:msthistory} and \cite{nesetril:history} for the full account)
begins in~1926 with the pioneering work of Bor\o{u}vka
\cite{boruvka:ojistem}\foot{See \cite{nesetril:boruvka} for an English translation with commentary.},
who studied primarily an Euclidean version of the problem related to planning
of electrical transmission lines (see \cite{boruvka:networks}), but gave an efficient
algorithm for the general version of the problem. As it was well before the dawn of graph
theory, the language of his paper was complicated, so we will better state the problem
in contemporary terminology:

\proclaim{Problem}Given an undirected graph~$G$ with weights $w:E(G)\rightarrow {\bb R}$,
find its minimum spanning tree, defined as follows:

\defn\id{mstdef}%
For a given graph~$G$ with weights $w:E(G)\rightarrow {\bb R}$:
\itemize\ibull
\:A~subgraph $H\subseteq G$ is called a \df{spanning subgraph} if $V(H)=V(G)$.
\:A~\df{spanning tree} of~$G$ is any spanning subgraph of~$G$ that is a tree.
\:For any subgraph $H\subseteq G$ we define its \df{weight} $w(H):=\sum_{e\in E(H)} w(e)$.
\:A~\df{minimum spanning tree (MST)} of~$G$ is a spanning tree~$T$ such that its weight $w(T)$
  is the smallest possible among all the spanning trees of~$G$.
\:For a disconnected graph, a \df{(minimum) spanning forest (MSF)} is defined as
  a union of (minimum) spanning trees of its connected components.
\endlist

Bor\o{u}vka's work was further extended by Jarn\'\i{}k \cite{jarnik:ojistem}, again in
mostly geometric setting, and he has discovered another efficient algorithm.
In the next 50 years, several significantly faster algorithms were published, ranging
from the $\O(m\timesbeta(m,n))$ time algorithm by Fredman and Tarjan \cite{ft:fibonacci},
over algorithms with inverse-Ackermann type complexity by Chazelle \cite{chazelle:ackermann}
and Pettie \cite{pettie:ackermann}, to an~algorithm by Pettie \cite{pettie:optimal}
whose time complexity is provably optimal.

Before we discuss the algorithms, let us review the basic properties of spanning trees.
We will mostly follow the theory developed by Tarjan in~\cite{tarjan:dsna} and show
that the weights on edges are not necessary for the definition of the MST.

\defnn{Heavy and light edges}\id{heavy}%
Let~$G$ be a~connected graph with edge weights~$w$ and $T$ its spanning tree. Then:
\itemize\ibull
\:For vertices $x$ and $y$, let $T[x,y]$ denote the (unique) path in~$T$ joining $x$ with~$y$.
\:For an edge $e=xy$ we will call $T[e]:=T[x,y]$ the \df{path covered by~$e$} and
  the edges of this path \df{edges covered by~$e$}.
\:An edge~$e$ is called \df{light with respect to~$T$} (or just \df{$T$-light}) if it covers a~heavier edge, i.e., if there
  is an~edge $f\in T[e]$ such that $w(f) > w(e)$.
\:An edge~$e$ is called \df{$T$-heavy} if it covers a~lighter edge.
\endlist

\thm
A~spanning tree~$T$ is minimum iff there is no $T$-light edge.

\thm
If all edge weights are distinct, then the minimum spanning tree is unique.

\para
When $G$ is a graph with distinct edge weights, we will use $\mst(G)$ to denote
its unique minimum spanning tree.
To simplify the description of MST algorithms, we will assume that the weights
of all edges are distinct and that instead of numeric weights we are given a~\df{comparison oracle.}
The oracle is a~function that answers questions of type ``Is $w(e)<w(f)$?'' in
constant time. This will conveniently shield us from problems with representation
of real numbers in algorithms and in the few cases where we need a more concrete
input, we will explicitly state so. In case the weights are not distinct, the ties
can be broken arbitrarily.

\section{Classical algorithms}

The characterization of MST's in terms of light edges makes it easy to develop
the Tarjan's Red-Blue meta-algorithm, which is based on the following properties:

\lemman{Blue lemma, also known as the Cut rule}\id{bluelemma}%
The lightest edge of every cut is contained in the MST.

\lemman{Red lemma, also known as the Cycle rule}\id{redlemma}%
An~edge~$e$ is not contained in the MST iff it is the heaviest on some cycle.

The algorithm repeatedly colors lightest edges of cuts blue and heaviest
edges of cycles red. We prove that no matter which order of the colorings
we use, the algorithm always stops and the blue edges form the MST.

All three classical MST algorithms (Bor\o{u}vka's, Jarn\'\i{}k's and Kruskal's)
can be then obtained as specializations of this procedure. We also calculate the
time complexity of standard implementations of these algorithms.

\algn{Bor\o{u}vka \cite{boruvka:ojistem}, Choquet \cite{choquet:mst}, Sollin \cite{sollin:mst}, and others}
\algo
\algin A~graph~$G$ with an edge comparison oracle.
\:$T\=$ a forest consisting of vertices of~$G$ and no edges.
\:While $T$ is not connected, perform a~\df{Bor\o{u}vka step:}
\::For each component $T_i$ of~$T$, choose the lightest edge $e_i$ from the cut
   separating $T_i$ from the rest of~$T$.
\::Add all $e_i$'s to~$T$.
\algout Minimum spanning tree~$T$.
\endalgo

\thm
The Bor\o{u}vka's algorithm finds the MST in time $\O(m\log n)$.

\algn{Jarn\'\i{}k \cite{jarnik:ojistem}, Prim \cite{prim:mst}, Dijkstra \cite{dijkstra:mst}}\id{jarnik}%
\algo
\algin A~graph~$G$ with an edge comparison oracle.
\:$T\=$ a single-vertex tree containing an~arbitrary vertex of~$G$.
\:While there are vertices outside $T$:
\::Pick the lightest edge $uv$ such that $u\in V(T)$ and $v\not\in V(T)$.
\::$T\=T+uv$.
\algout Minimum spanning tree~$T$.
\endalgo

\thm
The Jarn\'\i{}k's algorithm computes the MST of a~given graph in time $\O(m\log n)$.

\algn{Kruskal \cite{kruskal:mst}}
\algo
\algin A~graph~$G$ with an edge comparison oracle.
\:Sort edges of~$G$ by their increasing weights.
\:$T\=\hbox{an empty spanning subgraph}$.
\:For all edges $e$ in their sorted order:
\::If $T+e$ is acyclic, add~$e$ to~$T$.
\::Otherwise drop~$e$.
\algout Minimum spanning tree~$T$.
\endalgo

\thm
The Kruskal's algorithm finds the MST of a given graph in time $\O(m\log n)$.
If the edges are already sorted by their weights, the time drops to
$\O(m\timesalpha(m,n))$.\foot{Here $\alpha(m,n)$ is a~certain inverse of the Ackermann's
function.}

\section{Contractive algorithms}\id{contalg}%

While the classical algorithms are based on growing suitable trees, they
can be also reformulated in terms of edge contraction. Instead of keeping
a~forest of trees, we can keep each tree contracted to a single vertex.
This replaces the relatively complex tree-edge incidencies by simple
vertex-edge incidencies, potentially speeding up the calculation at the
expense of having to perform the contractions.
A~contractive version of the Bor\o{u}vka's algorithm is easy to formulate
and also to analyse:

\algn{Contractive version of Bor\o{u}vka's algorithm}\id{contbor}%
\algo
\algin A~graph~$G$ with an edge comparison oracle.
\:$T\=\emptyset$.
\:$\ell(e)\=e$ for all edges~$e$. \cmt{Initialize edge labels.}
\:While $n(G)>1$:
\::For each vertex $v_k$ of~$G$, let $e_k$ be the lightest edge incident to~$v_k$.
\::$T\=T\cup \{ \ell(e_1),\ldots,\ell(e_n) \}$.\hfil\break\cmt{Remember labels of all selected edges.}
\::Contract all edges $e_k$, inheriting labels and weights.\foot{In other words, we will ask the comparison oracle for the edge $\ell(e)$ instead of~$e$.}
\::Flatten $G$ (remove parallel edges and loops).
\algout Minimum spanning tree~$T$.
\endalgo

\thm
The Contractive Bor\o{u}vka's algorithm finds the MST of the input graph in
time $\O(\min(n^2,m\log n))$.

We also show that this time bound is tight --- we construct an~explicit
family of graphs on which the algorithm spends $\Theta(m\log n)$ steps.
Given a~planar graph, the algorithm however runs much faster (we get a~linear-time
algorithm much simpler than the one of Matsui \cite{matsui:planar}):

\thm
When the input graph is planar, the Contractive Bor\o{u}vka's algorithm runs in
time $\O(n)$.

Graph contractions are indeed a~very powerful tool and they can be used in other MST
algorithms as well. The following lemma shows the gist:

\lemman{Contraction lemma}\id{contlemma}%
Let $G$ be a weighted graph, $e$~an arbitrary edge of~$\mst(G)$, $G/e$ the multigraph
produced by contracting~$e$ in~$G$, and $\pi$ the bijection between edges of~$G-e$ and
their counterparts in~$G/e$. Then $\mst(G) = \pi^{-1}[\mst(G/e)] + e.$

\chapter{Fine Details of Computation}

\section{Models and machines}

Traditionally, computer scientists have been using a~variety of computational models
as a~formalism in which their algorithms are stated. If we were studying
NP-complete\-ness, we could safely assume that all these models are equivalent,
possibly up to polynomial slowdown which is negligible. In our case, the
differences between good and not-so-good algorithms are on a~much smaller
scale, so we need to state our computation models carefully and develop
a repertoire of basic data structures tailor-made for the fine details of the
models.

In recent decades, most researchers in the area of combinatorial algorithms
have been considering two computational models: the Random Access Machine and the Pointer
Machine. The former is closer to the programmer's view of a~real computer,
the latter is slightly more restricted and ``asymptotically safe.''
We will follow this practice and study our algorithms in both models.

The \df{Random Access Machine (RAM)} is not a~single coherent model, but rather a~family
of closely related machines (See Cook and Reckhow \cite{cook:ram} for one of the usual formal definitions
and Hagerup \cite{hagerup:wordram} for a~thorough description of the differences
between the RAM variants.) We will consider the variant usually called the \df{Word-RAM.}
It allows the ``C-language operators'', i.e., arithmetics and bitwise logical operations,
running in constant time on words of a~specified size.

The \df{Pointer Machine (PM)} also does not seem to have any well established definition.
The various kinds of pointer machines are examined by Ben-Amram in~\cite{benamram:pm},
but unlike the RAM's they turn out to be equivalent up to constant slowdown.
Our formal definition is closely related to the \em{linking automaton} proposed
by Knuth in~\cite{knuth:fundalg}.

\section{Pointer machine techniques}\id{bucketsort}%

In the Contractive Bor\o{u}vka's algorithm, we needed to contract a~given
set of edges in the current graph and then flatten the graph, all this in time $\O(m)$.
This can be easily handled on both the RAM and the PM by bucket sorting. We develop
a~bunch of pointer-based sorted techniques which can be summarized by the following
lemma:

\lemma
Partitioning of a~collection of sequences $S_1,\ldots,S_n$, whose elements are
arbitrary pointers and symbols from a~finite alphabet, to equality classes can
be performed on the Pointer Machine in time $\O(n + \sum_i \vert S_i \vert)$.

\para
A~direct consequence of this unification is a~linear-time algorithm for subtree
isomorphism, significantly simpler than the standard one due to Zemlayachenko (see \cite{zemlay:treeiso}
and also Dinitz et al.~\cite{dinitz:treeiso}). When we apply a~similar technique
to general graphs, we get the framework of topological graph computation
of Buchsbaum et al.~\cite{buchsbaum:verify}.

\defn
A~\df{graph computation} is a~function that takes a~\df{labeled undirected graph} as its input. The labels of
vertices and edges can be arbitrary symbols drawn from a~finite alphabet. The output
of the computation is another labeling of the same graph. This time, the vertices and
edges can be labeled with not only symbols of the alphabet, but also with pointers to the vertices
and edges of the input graph, and possibly also with pointers to outside objects.
A~graph computation is called \df{topological} if it produces isomorphic
outputs for isomorphic inputs. The isomorphism of course has to preserve not only
the structure of the graph, but also the labels in the obvious way.

\defn
For a~collection~$\C$ of graphs, we define $\vert\C\vert$ as the number of graphs in
the collection and $\Vert\C\Vert$ as their total size, i.e., $\Vert\C\Vert = \sum_{G\in\C} n(G) + m(G)$.

\thm
Suppose that we have a~topological graph computation~$\cal T$ that can be performed in time
$T(k)$ for graphs on $k$~vertices. Then we can run~$\cal T$ on a~collection~$\C$
of labeled graphs on~$k$ vertices in time $\O(\Vert\C\Vert + (k+s)^{k(k+2)}\cdot (T(k)+k^2))$,
where~$s$ is a~constant depending only on the number of symbols used as vertex/edge labels.

\section{Data structures on the RAM}\id{ramds}%

There is a~lot of data structures designed specifically for the RAM. These structures
take advantage of both indexing and arithmetics and they often surpass the known
lower bounds for the same problem on the~PM. In many cases, they achieve constant time
per operation, at least when either the magnitude of the values or the size of
the data structure is suitably bounded.

A~classical result of this type is the tree of van Emde Boas~\cite{boas:vebt}
which represent a~subset of the integers $\{0,\ldots,U-1\}$. It allows insertion,
deletion and order operations (minimum, maximum, successor etc.) in time $\O(\log\log U)$,
regardless of the size of the subset. If we replace the heap used in the Jarn\'\i{}k's
algorithm (\ref{jarnik}) by this structure, we immediately get an~algorithm
for finding the MST in integer-weighted graphs in time $\O(m\log\log w_{max})$,
where $w_{max}$ is the maximum weight.

A~real breakthrough has however been made by Fredman and Willard who introduced
the Fusion trees~\cite{fw:fusion}. They again perform membership and predecessor
operation on a~set of $n$~integers, but with time complexity $\O(\log_W n)$
per operation on a~Word-RAM with $W$-bit words. This of course assumes that
each element of the set fits in a~single word. As $W$ must at least~$\log n$,
the operations take $\O(\log n/\log\log n)$ time and thus we are able to sort $n$~integers
in time~$o(n\log n)$. This was a~beginning of a~long sequence of faster and
faster sorting algorithms, culminating with the work of Thorup and Han.
They have improved the time complexity of integer sorting to $\O(n\log\log n)$ deterministically~\cite{han:detsort}
and expected $\O(n\sqrt{\log\log n})$ for randomized algorithms~\cite{hanthor:randsort},
both in linear space.

The Fusion trees themselves have very limited use in graph algorithms, but the
principles behind them are ubiquitous in many other data structures and these
will serve us well and often. We are going to build the theory of Q-heaps,
which will later lead to a~linear-time MST algorithm for arbitrary integer weights.
Other such structures will help us in building linear-time RAM algorithms for computing the ranks
of various combinatorial structures in Chapter~\ref{rankchap}.

Outside our area, important consequences of RAM data structures include the
Thorup's $\O(m)$ algorithm for single-source shortest paths in undirected
graphs with positive integer weights \cite{thorup:usssp} and his $\O(m\log\log
n)$ algorithm for the same problem in directed graphs \cite{thorup:sssp}. Both
algorithms have been then significantly simplified by Hagerup
\cite{hagerup:sssp}.

Despite the progress in the recent years, the corner-stone of all RAM structures
is still the representation of combinatorial objects by integers introduced by
Fredman and Willard.

First of all, we observe that we can encode vectors in integers:

\notan{Bit strings}\id{bitnota}%
We will work with binary representations of natural numbers by strings over the
alphabet $\{\0,\1\}$: we will use $\(x)$ for the number~$x$ written in binary,
$\(x)_b$ for the same padded to exactly $b$ bits by adding leading zeroes,
and $x[k]$ for the value of the $k$-th bit of~$x$ (with a~numbering of bits such that $2^k[k]=1$).
The usual conventions for operations on strings will be utilized: When $s$
and~$t$ are strings, we write $st$ for their concatenation and
$s^k$ for the string~$s$ repeated $k$~times.
When the meaning is clear from the context,
we will use $x$ and $\(x)$ interchangeably to avoid outbreak of symbols.

\defn
The \df{bitwise encoding} of a~vector ${\bf x}=(x_0,\ldots,x_{d-1})$ of~$b$-bit numbers
is an~integer~$x$ such that $\(x)=\(x_{d-1})_b\0\(x_{d-2})_b\0\ldots\0\(x_0)_b$. In other
words, $x = \sum_i 2^{(b+1)i}\cdot x_i$. (We have interspersed the elements with \df{separator bits.})

\para
If we want to fit the whole vector in a~single machine word, the parameters $b$ and~$d$ must satisfy
the condition $(b+1)d\le W$ (where $W$~is the word size of the machine).
By using multiple-precision arithmetics, we can encode all vectors satisfying $bd=\O(W)$.
We describe how to translate simple vector manipulations to sequences of $\O(1)$ RAM operations
on their codes. For example, we can handle element-wise comparison of vectors, insertion
in a~sorted vector or shuffling elements of a~vector according to a~fixed permutation,
all in $\O(1)$ time. This also implies that several functions on numbers can be performed
in constant time, most notably binary logarithms.
The vector operations then serve as building blocks for construction of the Q-heaps. We get:

\thm
Let $W$ and~$k$ be positive integers such that $k=\O(W^{1/4})$. Let~$Q$
be a~Q-heap of at most $k$-elements of $W$~bits each. Then we can perform
Q-heap operations on~$Q$ (insertion, deletion, search for a~given value and search
for the $i$-th smallest element) in constant time on a~Word-RAM with word size~$W$,
after spending time $\O(2^{k^4})$ on the same RAM on precomputing of tables.

\cor
For every positive integer~$r$ and $\delta>0$ there exists a~data structure
capable of maintaining the minimum of a~set of at most~$r$ word-sized numbers
under insertions and deletions. Each operation takes $\O(1)$ time on a~Word-RAM
with word size $W=\Omega(r^{\delta})$, after spending time
$\O(2^{r^\delta})$ on precomputing of tables.

\chapter{Advanced MST Algorithms}

\section{Minor-closed graph classes}\id{minorclosed}%

The contractive algorithm given in Section~\ref{contalg} has been found to perform
well on planar graphs, but in the general case its time complexity was not linear.
Can we find any broader class of graphs where this algorithm is still efficient?
The right context turns out to be the minor-closed graph classes, which are
closed under contractions and have bounded density.

\defn\id{minordef}%
A~graph~$H$ is a \df{minor} of a~graph~$G$ (written as $H\minorof G$) iff it can be obtained
from a~subgraph of~$G$ by a sequence of simple graph contractions.

\defn
A~class~$\cal C$ of graphs is \df{minor-closed}, when for every $G\in\cal C$ and
every minor~$H$ of~$G$, the graph~$H$ lies in~$\cal C$ as well. A~class~$\cal C$ is called
\df{non-trivial} if at least one graph lies in~$\cal C$ and at least one lies outside~$\cal C$.

\example
Non-trivial minor-closed classes include:
\itemize\ibull
\:planar graphs,
\:graphs embeddable in any fixed surface (i.e., graphs of bounded genus),
\:graphs embeddable in~${\bb R}^3$ without knots or without interlocking cycles,
\:graphs of bounded tree-width or path-width.
\endlist

\para
Many of the nice structural properties of planar graphs extend to
minor-closed classes, too (see Lov\'asz \cite{lovasz:minors} for a~nice survey
of this theory and Diestel \cite{diestel:gt} for some of the deeper results).
For analysis of the contractive algorithm, we will make use of the bounded
density of minor-closed classes:

\defn\id{density}%
Let $G$ be a~graph and $\cal C$ be a class of graphs. We define the \df{edge density}
$\varrho(G)$ of~$G$ as the average number of edges per vertex, i.e., $m(G)/n(G)$. The
edge density $\varrho(\cal C)$ of the class is then defined as the infimum of $\varrho(G)$ over all $G\in\cal C$.

\thmn{Density of minor-closed classes, Mader~\cite{mader:dens}}
Every non-trivial minor-closed class of graphs has finite edge density.

\thmn{MST on minor-closed classes, Mare\v{s} \cite{mm:mst}}\id{mstmcc}%
For any fixed non-trivial minor-closed class~$\cal C$ of graphs, the Contractive Bor\o{u}vka's
algorithm (\ref{contbor}) finds the MST of any graph of this class in time
$\O(n)$. (The constant hidden in the~$\O$ depends on the class.)

\paran{Local contractions}\id{nobatch}%
The contractive algorithm uses ``batch processing'' to perform many contractions
in a single step. It is also possible to perform them one edge at a~time,
batching only the flattenings. A~contraction of an edge~$uv$ can be done in time~$\O(\deg(u))$,
so we have to make sure that there is a~steady supply of low-degree vertices.
It indeed is in minor-closed classes:

\lemman{Low-degree vertices}\id{lowdeg}%
Let $\cal C$ be a graph class with density~$\varrho$ and $G\in\cal C$ a~graph
with $n$~vertices. Then at least $n/2$ vertices of~$G$ have degree at most~$4\varrho$.

So we get the following algorithm:

\algn{Local Bor\o{u}vka's Algorithm, Mare\v{s} \cite{mm:mst}}%
\algo
\algin A~graph~$G$ with an edge comparison oracle and a~parameter~$t\in{\bb N}$.
\:$T\=\emptyset$.
\:$\ell(e)\=e$ for all edges~$e$.
\:While $n(G)>1$:
\::While there exists a~vertex~$v$ such that $\deg(v)\le t$:
\:::Select the lightest edge~$e$ incident with~$v$.
\:::Contract~$e$.
\:::$T\=T + \ell(e)$.
\::Flatten $G$, removing parallel edges and loops.
\algout Minimum spanning tree~$T$.
\endalgo

\thm
When $\cal C$ is a minor-closed class of graphs with density~$\varrho$, the
Local Bor\o{u}vka's Algorithm with the parameter~$t$ set to~$4\varrho$ 
finds the MST of any graph from this class in time $\O(n)$. (The constant
in the~$\O$ depends on~the class.)

\section{Iterated algorithms}\id{iteralg}%

We have seen that the Jarn\'\i{}k's Algorithm \ref{jarnik} runs in $\Theta(m\log n)$ time.
Fredman and Tarjan \cite{ft:fibonacci} have shown a~faster implementation using their Fibonacci
heaps, which runs in time $\O(m+n\log n)$. This is $\O(m)$ whenever the density of the
input graph reaches $\Omega(\log n)$. This suggests that we could combine the algorithm with
another MST algorithm, which identifies a~part of the MST edges and contracts
them to increase the density of the graph. For example, if we perform several Bor\o{u}vka
steps and then run the Jarn\'\i{}k's algorithm, we find the MST in time $\O(m\log\log n)$.

Actually, there is a~much better choice of the algorithms to combine: use the
Jarn\'\i{}k's algorithm with a~Fibonacci heap multiple times, each time stopping it after a~while.
A~good choice of the stopping condition is to place a~limit on the size of the heap.
We start with an~arbitrary vertex, grow the tree as usually and once the heap gets too large,
we conserve the current tree and start with a~different vertex and an~empty heap. When this
process runs out of vertices, it has identified a~sub-forest of the MST, so we can
contract the edges of~this forest and iterate.

\algn{Iterated Jarn\'\i{}k; Fredman and Tarjan \cite{ft:fibonacci}}\id{itjar}%
\algo
\algin A~graph~$G$ with an edge comparison oracle.
\:$T\=\emptyset$. \cmt{edges of the MST}
\:$\ell(e)\=e$ for all edges~$e$. \cmt{edge labels as usually}
\:$m_0\=m$.
\:While $n>1$: \cmt{We will call iterations of this loop \df{phases}.}
\::$F\=\emptyset$. \cmt{forest built in the current phase}
\::$t\=2^{\lceil 2m_0/n \rceil}$. \cmt{the limit on heap size}
\::While there is a~vertex $v_0\not\in F$:
\:::Run the Jarn\'\i{}k's algorithm from~$v_0$, stop when:
\::::all vertices have been processed, or
\::::a~vertex of~$F$ has been added to the tree, or
\::::the heap has grown to more than~$t$ elements.
\:::Denote the resulting tree~$R$.
\:::$F\=F\cup R$.
\::$T\=T\cup \ell[F]$. \cmt{Remember MST edges found in this phase.}
\::Contract all edges of~$F$ and flatten~$G$.
\algout Minimum spanning tree~$T$.
\endalgo

\thm\id{itjarthm}%
The Iterated Jarn\'\i{}k's algorithm finds the MST of the input graph in time
$\O(m\timesbeta(m,n))$, where $\beta(m,n):=\min\{ i \mid \log^{(i)}n \le m/n \}$.

\cor
The Iterated Jarn\'\i{}k's algorithm runs in time $\O(m\log^* n)$.

\paran{Integer weights}%
The algorithm spends most of the time in phases which have small heaps. Once the
heap grows to $\Omega(\log^{(k)} n)$ for any fixed~$k$, the graph gets dense enough
to guarantee that at most~$k$ phases remain. This means that if we are able to
construct a~heap of size $\Omega(\log^{(k)} n)$ with constant time per operation,
we can get a~linear-time algorithm for MST. This is the case when the weights are
integers (we can use the Q-heap trees from Section~\ref{ramds}.

\thmn{MST for integer weights, Fredman and Willard \cite{fw:transdich}}\id{intmst}%
MST of a~graph with integer edge weights can be found in time $\O(m)$ on the Word-RAM.

\section{Verification of minimality}\id{verifysect}%

Now we will turn our attention to a~slightly different problem: given a~spanning
tree, how to verify that it is minimum? We will show that this can be achieved
in linear time and it will serve as a~basis for a~randomized linear-time
MST algorithm in the next section.

MST verification has been studied by Koml\'os \cite{komlos:verify}, who has
proven that $\O(m)$ edge comparisons are sufficient, but his algorithm needed
super-linear time to find the edges to compare. Dixon, Rauch and Tarjan \cite{dixon:verify}
have later shown that the overhead can be reduced
to linear time on the RAM using preprocessing and table lookup on small
subtrees. Later, King has given a~simpler algorithm in \cite{king:verifytwo}.

To verify that a~spanning tree~$T$ is minimum, it is sufficient to check that all
edges outside~$T$ are $T$-heavy. For each edge $uv\in E\setminus T$, we will
find the heaviest edge of the tree path $T[u,v]$ (we will call it the \df{peak}
of the path) and compare its weight to $w(uv)$. We have therefore transformed
the MST verification to the problem of finding peaks for a~set of \df{query
paths} on a~given tree. By a~sequence of further transformations, we can even
assume that the given tree is \df{complete branching} (all vertices are on
the same level and internal vertices always have outdegree~2) and that the
query paths join a~vertex with one of its ancestors.

Koml\'os has given a~simple algorithm that traverses the complete branching
tree recursively. At each moment, it maintains an~array of peaks of the restrictions
of the query paths to the subtree below the current vertex. If we account for the
comparisons performed by this algorithm carefully and express the bound in terms
of the size of the original problem (before all the transformations), we get:

\thmn{Verification of the MST, Koml\'os \cite{komlos:verify}}\id{verify}%
For every weighted graph~$G$ and its spanning tree~$T$, it is sufficient to
perform $\O(m)$ comparisons of edge weights to determine whether~$T$ is minimum
and to find all $T$-light edges in~$G$.

It remains to demonstrate that the overhead of the algorithm needed to find
the required comparisons and infer the peaks from their results can be decreased,
so that it gets bounded by the number of comparisons and therefore also by $\O(m)$.
We will follow the idea of King from \cite{king:verifytwo}, but as we have the power
of the RAM data structures from Section~\ref{ramds} at our command, the low-level
details will be easier. Still, the construction is rather technical, so we omit
it in this abstract and state only the final theorem:

\thmn{Verification of MST on the RAM}\id{ramverify}%
There is a~RAM algorithm which for every weighted graph~$G$ and its spanning tree~$T$
determines whether~$T$ is minimum and finds all $T$-light edges in~$G$ in time $\O(m)$.

\rem
Buchsbaum et al.~\cite{buchsbaum:verify} have recently shown that linear-time
verification can be achieved even on the Pointer Machine.
They combine an~algorithm of time complexity $\O(m\timesalpha(m,n))$
based on the Disjoint Set Union data structure with the framework of topological graph
computations described in Section \ref{bucketsort}.
The online version of this problem has turned out to be more difficult: there
is a~super-linear lower bound for it due to Pettie \cite{pettie:onlineverify}.

\section{A randomized algorithm}\id{randmst}%

When we analysed the Contractive Bor\o{u}vka's algorithm in Section~\ref{contalg},
we observed that while the number of vertices per iteration decreases exponentially,
the number of edges generally does not, so we spend $\Theta(m)$ time on every phase.
Karger, Klein and Tarjan \cite{karger:randomized} have overcome this problem by
combining the Bor\o{u}vka's algorithm with filtering based on random sampling.
This leads to a~randomized algorithm which runs in linear expected time.

The principle of the filtering is simple: Let us consider any spanning tree~$T$
of the input graph~$G$. Each edge of~$G$ that is $T$-heavy is the heaviest edge
of some cycle, so by the Red lemma it cannot participate in
the MST of~$G$. We can therefore discard all $T$-heavy edges and continue with
finding the MST on the reduced graph. Of course, not all choices of~$T$ are equally
good, but it will soon turn out that when we take~$T$ as the MST of a~randomly selected
subgraph, only a~small expected number of edges remains:

\lemman{Random sampling, Karger \cite{karger:sampling}}\id{samplemma}%
Let $H$~be a~subgraph of~$G$ obtained by including each edge independently
with probability~$p$. Let further $F$~be the minimum spanning forest of~$H$. Then the
expected number of $F$-nonheavy\foot{That is, $F$-light edges and also edges of~$F$ itself.}
edges in~$G$ is at most $n/p$.

\para
We will formulate the algorithm as a~doubly-recursive procedure. It alternatively
performs steps of the Bor\o{u}vka's algorithm and filtering based on the above lemma.
The first recursive call computes the MSF of the sampled subgraph, the second one
finds the MSF of the original graph, but without the heavy edges.

\algn{MSF by random sampling --- the KKT algorithm}\id{kkt}%
\algo
\algin A~graph $G$ with an~edge comparison oracle.
\:Remove isolated vertices from~$G$. If no vertices remain, stop and return an~empty forest.
\:Perform two Bor\o{u}vka steps (iterations of Algorithm \ref{contbor}) on~$G$ and
  remember the set~$B$ of the edges having been contracted.
\:Select a~subgraph~$H\subseteq G$ by including each edge independently with
  probability $1/2$.
\:$F\=\msf(H)$ calculated recursively.
\:Construct $G'\subseteq G$ by removing all $F$-heavy edges of~$G$.
\:$R\=\msf(G')$ calculated recursively.
\:Return $R\cup B$.
\algout The minimum spanning forest of~$G$.
\endalgo

A~careful analysis of this algorithm, based on properties of its recursion tree
and the peak-finding algorithm of the previous section yields the following time bounds:

\thm
The KKT algorithm runs in time $\O(\min(n^2,m\log n))$ in the worst case on the RAM.

\thm
The expected time complexity of the KKT algorithm on the RAM is $\O(m)$.

\chapter{Approaching Optimality}\id{optchap}%

\section{Soft heaps}\id{shsect}%

A~vast majority of MST algorithms that we have encountered so far is based on
the Tarjan's Blue rule (Lemma \ref{bluelemma}). The rule serves to identify
edges that belong to the MST, while all other edges are left in the process. This
unfortunately means that the later stages of computation spend most of
their time on these edges that never enter the MSF. A~notable exception is the randomized
algorithm of Karger, Klein and Tarjan. It adds an~important ingredient: it uses
the Red rule (Lemma \ref{redlemma}) to filter out edges that are guaranteed to stay
outside the MST, so that the graphs with which the algorithm works get smaller
with time.

Recently, Chazelle \cite{chazelle:ackermann} and Pettie \cite{pettie:ackermann}
have presented new deterministic algorithms for the MST which are also based
on the combination of both rules. They have reached worst-case time complexity
$\O(m\timesalpha(m,n))$ on the Pointer Machine. We will devote this chapter to their results
and especially to another algorithm by Pettie and Ramachandran \cite{pettie:optimal}
which is provably optimal.

At the very heart of all these algorithms lies the \df{soft heap} discovered by
Chazelle \cite{chazelle:softheap}. It is a~meldable priority queue, roughly
similar to the Vuillemin's binomial heaps \cite{vuillemin:binheap} or Fredman's
and Tarjan's Fibonacci heaps \cite{ft:fibonacci}. The soft heaps run faster at
the expense of \df{corrupting} a~fraction of the inserted elements by raising
their values (the values are however never lowered). This allows for
a~trade-off between accuracy and speed, controlled by a~parameter~$\varepsilon$.
The heap operations take $\O(\log(1/\varepsilon))$ amortized time and at every
moment at most~$\varepsilon n$ elements of the $n$~elements inserted can be
corrupted.

\defnn{Soft heap interface}%
The \df{soft heap} contains a~set of distinct items from a~totally ordered universe and it
supports the following operations:
\itemize\ibull
\:$\<Create>(\varepsilon)$ --- Create an~empty soft heap with the given accuracy parameter~$\varepsilon$.
\:$\<Insert>(H,x)$ --- Insert a~new item~$x$ into the heap~$H$.
\:$\<Meld>(P,Q)$ --- Merge two heaps into one, more precisely move all items of a~heap~$Q$
  to the heap~$P$, destroying~$Q$ in the process. Both heaps must have the same~$\varepsilon$.
\:$\<DeleteMin>(H)$ --- Delete the minimum item of the heap~$H$ and return its value
  (optionally signalling that the value has been corrupted).
\:$\<Explode>(H)$ --- Destroy the heap and return a~list of all items contained in it
  (again optionally marking those corrupted).
\endlist

\>We describe the exact mechanics of the soft heaps and analyse its complexity.
The important properties are summarized in the following theorem:

\thmn{Performance of soft heaps, Chazelle \cite{chazelle:softheap}}\id{softheap}%
A~soft heap with error rate~$\varepsilon$ ($0<\varepsilon\le 1/2$) processes
a~sequence of operations starting with an~empty heap and containing $n$~\<Insert>s
in time $\O(n\log(1/\varepsilon))$ on the Pointer Machine. At every moment, the
heap contains at most $\varepsilon n$ corrupted items.

\section{Robust contractions}

Having the soft heaps at hand, we would like to use them in a~conventional MST
algorithm in place of a~normal heap. The most efficient specimen of a~heap-based
algorithm we have seen so far is the Jarn\'\i{}k's algorithm.
We can try implanting the soft heap in this algorithm, preferably in the earlier
version without Fibonacci heaps as the soft heap lacks the \<Decrease> operation.
This brave, but somewhat simple-minded attempt is however doomed to
fail. The reason is of course the corruption of items inside the heap, which
leads to increase of weights of some subset of edges. In presence of corrupted
edges, most of the theory we have so carefully built breaks down. There is
fortunately some light in this darkness. While the basic structural properties
of MST's no longer hold, there is a~weaker form of the Contraction lemma that
takes the corrupted edges into account. Before we prove this lemma, we expand
our awareness of subgraphs which can be contracted.

\defn
A~subgraph $C\subseteq G$ is \df{contractible} iff for every pair of edges $e,f\in\delta(C)$\foot{That is,
of~$G$'s edges with exactly one endpoint in~$C$.} there exists a~path in~$C$ connecting the endpoints
of the edges $e,f$ such that all edges on this path are lighter than either $e$ or~$f$.

For example, when we stop the Jarn\'\i{}k's algorithm at some moment and
we take a~subgraph~$C$ induced by the constructed tree, this subgraph is contractible.
We can now easily reformulate the Contraction lemma (\ref{contlemma}) in the language
of contractible subgraphs:

\lemman{Generalized contraction}
When~$C\subseteq G$ is a~contractible subgraph, then $\msf(G)=\msf(C) \cup \msf(G/C)$.

We can now bring corruption back to the game and state a~``robust'' version
of this lemma. A~notation for corrupted graphs will be handy:

\nota\id{corrnota}%
When~$G$ is a~weighted graph and~$R$ a~subset of its edges, we will use $G\crpt
R$ to denote an arbitrary graph obtained from~$G$ by increasing the weights of
some of the edges in~$R$.
Whenever~$C$ is a~subgraph of~$G$, we will use $R^C$ to refer to the edges of~$R$ with
exactly one endpoint in~$C$ (i.e., $R^C = R\cap \delta(C)$).

\lemman{Robust contraction, Chazelle \cite{chazelle:almostacker}}\id{robcont}%
Let $G$ be a~weighted graph and $C$~its subgraph contractible in~$G\crpt R$
for some set~$R$ of edges. Then $\msf(G) \subseteq \msf(C) \cup \msf((G/C) \setminus R^C) \cup R^C$.

\para
We will mimic the Iterated Jarn\'\i{}k's algorithm. We will partition the given graph to a~collection~$\C$
of non-overlapping contractible subgraphs called \df{clusters} and we put aside all edges that got corrupted in the process.
We recursively compute the MSF of those subgraphs and of the contracted graph. Then we take the
union of these MSF's and add the corrupted edges. According to the previous lemma, this does not produce
the MSF of~$G$, but a~sparser graph containing it, on which we can continue.

\thmn{Partitioning to contractible clusters, Chazelle \cite{chazelle:almostacker}}\id{partthm}%
Given a~weighted graph~$G$ and parameters $\varepsilon$ ($0<\varepsilon\le 1/2$)
and~$t$, we can construct a~collection $\C=\{C_1,\ldots,C_k\}$ of clusters and a~set~$R^\C$ of edges such that:

\numlist\ndotted
\:All the clusters and the set~$R^\C$ are mutually edge-disjoint.
\:Each cluster contains at most~$t$ vertices.
\:Each vertex of~$G$ is contained in at least one cluster.
\:The connected components of the union of all clusters have at least~$t$ vertices each,
  except perhaps for those which are equal to a~connected component of $G\setminus R^\C$.
\:$\vert R^\C\vert \le 2\varepsilon m$.
\:$\msf(G) \subseteq \bigcup_i \msf(C_i) \cup \msf\bigl((G / \bigcup_i C_i) \setminus R^\C\bigr) \cup R^\C$.
\:The construction takes $\O(n+m\log (1/\varepsilon))$ time.
\endlist

\section{Decision trees}\id{dtsect}%

The Pettie's and Ramachandran's algorithm combines the idea of robust partitioning with optimal decision
trees constructed by brute force for very small subgraphs.
Formally, the decision trees are defined as follows:

\defnn{Decision trees and their complexity}\id{decdef}%
A~\df{MSF decision tree} for a~graph~$G$ is a~binary tree. Its internal vertices
are labeled with pairs of $G$'s edges to be compared, each of the two outgoing tree edges
corresponds to one possible result of the comparison.
Leaves of the tree are labeled with spanning trees of the graph~$G$.

A~\df{computation} of the decision tree on a~specific permutation of edge weights
in~$G$ is the path from the root to a~leaf such that the outcome of every comparison
agrees with the edge weights. The result of the computation is the spanning tree
assigned to its final leaf.
A~decision tree is \df{correct} iff for every permutation the corresponding
computation results in the real MSF of~$G$ with the particular weights.

The \df{time complexity} of a~decision tree is defined as its depth. It therefore
bounds the number of comparisons spent on every path. (It need not be equal since
some paths need not correspond to an~actual computation --- the sequence of outcomes
on the path could be unsatisfiable.)

A~decision tree is called \df{optimal} if it is correct and its depth is minimum possible
among the correct decision trees for the given graph.
We will denote an~arbitrary optimal decision tree for~$G$ by~${\cal D}(G)$ and its
complexity by~$D(G)$.

The \df{decision tree complexity} $D(m,n)$ of the MSF problem is the maximum of~$D(G)$
over all graphs~$G$ with $n$~vertices and~$m$ edges.

\obs
Decision trees are the most general deterministic comparison-based computation model possible.
The only operations that count in its time complexity are comparisons. All
other computation is free, including solving NP-complete problems or having
access to an~unlimited source of non-uniform constants. The decision tree
complexity is therefore an~obvious lower bound on the time complexity of the
problem in all other comparison-based models.

The downside is that we do not know any explicit construction of the optimal
decision trees, or at least a~non-constructive proof of their complexity.
On the other hand, the complexity of any existing comparison-based algorithm
can be used as an~upper bound on the decision tree complexity. Also, we can
construct an~optimal decision tree using brute force:

\lemma
An~optimal MST decision tree for a~graph~$G$ on~$n$ vertices can be constructed on
the Pointer Machine in time $\O(2^{2^{4n^2}})$.

\section{An optimal algorithm}\id{optalgsect}%

Once we have developed the soft heaps, partitioning and MST decision trees,
it is now simple to state the Pettie's and Ramachandran's MST algorithm
and prove that it is asymptotically optimal among all MST algorithms in
comparison-based models. Several standard MST algorithms from the previous
chapters will also play their roles.

We will describe the algorithm as a~recursive procedure. When the procedure is
called on a~graph~$G$, it sets the parameter~$t$ to roughly $\log^{(3)} n$ and
it calls the \<Partition> procedure to split the graph into a~collection of
clusters of size~$t$ and a~set of corrupted edges. Then it uses precomputed decision
trees to find the MSF of the clusters. The graph obtained by contracting
the clusters is on the other hand dense enough, so that the Iterated Jarn\'\i{}k's
algorithm runs on it in linear time. Afterwards we combine the MSF's of the clusters
and of the contracted graphs, we mix in the corrupted edges and run two iterations
of the Contractive Bor\o{u}vka's algorithm. This guarantees reduction in the number of
both vertices and edges by a~constant factor, so we can efficiently recurse on the
resulting graph.

\algn{Optimal MST algorithm, Pettie and Ramachandran \cite{pettie:optimal}}\id{optimal}%
\algo
\algin A~connected graph~$G$ with an~edge comparison oracle.
\:If $G$ has no edges, return an~empty tree.
\:$t\=\lfloor\log^{(3)} n\rfloor$. \cmt{the size of clusters}
\:Call \<Partition> (\ref{partition}) on $G$ and $t$ with $\varepsilon=1/8$. It returns
  a~collection~$\C=\{C_1,\ldots,C_k\}$ of clusters and a~set~$R^\C$ of corrupted edges.
\:$F_i \= \mst(C_i)$ for all~$i$, obtained using optimal decision trees.
\:$G_A \= (G / \bigcup_i C_i) \setminus R^\C$. \cmt{the contracted graph}
\:$F_A \= \msf(G_A)$ calculated by the Iterated Jarn\'\i{}k's algorithm (\ref{itjar}).
\:$G_B \= \bigcup_i F_i \cup F_A \cup R^\C$. \cmt{combine subtrees with corrupted edges}
\:Run two Bor\o{u}vka steps (iterations of the Contractive Bor\o{u}vka's algorithm, \ref{contbor}) on~$G_B$,
  getting a~contracted graph~$G_C$ and a~set~$F_B$ of MST edges.
\:$F_C \= \mst(G_C)$ obtained by a~recursive call to this algorithm.
\:Return $F_B \cup F_C$.
\algout The minimum spanning tree of~$G$.
\endalgo

Correctness of this algorithm immediately follows from the Partitioning theorem (\ref{partthm})
and from the proofs of the respective algorithms used as subroutines. As for time complexity:

\lemma\id{optlemma}%
The time complexity $T(m,n)$ of the Optimal algorithm satisfies the following recurrence:
$$
T(m,n) \le \sum_i c_1 D(C_i) + T(m/2, n/4) + c_2 m,
$$
where~$c_1$ and~$c_2$ are some positive constants and $D$~is the decision tree complexity
from the previous section.

It turns out that the recurrence is satisfied by the decision tree complexity function
$D(m,n)$ itself, so we can prove the following theorem by induction:

\thm
The time complexity of the Optimal algorithm is $\Theta(D(m,n))$.

\paran{Complexity of MST}%
As we have already noted, the exact decision tree complexity $D(m,n)$ of the MST problem
is still open and so therefore is the time complexity of the optimal algorithm. However,
every time we come up with another comparison-based algorithm, we can use its complexity
(or more specifically the number of comparisons it performs, which can be even lower)
as an~upper bound on the optimal algorithm.
The best explicit comparison-based algorithm known to date has been discovered by Chazelle
\cite{chazelle:ackermann} and independently by Pettie \cite{pettie:ackermann}. It achieves complexity $\O(m\timesalpha(m,n))$.
Using any of these results, we can prove an~Ackermannian upper bound on the
optimal algorithm:

\thm
The time complexity of the Optimal algorithm is $\O(m\alpha(m,n))$.

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

\chapter{Ranking Combinatorial Structures}\id{rankchap}%

\chapter{Bibliography}

\dumpbib

\vfill\eject
\ifodd\pageno\else\hbox{}\fi

\bye
