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

\chapter{Notation}\id{notapp}%

\section{Symbols}

{\obeylines\parskip=0pt
\def\n#1#2{\>\hangindent=6em\hangafter=1 \hbox to 6em{#1 \dotfill~}#2}
\def\[#1]{~{\it(\ref{#1})}}

\n{$A(x,y)$}{Ackermann's function \[ackerdef]}
\n{$A(x)$}{diagonal Ackermann's function \[ackerdef]}
\n{$\band$}{bitwise conjunction: $(x\band y)[i]=1$ iff $x[i]=1 \land y[i]=1$}
\n{$C_k$}{cycle on~$k$ vertices}
\n{${\cal D}(G)$}{optimal MSF decision tree for a~graph~$G$ \[decdef]}
\n{$D(G)$}{depth of ${\cal D}(G)$ \[decdef]}
\n{$D(m,n)$}{decision tree complexity of MSF for $m$~edges and $n$~vertices \[decdef]}
\n{$D_n$}{$n\times n$ matrix with 0's on the main diagonal and 1's elsewhere \[hatrank]}
\n{$\deg_G(v)$}{degree of vertex~$v$ in graph~$G$; we omit $G$ if it is clear from context}
\n{$E(G)$}{set of edges of a graph~$G$}
\n{$E$}{$E(G)$ when the graph~$G$ is clear from context}
\n{${\E}X$}{expected value of a~random variable~$X$}
\n{$K_k$}{complete graph on~$k$ vertices}
\n{$L(\pi,A)$}{lexicographic ranking function for permutations on a~set~$A\subseteq{\bb N}$ \[brackets]}
\n{$L^{-1}(i,A)$}{lexicographic unranking function, the inverse of~$L$ \[brackets]}
\n{$\log n$}{binary logarithm of the number~$n$}
\n{$\log^* n$}{iterated logarithm: $\log^*n := \min\{i \mid \log^{(i)}n \le 1\}$; the inverse of~$2\tower n$}
\n{$\<LSB>(x)$}{position of the lowest bit set in~$x$ \[lsbmsb]}
\n{$\<MSB>(x)$}{position of the highest bit set in~$x$ \[lsbmsb]}
\n{MSF}{minimum spanning forest \[mstdef]}
\n{$\msf(G)$}{the unique minimum spanning forest of a graph~$G$ \[mstnota]}
\n{MST}{minimum spanning tree \[mstdef]}
\n{$\mst(G)$}{the unique minimum spanning tree of a graph~$G$ \[mstnota]}
\n{$m(G)$}{number of edges of a graph~$G$, that is $\vert E(G)\vert$}
\n{$m$}{$m(G)$ when the graph~$G$ is clear from context}
\n{$\bb N$}{set of all non-negative integers}
\n{${\bb N}^+$}{set of all positive integers}
\n{$N_0(M)$}{number of permutations satisfying the restrictions~$M$ \[restnota]}
\n{$n(G)$}{number of vertices of a graph~$G$, that is $\vert V(G)\vert$}
\n{$n$}{$n(G)$ when the graph~$G$ is clear from context}
\n{$\bnot$}{bitwise negation: $(\bnot x)[i]=1-x[i]$}
\n{$\O(g)$}{asymptotic~$O$: $f=\O(g)$ iff $\exists c>0: f(n)\le g(n)$ for all~$n\ge n_0$}
\n{$\widetilde\O(g)$}{$f=\widetilde\O(g)$ iff $f=\O(g\cdot\log^{\O(1)} g)$}
\n{$\bor$}{bitwise disjunction: $(x\bor y)[i]=1$ iff $x[i]=1 \lor y[i]=1$}
\n{${\cal P}_A$}{set of all permutations on a~set~$A$ \[restnota]}
\n{${\cal P}_{A,M}$}{set of all permutations on~$A$ satisfying the restrictions~$M$ \[restnota]}
\n{$\per M$}{permanent of a~square matrix~$M$}
\n{$\poly(n)$}{$f=\poly(n)$ iff $f=\O(n^c)$ for some $c$}
\n{${\rm Pr}[\varphi]$}{probability that a predicate~$\varphi$ is true}
\n{$\bb R$}{set of all real numbers}
\n{$R_{C,\prec}(x)$}{rank of~$x$ in a~set~$C$ ordered by~$\prec$ \[rankdef]}
\n{$R^{-1}_{C,\prec}(i)$}{unrank of~$i$: the $i$-th smallest element of a~set~$C$ ordered by~$\prec$ \[rankdef]}
\n{$V(G)$}{set of vertices of a graph~$G$}
\n{$V$}{$V(G)$ when the graph~$G$ is clear from context}
\n{$W$}{word size of the RAM \[wordsize]}
\n{$w(e)$}{weight of an edge $e$}
\n{$\bxor$}{bitwise non-equivalence: $(x\bxor y)[i]=1$ iff $x[i]\ne y[i]$}

\n{$\alpha(n)$}{diagonal inverse of the Ackermann's function \[ackerinv]}
\n{$\alpha(m,n)$}{$\alpha(m,n) := \min\{ x\ge 1 \mid A(x,4\lceil m/n\rceil) > \log n \}$ \[ackerinv]}
\n{$\beta(m,n)$}{$\beta(m,n) := \min\{i \mid \log^{(i)}n \le m/n \}$ \[itjarthm]}
\n{$\delta_G(U)$}{the cut separating $U\subset V(G)$ from $V(G)\setminus U$ \[deltanota]}
\n{$\delta_G(v)$}{edges of a one-vertex cut, i.e., $\delta_G(\{v\})$ \[deltanota]}
\n{$\Theta(g)$}{asymptotic~$\Theta$: $f=\Theta(g)$ iff $f=\O(g)$ and $f=\Omega(g)$}
\n{$\lambda_i(n)$}{inverse of the $i$-th row of the Ackermann's function \[ackerinv]}
\n{$\varrho({\cal C})$}{edge density of a graph class~$\cal C$ \[density]}
\n{$\Omega(g)$}{asymptotic~$\Omega$: $f=\Omega(g)$ iff $\exists c>0: f(n)\ge g(n)$ for all~$n\ge n_0$}

%%\n{$x := y$}{$x$ is defined as~$y$}
\n{$T[u,v]$}{the path in a tree~$T$ joining vertices $u$ and $v$ \[heavy]}
\n{$T[e]$}{the path in a tree~$T$ joining the endpoints of an~edge~$e$ \[heavy]}
\n{$A\symdiff B$}{symetric difference of sets: $(A\setminus B) \cup (B\setminus A)$}
\n{$G-e$}{graph $G$ with edge $e$ removed}
\n{$G+e$}{graph $G$ with edge $e$ added}
\n{$G[U]$}{subgraph induced by a~set $U\subset V(G)$}
\n{$X \choose k$}{the set of all $k$-element subsets of a set~$X$}
\n{$G/e$}{multigraph contraction \[contract]}
\n{$G\sgc e$}{simple graph contraction \[simpcont]}
\n{$G/X$, $G\sgc X$}{contraction by a~set $X$ of vertices or edges \[setcont]}
\n{$f[X]$}{function applied to a set: $f[X]:=\{ f(x) \mid x\in X \}$}
\n{$f[e]$}{as edges are two-element sets, $f[e]$ maps both endpoints of an edge~$e$}
\n{$f^{(i)}$}{function~$f$ iterated $i$~times: $f^{(0)}(x):=x$, $f^{(i+1)}(x):=f(f^{(i)}(x))$}
\n{$2\tower n$}{the tower function (iterated exponential): $2\tower 0:=1$, $2\tower (n+1):=2^{2\tower n}$}
\n{$\(x)$}{number~$x\in{\bb N}$ written in binary \[bitnota]}
\n{$\(x)_b$}{$\(x)$ zero-padded to exactly $b$ bits \[bitnota]}
\n{$x[i]$}{when $x\in{\bb N}$: the value of the $i$-th bit of~$x$ \[bitnota]}
\n{$x[B]$}{when $x\in{\bb N}$: the values of the bits at positions in the set~$B$ \[qhnota]}
\n{$\pi[i]$}{when $\pi$ is a~sequence: the $i$-th element of~$\pi$, starting with $\pi[1]$ \[brackets]}
\n{$\pi[i\ldots j]$}{the subsequence $\pi[i], \pi[i+1], \ldots, \pi[j]$}
\n{$\sigma^k$}{the string~$\sigma$ repeated $k$~times \[bitnota]}
\n{$\0$, $\1$}{bits in a~bit string \[bitnota]}
\n{$\equiv$}{congruence modulo a~given number}
\n{$\bf x$}{vector with elements $x_1,\ldots,x_d$; $x$ is its bitwise encoding \[vecnota]}
\n{$x \shl n$}{bitwise shift of~$x$ by $n$~positions to the left: $x\shl n = x\cdot 2^n$}
\n{$x \shr n$}{bitwise shift of~$x$ by $n$~positions to the right: $x\shr n = \lfloor x/2^n \rfloor$}
\n{$[n]$}{the set $\{1,2,\ldots,n\}$ \[pranksect]}
\n{$n^{\underline k}$}{$k$-th falling factorial power: $n\cdot(n-1)\cdot\ldots\cdot(n-k+1)$ \[kpranksect]}
\n{$H\minorof G$}{$H$ is a~minor of~$G$ \[minordef]}
\n{$G\crpt R$}{graph~$G$ with edges in~$R$ corrupted \[corrnota]}
\n{$R^C$}{$R^C = R\cap \delta(C)$ \[corrnota]}
\n{$M^{i,j}$}{the matrix $M$ with $i$-th row and $j$-th column deleted \[restnota]}

}

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

\section{Multigraphs and contractions}

Since the formalism of multigraphs is not fixed in the literature, we will
better define it carefully, following \cite{diestel:gt}:

\defn A~\df{multigraph} is an ordered triple $(V,E,M)$, where $V$~is the
set of vertices, $E$~is the set of edges, taken as abstract objects disjoint
with the vertices, and $M$ is a mapping $E\rightarrow {V \choose 2} \cup {V \choose 1}$
which assigns to each edge either a pair of vertices or a single vertex
(if the edge is a loop).

\proclaim{Notation}%
When the meaning is clear from the context, we use the standard graph notation
even for multigraphs. For example, $xy\in E(G)$ becomes a
shorthand for $\exists e\in E(G)$ such that $M(G)(e) = \{x,y\}$. Also, we
consider multigraphs with no multiple edges nor loops and simple graphs to be
the same objects, although they formally differ.

\defn\id{contract}%
Let $G=(V,E,M)$ be a multigraph and $e=xy$ its arbitrary edge.
The \df{(multigraph) contraction of~$e$ in~$G$}
produces a multigraph $G/e=(V',E',M')$ such that:
$$\eqalign{
V' &= (V(G) \setminus \{x,y\}) \cup \{v_e\},\quad\hbox{where $v_e$ is a new vertex,}\cr
E' &= E(G) - \{e\},\cr
M'(f) &= \{ m(v) \mid v\in M(f) \} \quad\hbox{for every $f\in E'$, and}\cr
m(x) &= \cases{v_e & \hbox{for $v=x,y,$}\cr \noalign{\vskip5pt} v & \hbox{otherwise.}} \cr
}$$

We sometimes also need to contract edges in simple graphs. It is equivalent to performing
the multigraph contraction and then unifying parallel edges and deleting loops.

\defn\id{simpcont}%
Let $G=(V,E)$ a simple graph and $e=xy$ its arbitrary edge.
The \df{(simple graph) contraction of~$e$ in~$G$}
produces a graph $G\sgc e=(V',E')$ such that:
$$\eqalign{
V' &= (V(G) \setminus \{x,y\}) \cup \{v_e\},\quad\hbox{where $v_e$ is a new vertex,}\cr
E' &= \{ \{m(x),m(y)\} \mid xy\in E \land m(x)\ne m(y) \},\cr
m(x) &= \cases{v_e & \hbox{for $v=x,y,$}\cr \noalign{\vskip5pt} v & \hbox{otherwise.}} \cr
}$$

\defn\id{setcont}%
We can also extend the above definitions to contractions of a~set of vertices or edges.
For $F\subseteq E(G)$, the graph $G/F$ is defined as $(G/f_1)/f_2/\ldots/f_k$ where
$f_1,\ldots,f_k$ are the elements of~$F$ (the result obviously does not depend on the order of edges).
For $U\subseteq V(G)$, we define $G/U$
as the graph with all vertices of~$U$ merged to a~single vertex, that is $(G\cup U^*)/U^*$,
where $U^*$ is the complete graph on~$U$. Similarly for $G\sgc F$ and $G\sgc U$.

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

\section{Ackermann's function and its inverses}\id{ackersec}%

The Ackermann's function is an~extremely quickly growing function which has been
introduced by Ackermann \cite{ackermann:function} in the context of
computability theory. Its original purpose was to demonstrate that not every recursive
function is also primitive recursive. At the first sight, it does not
seem related to efficient algorithms at all. Its various inverses however occur in
analyses of algorithms and mathematical structures surprisingly often:
We meet them in Section \ref{classalg} in the time complexity of the Disjoint Set Union
data structure and also in the best known upper bound on the decision tree
complexity of minimum spanning trees in Section \ref{optalgsect}. Another
important application is in the complexity of Davenport-Schinzel sequences (see
Klazar's survey \cite{klazar:gdss}), but as far as we know, these are not otherwise
related to the topic of our study.

Various sources differ in the exact definition of both the Ackermann's
function and its inverse, but most of these differences are in factors that
are negligible in the light of the giant asymptotic growth of the function.\foot{%
To quote Pettie \cite{pettie:onlineverify}: ``In the field of algorithms \& complexity,
Ackermann's function is rarely defined the same way twice. We would not presume to buck
such a~well-established precedent. Here is a~slight variant.''}
We will use the definition by double recursion given by Tarjan \cite{tarjan:setunion},
which is predominant in the literature on graph algorithms.

\defn\id{ackerdef}%
The \df{Ackermann's function} $A(x,y)$ is a~function on non-negative integers defined as follows:
$$\eqalign{
A(0,y) &:= 2y, \cr
A(x,0) &:= 0, \cr
A(x,1) &:= 2 \quad \hbox{for $x\ge 1$}, \cr
A(x,y) &:= A(x-1, A(x,y-1)) \quad \hbox{for $x\ge 1$, $y\ge 2$}. \cr
}$$
The functions $A(x,\cdot)$ are called the \df{rows} of $A(x,y)$, similarly $A(\cdot,y)$ are
its \df{columns.}

Sometimes, a~single-parameter version of this function is also used. It is defined
as the diagonal of the previous function, i.e., $A(x):=A(x,x)$.

\example
We can try evaluating $A(x,y)$ in some points:
$$\eqalign{
A(x,2) &= A(x-1, A(x,1)) = A(x-1,2) = A(0,2) = 4, \cr
A(1,y) &= A(0, A(1,y-1)) = 2A(1,y-1) = 2^{y-1}A(1,1) = 2^y, \cr
A(2,y) &= A(1, A(2,y-1)) = 2^{A(2,y-1)} = 2\tower y \hbox{~~(the tower of exponentials),} \cr
A(3,y) &= \hbox{the tower function iterated $y$~times,} \cr
A(4,3) &= A(3,A(4,2)) = A(3,4) = A(2,A(3,3)) = A(2,A(2,A(3,2))) = \cr
       &= A(2,A(2,4)) = 2\tower(2\tower 4) = 2\tower 65536. \cr
}$$

\paran{Inverses}%
As common for functions of multiple parameters, there is no single function which
could claim the title of the only true Inverse Ackermann's function.
The following three functions related to the inverse of the function~$A$ are often considered:

\defn\id{ackerinv}%
The \df{$i$-th row inverse} $\lambda_i(n)$ of the Ackermann's function is defined by:
$$
\lambda_i(n) := \min\{ y \mid A(i,y) > \log n \}.
$$
The \df{diagonal inverse} $\alpha(n)$ is defined by:
$$
\alpha(n) := \min\{ x \mid A(x) > \log n \}.
$$
The two-parameter \df{alpha function} $\alpha(m,n)$ is defined for $m\ge n$ by:
$$
\alpha(m,n) :=  \min\{ x\ge 1 \mid A(x,4\lceil m/n\rceil) > \log n \}.
$$

\example
$\lambda_1(n) = \O(\log\log n)$, $\lambda_2(n) = \O(\log^* n)$, $\lambda_3(n)$ grows even slower
and $\alpha(n)$ is asymptotically smaller than $\lambda_i(n)$ for any fixed~$i$.

\obs
It is easy to verify that all the rows are strictly increasing and so are all
columns, except the first three columns which are constant. Therefore for a~fixed~$n$,
$\alpha(m,n)$ is maximized at $m=n$. So $\alpha(m,n) \le 3$ when $\log n < A(3,4)$,
which covers all values of~$m$ that are likely to occur in practice.

\lemma
$\alpha(m,n) \le \alpha(n)+1$.

\proof
We know that
$A(x,4\lceil m/n\rceil) \ge A(x,4) = A(x-1,A(x,3)) \ge A(x-1,x-1)$, so $A(x,4\lceil m/n\rceil)$
rises above $\log n$ no later than $A(x-1,x-1)$ does.
\qed

\lemma\id{alphaconst}%
When $i$~is a~fixed constant and $m\ge n\cdot \lambda_i(n)$, then $\alpha(m,n) \le i$.

\proof
The choice of~$m$ guarantees that $A(x,4\lceil m/n\rceil) \ge A(x,\lambda_i(n))$, which
is greater than $\log n$ for all $x \ge i$.
\qed

\endpart
