\input macros.tex
\fi
-\chapter{Notation}
+\chapter{Notation}\id{notapp}%
+
+\section{Symbols}
{\obeylines\parskip=0pt
-\def\n#1#2{\>\hbox to 6em{#1 \dotfill} #2}
+\def\n#1#2{\>\hangindent=6em\hangafter=1 \hbox to 6em{#1 \dotfill~}#2}
\def\[#1]{~{\it(\ref{#1})}}
-\n{$\bb R$}{the set of all real numbers}
-\n{$\bb N$}{the set of all natural numbers, including 0}
-\n{${\bb N}^+$}{the set of all positive integers}
+
+\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{$\Omega(g)$}{asymptotic~$\Omega$: $f=\Omega(g)$ iff $\exists c>0: f(n)\ge g(n)$ for all~$n\ge n_0$}
-\n{$\Theta(g)$}{asymptotic~$\Theta$: $f=\Theta(g)$ iff $f=\O(g)$ and $f=\Omega(g)$}
\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{$w(e)$}{weight of an edge $e$}
-\n{$V(G)$}{set of vertices of a graph~$G$}
-\n{$E(G)$}{set of edges of a graph~$G$}
-\n{$n(G)$}{number of vertices of a graph~$G$, that is $\vert V(G)\vert$}
-\n{$m(G)$}{number of edges of a graph~$G$, that is $\vert E(G)\vert$}
-\n{$V,E,n,m$}{when used without $(G)$, they refer to the input of the current algorithm}
\n{$G[U]$}{subgraph induced by a~set $U\subset V(G)$}
-\n{$\delta_G(U)$}{all edges connecting $U\subset V(G)$ with $V(G)\setminus U$; we usually omit the~$G$}
-\n{$\delta_G(v)$}{the edges of a one-vertex cut, i.e., $\delta_G(\{v\})$}
-\n{MST}{minimum spanning tree \[mstdef]}
-\n{MSF}{minimum spanning forest \[mstdef]}
-\n{$\mst(G)$}{the unique minimum spanning tree of a graph~$G$ \[mstnota]}
-\n{$\msf(G)$}{the unique minimum spanning forest of a graph~$G$ \[mstnota]}
\n{$X \choose k$}{the set of all $k$-element subsets of a set~$X$}
\n{$G/e$}{multigraph contraction \[contract]}
-\n{$G.e$}{simple graph contraction \[simpcont]}
-\n{$G/X$, $G.X$}{contraction by a~set $X$ of vertices or edges \[setcont]}
-\n{$f[X]$}{function applied to a set: $f[X]:=\{ f(x) ; x\in X \}$}
+\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{$\varrho({\cal C})$}{edge density of a graph class~$\cal C$ \[density]}
-\n{$\deg_G(v)$}{degree of vertex~$v$ in graph~$G$; we omit $G$ if it is clear from context}
-\n{${\E}X$}{expected value of a~random variable~$X$}
-\n{${\rm Pr}[\varphi]$}{probability that a predicate~$\varphi$ is true}
-\n{$\log n$}{a binary logarithm of the number~$n$}
\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{$\log^* n$}{the iterated logarithm: $\log^*n := \min\{i: \log^{(i)}n \le 1\}$; the inverse of~$2\tower n$}
-\n{$\beta(m,n)$}{$\beta(m,n) := \min\{i: \log^{(i)}n \le m/n \}$ \[itjarthm]}
-\n{$W$}{word size of the RAM \[wordsize]}
\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{$\sigma^k$}{the string~$\sigma$ repeated $k$~times \[bitnota]}
\n{$\0$, $\1$}{bits in a~bit string \[bitnota]}
\n{$\equiv$}{congruence modulo a~given number}
-\n{$\<LSB>(x)$}{the position of the lowest bit set in~$x$ \[lsbmsb]}
-\n{$\<MSB>(x)$}{the position of the highest bit set in~$x$ \[lsbmsb]}
-\n{$\bf x$}{a~vector with elements $x_1,\ldots,x_d$; $x$ is its bitwise encoding \[vecnota]}
-\n{$\band$}{bitwise conjunction: $(x\band y)[i]=1$ iff $x[i]=1 \land y[i]=1$}
-\n{$\bor$}{bitwise disjunction: $(x\bor y)[i]=1$ iff $x[i]=1 \lor y[i]=1$}
-\n{$\bnot$}{bitwise negation: $(\bnot x)[i]=1-x[i]$}
-\n{$\bxor$}{bitwise non-equivalence: $(x\bxor y)[i]=1$ iff $x[i]\ne y[i]$}
+\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{$R_{C,\prec}(x)$}{the rank of~$x$ in a~set~$C$ ordered by~$\prec$ \[rankdef]}
-\n{$R^{-1}_{C,\prec}(i)$}{the unrank of~$i$: the $i$-th smallest element of a~set~$C$ ordered by~$\prec$ \[rankdef]}
\n{$[n]$}{the set $\{1,2,\ldots,n\}$ \[pranksect]}
-\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{$n^{\underline k}$}{the $k$-th falling factorial power: $n\cdot(n-1)\cdot\ldots\cdot(n-k+1)$ \[kpranksect]}
+\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{$K_k$}{the complete graph on~$k$ vertices}
-\n{$C_k$}{the cycle on~$k$ vertices}
-\n{${\cal P}_A$}{the set of all permutations on a~set~$A$ \[restnota]}
-\n{${\cal P}_{A,M}$}{the set of all permutations on~$A$ satisfying the restrictions~$M$ \[restnota]}
-\n{$N_0(M)$}{the number of permutations satisfying the restrictions~$M$ \[restnota]}
-\n{$M^{i,j}$}{the matrix $M$ with $i$-th row and $j$-th column deleted \[restnota]}
-\n{$D_n$}{the $n\times n$ matrix with $D[i,i]=0$ for all~$i$ and ones elsewhere else \[hatrank]}
-\n{$\per M$}{the permanent of a~square matrix~$M$}
\n{$G\crpt R$}{graph~$G$ with edges in~$R$ corrupted \[corrnota]}
\n{$R^C$}{$R^C = R\cap \delta(C)$ \[corrnota]}
-\n{${\cal D}(G)$}{The optimal MSF decision tree for a~graph~$G$ \[decdef]}
-\n{$D(G)$}{The depth of ${\cal D}(G)$ \[decdef]}
-\n{$D(m,n)$}{Decision tree complexity of MSF \[decdef]}
-\n{$A(x,y)$}{The Ackermann's function \[ackerdef]}
-\n{$A(x)$}{The diagonal Ackermann's function \[ackerdef]}
-\n{$a(x,n)$}{The inverse of the $x$-th row of the Ackermann's function \[ackerinv]}
-\n{$a(n)$}{The 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{$\Eul(T)$}{The Eulerian tour sequence for a~tree~$T$ \[eulseq]}
+\n{$M^{i,j}$}{the matrix $M$ with $i$-th row and $j$-th column deleted \[restnota]}
+
}
%--------------------------------------------------------------------------------
(if the edge is a loop).
\proclaim{Notation}%
-When the meaning is clear from the context, we use our notation originally
-defined for graphs even for multigraphs. For example, $xy\in E(G)$ becomes a
+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 edge. \df{(Multigraph) contraction of~$G$ along~$e$}
+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:
$$\eqalign{
V' &= (V(G) \setminus \{x,y\}) \cup \{v_e\},\quad\hbox{where $v_e$ is a new vertex,}\cr
E' &= E(G) - \{e\},\cr
-M'(f) &= \{ m(v) ; v\in M(f) \} \quad\hbox{for every $f=\in E'$, and}\cr
-m(x) &= \cases{v_e & \hbox{for $v=x,y,$}\cr v & \hbox{otherwise.}} \cr
+M'(f) &= \{ m(v) \mid v\in M(f) \} \quad\hbox{for every $f\in E'$, and}\cr
+m(x) &= \cases{v_e & \hbox{for $v=x,y,$}\cr \noalign{\vskip5pt} v & \hbox{otherwise.}} \cr
}$$
-Sometimes we need contraction for simple graphs as well. It corresponds to performing
-the multigraph contraction, unifying parallel edges and deleting loops.
+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 edge. \df{(Simple graph) contraction of~$G$ along~$e$}
-produces a graph $G.e=(V',E')$ such that:
+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:
$$\eqalign{
V' &= (V(G) \setminus \{x,y\}) \cup \{v_e\},\quad\hbox{where $v_e$ is a new vertex,}\cr
-E' &= \{ \{m(x),m(y)\} ; xy\in E \land m(x)\ne m(y) \},\cr
-m(x) &= \cases{v_e & \hbox{for $v=x,y,$}\cr v & \hbox{otherwise.}} \cr
+E' &= \{ \{m(x),m(y)\} \mid xy\in E \land m(x)\ne m(y) \},\cr
+m(x) &= \cases{v_e & \hbox{for $v=x,y,$}\cr \noalign{\vskip5pt} v & \hbox{otherwise.}} \cr
}$$
\defn\id{setcont}%
-We can also extend the above definitions to contractions by a~set of vertices or edges.
+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~$F$ (you can observe that the result
-does not depend on the order of edges). For $U\subseteq V(G)$, we define $G/U$
+$f_1,\ldots,f_k$ 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 $G.F$ and $G.U$.
+where $U^*$ is the complete graph on~$U$. Similarly for $G\sgc F$ and $G\sgc U$.
%--------------------------------------------------------------------------------
-\section{Ackermann's function and its inverse}\id{ackersec}%
+\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 various algorithms and mathematical structures surprisingly often:
+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
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 the differences are in factors that
-are negligible in the light of the giant asymptotic growth of the function.
+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:
+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:
&= A(2,A(2,4)) = 2\tower(2\tower 4) = 2\tower 65536. \cr
}$$
-\para
-Three functions related to the inverse of the function~$A$ are usually considered:
+\paran{Inverses}%
+As common for functions of multiple parameters, there is no single function which
+could claim the title of the only true Inverse Ackermann's function.
+The following three functions related to the inverse of the function~$A$ are often considered:
\defn\id{ackerinv}%
-The \df{row inverse} $a(x,y)$ of the Ackermann's function is defined by:
+The \df{$i$-th row inverse} $\lambda_i(n)$ of the Ackermann's function is defined by:
$$
-a(x,n) := \min\{ y \mid A(x,y) > \log n \}.
+\lambda_i(n) := \min\{ y \mid A(i,y) > \log n \}.
$$
-The \df{diagonal inverse} $a(n)$ is defined by:
+The \df{diagonal inverse} $\alpha(n)$ is defined by:
$$
-a(n) := \min\{ x \mid A(x) > \log n \}.
+\alpha(n) := \min\{ x \mid A(x) > \log n \}.
$$
-The \df{alpha function} $\alpha(m,n)$ is defined for $m\ge n$ by:
+The two-parameter \df{alpha function} $\alpha(m,n)$ is defined for $m\ge n$ by:
$$
\alpha(m,n) := \min\{ x\ge 1 \mid A(x,4\lceil m/n\rceil) > \log n \}.
$$
\example
-$a(1,n) = \O(\log\log n)$, $a(2,n) = \O(\log^* n)$, $a(3,n)$ grows even slower
-and $a(n)$ is asymptotically smaller than $a(x,n)$ for any fixed~$x$.
+$\lambda_1(n) = \O(\log\log n)$, $\lambda_2(n) = \O(\log^* n)$, $\lambda_3(n)$ 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
which covers all values of~$m$ that are likely to occur in practice.
\lemma
-$\alpha(m,n) \le a(n)+1$.
+$\alpha(m,n) \le \alpha(n)+1$.
\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 $k$~is a~fixed constant and $m\ge n\cdot a(k,n)$, then $\alpha(m,n) \le k$.
+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 $A(x,4\lceil m/n\rceil) \ge A(x,a(k,n))$, which
-is greater than $\log n$ for all $x \ge k$.
+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
projects / saga.git / blobdiff