{\obeylines\parskip=0pt
\def\n#1#2{\>\hbox to 6em{#1 \dotfill} #2}
-\def\[#1]{[\ref{#1}]}
+\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{$\O(g)$}{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{$\poly(n)$}{$f=\poly(n)$ iff $f=\O(n^c)$ for some $c$}
\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{$n(G)$}{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{$X \choose k$}{a set of all $k$-element subsets of a set~$X$}
+\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{$\alpha(n)$}{the inverse Ackermann's function}
-\n{$f[X]$}{function applied to a set: $f[X]:=\{ f(x) ; x\in X \}$}
+\n{$G/X$, $G.X$}{contraction by a~set $X$ of vertices or edges \[setcont]}
+\n{$f[X]$}{function applied to a set: $f[X]:=\{ f(x) \mid x\in X \}$}
\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{${\bb E}X$}{expected value of a~random variable~$X$}
+\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{$\log^* n$}{the iterated logarithm: $\log^*n := \min\{i \mid \log^{(i)}n \le 1\}$; the inverse of~$2\tower n$}
+\n{$\beta(m,n)$}{$\beta(m,n) := \min\{i \mid \log^{(i)}n \le m/n \}$ \[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{$x[B]$}{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{$\bxor$}{bitwise non-equivalence: $(x\bxor y)[i]=1$ iff $x[i]\ne y[i]$}
\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{$\prec$}{an~arbitrary linear order}
\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{$H\minorof G$}{$H$ 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{$\lambda_i(n)$}{The inverse of the $i$-th row of the Ackermann's function \[ackerinv]}
+\n{$\alpha(n)$}{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]}
}
%--------------------------------------------------------------------------------
\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 $E\mapsto V \cup {V \choose 2}$
+with the vertices, and $M$ is a mapping $E\rightarrow V \cup {V \choose 2}$
which assigns to each edge either a pair of vertices or a single vertex
(if the edge is a loop).
$$\eqalign{
V' &= (V(G) \setminus \{x,y\}) \cup \{v_e\},\quad\hbox{where $v_e$ is a new vertex,}\cr
E' &= E(G) - \{e\},\cr
-M'(f) &= \{ m(v) ; v\in M(f) \} \quad\hbox{for every $f=\in E'$, and}\cr
+M'(f) &= \{ m(v) \mid v\in M(f) \} \quad\hbox{for every $f=\in E'$, and}\cr
m(x) &= \cases{v_e & \hbox{for $v=x,y,$}\cr v & \hbox{otherwise.}} \cr
}$$
produces a graph $G.e=(V',E')$ such that:
$$\eqalign{
V' &= (V(G) \setminus \{x,y\}) \cup \{v_e\},\quad\hbox{where $v_e$ is a new vertex,}\cr
-E' &= \{ \{m(x),m(y)\} ; xy\in E \land m(x)\ne m(y) \},\cr
+E' &= \{ \{m(x),m(y)\} \mid xy\in E \land m(x)\ne m(y) \},\cr
m(x) &= \cases{v_e & \hbox{for $v=x,y,$}\cr v & \hbox{otherwise.}} \cr
}$$
+\defn\id{setcont}%
+We can also extend the above definitions to contractions by 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 $G/U$
+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 $G.U$.
+
+%--------------------------------------------------------------------------------
+
+\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:
+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 the 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:
+$$\eqalign{
+A(x,2) &= A(x-1, A(x,1)) = A(x-1,2) = A(0,2) = 4, \cr
+A(1,y) &= A(0, A(1,y-1)) = 2A(1,y-1) = 2^{y-1}A(1,1) = 2^y, \cr
+A(2,y) &= A(1, A(2,y-1)) = 2^{A(2,y-1)} = 2\tower y \hbox{~~(the tower of exponentials),} \cr
+A(3,y) &= \hbox{the tower function iterated $y$~times,} \cr
+A(4,3) &= A(3,A(4,2)) = A(3,4) = A(2,A(3,3)) = A(2,A(2,A(3,2))) = \cr
+ &= A(2,A(2,4)) = 2\tower(2\tower 4) = 2\tower 65536. \cr
+}$$
+
+\para
+Three functions related to the inverse of the function~$A$ are usually 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
+$A(x,4\lceil m/n\rceil) \ge A(x,4) = A(x-1,A(x,3)) \ge A(x-1,x-1)$, 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
projects / saga.git / blobdiff