X-Git-Url: http://mj.ucw.cz/gitweb/?a=blobdiff_plain;ds=inline;f=notation.tex;h=e1e424f4ea29e21399375eac188f26167d2424a2;hb=2673d7b09e7049aa35f3f81b6e7b395c934bdfbd;hp=bebbe7283a9f33792ad9179bb9efc28c54f855a0;hpb=bf3201fca03d66baf065656d818e697dc8afe470;p=saga.git diff --git a/notation.tex b/notation.tex index bebbe72..e1e424f 100644 --- a/notation.tex +++ b/notation.tex @@ -2,53 +2,86 @@ \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 \[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{$\log n$}{a 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{$\(x)$}{position of the lowest bit set in~$x$ \[lsbmsb]} +\n{$\(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 natural numbers, including 0} +\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{$\delta_G(U)$}{all edges connecting $U\subset V(G)$ with $V(G)\setminus U$; we usually omit the~$G$} +\n{$\delta_G(v)$}{edges of a one-vertex cut, i.e., $\delta_G(\{v\})$} +\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{$T[u,v]$}{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{$G\sgc e$}{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: $f[X]:=\{ f(x) \mid x\in X \}$} \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: $\log^*n := \min\{i \mid \log^{(i)}n \le 1\}$; 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]} @@ -58,40 +91,16 @@ \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{$\(x)$}{the position of the lowest bit set in~$x$ \[lsbmsb]} -\n{$\(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{$x \shl n$}{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{$H\minorof G$}{$H$ 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]} +\n{$M^{i,j}$}{the matrix $M$ with $i$-th row and $j$-th column deleted \[restnota]} + } %-------------------------------------------------------------------------------- @@ -115,7 +124,8 @@ 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: $$\eqalign{ V' &= (V(G) \setminus \{x,y\}) \cup \{v_e\},\quad\hbox{where $v_e$ is a new vertex,}\cr @@ -124,25 +134,26 @@ M'(f) &= \{ m(v) \mid v\in M(f) \} \quad\hbox{for every $f=\in E'$, and}\cr m(x) &= \cases{v_e & \hbox{for $v=x,y,$}\cr v & \hbox{otherwise.}} \cr }$$ -Sometimes we need contraction for simple graphs as well. It corresponds to performing -the multigraph contraction, unifying parallel edges and deleting loops. +Sometimes we need contraction for simple graphs as well. 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: $$\eqalign{ V' &= (V(G) \setminus \{x,y\}) \cup \{v_e\},\quad\hbox{where $v_e$ is a new vertex,}\cr E' &= \{ \{m(x),m(y)\} \mid xy\in E \land m(x)\ne m(y) \},\cr -m(x) &= \cases{v_e & \hbox{for $v=x,y,$}\cr v & \hbox{otherwise.}} \cr +m(x) &= \cases{v_e & \hbox{for $v=x,y,$}\cr \noalign{\vskip5pt} v & \hbox{otherwise.}} \cr }$$ \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 $G/U$ 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 $G\sgc U$. %-------------------------------------------------------------------------------- @@ -226,6 +237,7 @@ which covers all values of~$m$ that are likely to occur in practice. $\alpha(m,n) \le \alpha(n)+1$. \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