From: Martin Mares Date: Sat, 3 May 2008 18:40:30 +0000 (+0200) Subject: Corrections: Appendix A. X-Git-Tag: printed~21 X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=cd18ac99841929c27410589fe06b79bc1eb41382;p=saga.git Corrections: Appendix A. --- diff --git a/notation.tex b/notation.tex index 7c9d525..81efb49 100644 --- a/notation.tex +++ b/notation.tex @@ -16,7 +16,7 @@ \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{$D(m,n)$}{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~$G$} @@ -25,7 +25,7 @@ \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{$\log n$}{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]} @@ -35,7 +35,7 @@ \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 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 $\vert V(G)\vert$} @@ -78,7 +78,7 @@ \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{$G/X$, $G.X$}{contraction by a~set $X$ of vertices or edges \[setcont]} +\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: $f^{(0)}(x):=x$, $f^{(i+1)}(x):=f(f^{(i)}(x))$} @@ -92,11 +92,11 @@ \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{$\bf x$}{a~vector with elements $x_1,\ldots,x_d$; $x$ is its bitwise encoding \[vecnota]} +\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{$n^{\underline k}$}{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{$G\crpt R$}{graph~$G$ with edges in~$R$ corrupted \[corrnota]} \n{$R^C$}{$R^C = R\cap \delta(C)$ \[corrnota]} @@ -118,8 +118,8 @@ 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 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. @@ -131,11 +131,11 @@ produces a multigraph $G/e=(V',E',M')$ such that: $$\eqalign{ V' &= (V(G) \setminus \{x,y\}) \cup \{v_e\},\quad\hbox{where $v_e$ is a new vertex,}\cr E' &= E(G) - \{e\},\cr -M'(f) &= \{ m(v) \mid v\in M(f) \} \quad\hbox{for every $f=\in E'$, and}\cr -m(x) &= \cases{v_e & \hbox{for $v=x,y,$}\cr v & \hbox{otherwise.}} \cr +M'(f) &= \{ m(v) \mid v\in M(f) \} \quad\hbox{for every $f\in E'$, and}\cr +m(x) &= \cases{v_e & \hbox{for $v=x,y,$}\cr \noalign{\vskip5pt} v & \hbox{otherwise.}} \cr }$$ -Sometimes we need contraction for simple graphs as well. It is equivalent to performing +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}% @@ -151,8 +151,8 @@ m(x) &= \cases{v_e & \hbox{for $v=x,y,$}\cr \noalign{\vskip5pt} v & \hbox{otherw \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~$F$ (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 $G\sgc F$ and $G\sgc U$.