\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{$\mst(G)$}{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{$\beta(m,n)$}{$\beta(m,n) := \min\{i \mid \log^{(i)}n \le m/n \}$ \[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{$\(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]}
projects / saga.git / commitdiff