- impedance mismatch in terminology: contraction of G along e vs. contraction of e.
- use \delta(X) notation
- unify use of n(G) vs. n
-- use calligraphic letters from ams?
- change the notation for contractions -- use double slash instead of the dot?
- introduce \widehat\O early
-- unify { x ; ... }, { x | ...} and { x : ... }
- Ackermann: which of the Tarjan's set union papers should we cite?
-- Ackermann function vs. Ackermann's function
-Varia:
+Typography:
-- cite GA booklet
- formatting of multi-line \algin, \algout
+- use calligraphic letters from ams?
+
+Global:
+
+- Intro: cite GA booklet
- each chapter should make clear in which model we work
+- clean up bibliography
\thm\id{itjarthm}%
The Iterated Jarn\'\i{}k's algorithm finds the MST of the input graph in time
-$\O(m\timesbeta(m,n))$, where $\beta(m,n):=\min\{ i: \log^{(i)}n \le m/n \}$.
+$\O(m\timesbeta(m,n))$, where $\beta(m,n):=\min\{ i \mid \log^{(i)}n \le m/n \}$.
\proof
Phases are finite and in every phase at least one edge is contracted, so the outer
to $w(uv)$. It is therefore sufficient to solve the following problem:
\problem
-Given a~weighted tree~$T$ and a~set of \df{query paths} $Q \subseteq \{ T[u,v] ; u,v\in V(T) \}$
+Given a~weighted tree~$T$ and a~set of \df{query paths} $Q \subseteq \{ T[u,v] \mid u,v\in V(T) \}$
specified by their endpoints, find the heaviest edge (\df{peak}) for every path in~$Q$.
\paran{Bor\o{u}vka trees}%
\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{$f[X]$}{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{$\log n$}{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]}
$$\eqalign{
V' &= (V(G) \setminus \{x,y\}) \cup \{v_e\},\quad\hbox{where $v_e$ is a new vertex,}\cr
E' &= E(G) - \{e\},\cr
-M'(f) &= \{ m(v) ; v\in M(f) \} \quad\hbox{for every $f=\in E'$, and}\cr
+M'(f) &= \{ m(v) \mid v\in M(f) \} \quad\hbox{for every $f=\in E'$, and}\cr
m(x) &= \cases{v_e & \hbox{for $v=x,y,$}\cr v & \hbox{otherwise.}} \cr
}$$
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
For our set~$X$, we define~$T$ as a~compressed trie for the set of binary
-encodings of the numbers~$x_i$, padded to exactly $W$~bits, i.e., for $S = \{ \(x)_W ; x\in X \}$.
+encodings of the numbers~$x_i$, padded to exactly $W$~bits, i.e., for $S = \{ \(x)_W \mid x\in X \}$.
\obs
The trie~$T$ has several interesting properties. Since all words in~$S$ have the same
Let~$H\subseteq B$ be any set of holes in the board. Then:
\itemize\ibull
\:$N_j$ denotes the number of placements of $n$~rooks on the board such that exactly~$j$ of the rooks
-stand on holes. That is, $N_j := \#\{ \pi\in{\cal P}: \#(H\cap T(\pi)) = j \}$.
+stand on holes. That is, $N_j := \#\{ \pi\in{\cal P} \mid \#(H\cap T(\pi)) = j \}$.
\:$r_k$ is the number of ways how to place $k$~rooks on the holes. In other words,
this is the number of $k$-element subsets of~$H$ such that no two elements share
a~common row or column.
\example\id{hatcheck}%
Let us apply this theory to the hatcheck lady problem. The set~$H$ of holes is the main diagonal
-of the board: $H=\{ (i,i) : i\in[n] \}$. When we want to place $k$~rooks on the holes,
+of the board: $H=\{ (i,i) \mid i\in[n] \}$. When we want to place $k$~rooks on the holes,
we can do that in $r_k={n\choose k}$ ways. By the previous corollary, the number of
derangements is:
$$
\nota\id{hatrank}%
As we already know, the hatcheck permutations correspond to restriction
matrices that contain zeroes only on the main diagonal and graphs that are
-complete bipartite with the matching $\{(i,i) : i\in[n]\}$ deleted. For
+complete bipartite with the matching $\{(i,i) \mid i\in[n]\}$ deleted. For
a~given order~$n$, we will call this matrix~$D_n$ and the graph~$G_n$ and
we will show that the submatrices of~$D_n$ share several nice properties:
projects / saga.git / commitdiff