text:
\section{Introduction}

In this paper we focus on exact exponential-time algorithms for the {\sc Bandwidth} problem.
Let $G = (V,E)$ be an undirected graph, where $n = |V|$ and $m = |E|$. For a given one-to-one function
$\pi: V \to \{1,2\ldots,n\}$ (called {\it ordering}) its {\it bandwidth}   is the maximum difference
between positions of adjacent vertices, i.e. $\max_{uv \in E}|\pi(u)-\pi(v)|$. The {\it bandwidth} of the graph,
denoted by ${\rm bw}(G)$, is the minimum bandwidth over all orderings. The {\sc Bandwidth} problem asks
to find an ordering with bandwidth ${\rm bw}(G)$. One can consider a decision version of the {\sc Bandwidth} problem.
Precisely, we assume that the input for our problem contains
additionally an integer $b$, $1 \le b < n$. An ordering of $V$ with bandwidth at most $b$ is called a {\em
$b$-ordering}. In the decision version we ask if there exists a $b$-ordering and if that is the
case, finding it.

A very short summary of what is known about the {\sc Bandwidth} problem follows.
On special families of graphs ${\rm bw}(G)$ can be computed in polynomial time~\cite{assman,kleitman}. However,
in general {\sc Bandwidth} is NP-hard even on some subfamilies of trees~\cite{Garey,Monien}. Moreover
Unger~\cite{unger:bandwidth} showed that {\sc Bandwidth} problem does not belong to APX even in a
very restricted case when $G$ is a caterpillar, i.e. a very simple tree.
It is also hard for any fixed level of the W hierarchy~\cite{fellows:hardness}.
The best known polynomial-time
approximation, due to Feige \cite{feige-bandwidth-approx}, has $O(\log ^3n\sqrt{\log n \log \log n})$ approximation guarantee.
On WG 2008 we presented~\cite{naszewg} exact algorithm that runs in $O^*(5^n)$ time and $O^*(2^n)$ space.

In this paper we enhance the previous algorithm to make it run in $O(4.83^n)$ time. However, the cost of this
change is $O^*(4^n)$ space complexity, which makes this result purely theoretical. In Section \ref{s:algorytm} we
describe the enhanced algorithm. In Section \ref{s:analiza} we do Measure \& Conquer analysis
(method introduced by Fomin et al. in \cite{fgk:mis}) to obtain $O(4.83^n)$ time bound. We find this analysis
interesting, because Measure \& Conquer method at the first glance does not fit for the {\sc Bandwidth} problem at all.

\section{The algorithm}\label{s:algorytm}

The algorithm consists of two phases. First, we generate partial assignments of vertices to positions:
we do not assign precise position to a vertex, but a segment of length $2(b+1)$ or $4(b+1)$ of possible positions. We do this in every possible way.
In the second phase for every generated segment assignment we check whether there exists a precise assignment of vertices to positions (i.e., an ordering), consistent
with the partial assignment.

From this point we assume, that the graph $G$ has at least two vertices
and it is connected (if $G$ is not connected we may find $b$-orderings
of each connected component of $G$ in an independent manner).
Let us choose any (but fixed for the whole algorithm) spanning tree $\ensuremath{\mathcal{D}}$ of the graph $G$.

\subsection{First phase: generating segment assignments}

\begin{definition}\label{def:segment}
A {\em{segment}} is a nonempty set of consecutive positions which has form of $\{i(b+1)+1, i(b+1)+2, \ldots, j(b+1)\} \cap \{1,2,\ldots, n\}$ for
some integers $i < j$. We say that this segment has index $(i, j)$ and denote it as $\Theta_{(i,j)}$.
For the sake of simplicity we define $\Theta_i = \Theta_{(i,i+1)}$ and call such segments {\em{base segments}}.
\end{definition}

\begin{definition}
A {\em{segment assignment}} is a function $\phi$ assigning a segment to every vertex such that the following conditions hold:
\begin{enumerate}
\item Every leaf of the spanning tree $\ensuremath{\mathcal{D}}$ is assigned to a segment of size $4(b+1)$, i.e., segment $\Theta_{(i, i+4)}$ for some integer $i$.
\item Every inner vertex of $\ensuremath{\mathcal{D}}$ is assigned to a segment of size $2(b+1)$, i.e., segment $\Theta_{(i, i+2)}$ for some integer $i$.
\item For every edge $uv$ in $\ensuremath{\mathcal{D}}$, if vertex $u$ is the parent of the vertex $v$ and $v$ is an inner vertex, where $\phi(u) = \Theta_{(i, i+2)}$ and $\phi(v) = \Theta_{(j, j+2)}$,
  then $i$ and $j$ differ by exactly one.
\item For every leaf $v$, if $u$ is the parent of $v$ in $\ensuremath{\mathcal{D}}$ and $\phi(v) = \Theta_{(i, i+2)}$, then $\phi(u) = \Theta_{(i-1, i+3)}$.
\end{enumerate}
We say that a segment assignment is {\em{consistent}} with an ordering $\pi$ if for every vertex $v$ position $\pi(v)$ belongs to the segment $\phi(v)$.
\end{definition}

\begin{lemma}\label{dalekie}
Let $\pi$ be a $b$-ordering. In any segment assignment $\phi$ consistent with the ordering $\pi$, for every edge $uv$, if
$\phi(u) = \Theta_{(i, j)}$ and $\phi(v) = \Theta_{(k, l)}$ then $j \geq k$ and $l \geq i$.
\end{lemma}

\begin{proof}
If $j < k$, then there is a gap of size at least $b+1$ between $\phi(u)$ and $\phi(v)$. Since $\pi$ is consistent with
the segment assignment, edge $uv$ is longer than $b$, contradiction. Similar argument proves that $l \geq i$.
\end{proof}

\begin{lemma}\label{istniejasegmenty}
Let $\pi$ be a $b$-ordering. There exists a segment assignment consistent with $\pi$.
\end{lemma}

\begin{proof}
As a proof we present simple construction of the segment assignment. Let $u_0$ be the root of $\ensuremath{\mathcal{D}}$. Lets assign it to any segment of length $2(b+1)$ containing $\pi(u_0)$ (there
are exactly two possible segments). Then assign segments to vertices in the root-to-leaf order. Let $u$ be an unassigned vertex with parent $v$ and let
$T_{(i, i+2)}$ be the segment assigned to $v$. Since $i(b+1) + 1 \leq \pi(v) \leq (i+2)(b+1)$ and $\pi$ is a $b$-ordering,
$$\pi(u) \in \Theta_{(i-1, i+3)} = \Theta_{(i-1, i+1)} \cup \Theta_{(i+1, i+3)}.$$
We can assign $v$ to $\Theta_{(i-1, i+3)}$ or to one of the segments $\Theta_{(i-1, i+1)}$ and $\Theta_{(i+1, i+3)}$, containing $\pi(u)$, depending
whether $v$ is an inner vertex or a leaf.
\end{proof}

Our goal in the first phase is to generate a set of segment assignments such that for every $b$-ordering there exists generated segment assignment
consistent with it. In the second phase we check for every segment assignment whether consistent $b$-ordering exists. As a result we
check if there exists any $b$-ordering of the given graph $G$.

The first phase is as follows:
\begin{enumerate}
\item Assign root $u_0$ of the tree $\ensuremath{\mathcal{D}}$ to any valid segment of size $2(b+1)$.
\item Recursively assign other vertices in the root-to-leaf order in tree $\ensuremath{\mathcal{D}}$. Given vertex $u$ with parent $v$ assigned to segment $\Theta_{(i, i+2)}$ assign $u$
to one of the segments $\Theta_{(i-1, i+1)}$, $\Theta_{(i+1, i+3)}$ if $u$ is an inner vertex, or to $\Theta_{(i-1, i+3)}$ when $u$ is a leaf (see Figure~\ref{rys:podzialy_podwojne}).\label{alg:rek}
\item For every generated assignment check condition from Lemma \ref{dalekie}: in other words accept assignment iff for every edge $uv$ in $G$ segments
assigned to vertices $u$ and $v$ are not too far from each other.
\end{enumerate}

\begin{figure}[htp]
\begin{center}
\begin{tikzpicture}

    \definecolor{gray1}{rgb}{0.7,0.7,0.7}
    \definecolor{gray2}{rgb}{0.3,0.3,0.3}
    \fill[gray1] (0,0) rectangle (6,0.5);
    \fill[gray2] (6,0) rectangle (12,0.5);
    \draw[very thin,step=.5cm] (0,0) grid (6,0.5);
    \draw[very thin,step=.5cm] (6,0) grid (12,0.5);
    \draw[very thin,step=.5cm] (3,1.5) grid (9,2);
    \draw[very thin] (3,1.5) -- (3,2);
    \draw[very thin] (3,1.5) -- (9,1.5);

    \foreach \x in {3cm, 6cm, 9cm}
    {
      \draw[very thick] (\x,1.2cm) -- (\x,2.3cm);
    }

    \foreach \x in {0cm, 3cm, 6cm, 9cm, 12cm}
    {
      \draw[very thick] (\x,-0.3cm) -- (\x,0.8cm);
    }
    \draw[thin, dashed,<->,stealth-stealth] (3,2.3) -- (6,2.3);
    \draw[thin, dashed,<->,stealth-stealth] (6,2.3) -- (9,2.3);
    \draw (4.5, 2.5) node {$b+1$};
    \draw (7.5, 2.5) node {$b+1$};

    \draw (2.75, 1.75) node {$...$};
    \draw (9.25, 1.75) node {$...$};
    \draw (-0.25, 0.25) node {$...$};
    \draw (12.25, 0.25) node {$...$};
\end{tikzpicture}
\caption{Upper part of the picture shows the segment $\Theta_{i,i+2}$ assigned to the vertex $v$. Lower part contains
  possible positions for the vertex $u$ (child of $v$ in the tree $D$) covered by two segments $\Theta_{i-1,i+1}$ and $\Theta_{i+1,i+3}$.}
\label{rys:podzialy_podwojne}
\end{center}
\end{figure}

Note that Lemma \ref{istniejasegmenty} implies that for every $b$-ordering $\pi$ there is a generated segment assignment consistent with $\pi$.
Note that this is exactly the assignment described in the proof of Lemma \ref{istniejasegmenty}.

The second phase is fed with generated segment assignments on-line. Therefore the first phase uses polynomial space.
In Step \ref{alg:rek} of the algorithm we have two possibilities for every inner non-root vertex. For the root we have
$\lceil\frac{n}{b+1}\rceil + 1 \leq n+1$ possibilities (additional $1$ comes from segment $\Theta_{-1,1}$, which is also
correct segment according to the Definition~\ref{def:segment}). We do not have a choice for leaves,
thus the algorithm generates at most $(n+1)2^{n-1}$ assignments. This algorithm uses polynomial time
for every generated assignment, so this phase works in $O^*(2^n)$ time.

\subsection{Second phase: depth-first search}

In this phase for every generated segment assignment we check if there exists a $b$-ordering consistent with the segment assignment. Let us denote this
segment assignment by $\phi$.
This phase is very similar to the second phase of our $O^*(5^n)$ algorithm \cite{naszewg},
but adapted to the segments of size $2(b+1)$ and $4(b+1)$. This difference allows us to use Measure \& Conquer method in complexity analysis.
First, we recall some key definitions and facts from
the $O^*(5^n)$ algorithm.

We assign a vertex to each position one by one, but
the main idea is the order in which we fill those positions in.
For every position $i$, let
$\ensuremath{\mathtt{segment}}(i) = \lceil \frac{i}{b+1} \rceil$ be the base segment number
of this position, and let $\ensuremath{\mathtt{color}}(i) = ((i-1) \mod (b+1)) + 1$ be
the index of the position in its base segment, which we call the {\em{color}}
of this position. Note that the color of position is the remainder of this
number modulo $b+1$, but in the range $[1,b+1]$ instead of $[0,b]$.

Let us sort positions lexicographically according to pairs $(\ensuremath{\mathtt{color}}(i), \ensuremath{\mathtt{segment}}(i))$.
To each of those positions we assign a vertex, in exactly this order. We
call this ordering {\em{the color order}} of positions.

\begin{figure}[htbp]
\begin{center}
{\small{
\begin{tikzpicture}[scale=0.7]

    \foreach \x in {1, ..., 4}
    {
      \fill[darkgray] (\x * 4.0 - 4.0, 0.0) rectangle (\x * 4.0 - 3.0, 1.0);
      \draw[white] (\x * 4.0 - 3.5, 0.5) node {$\x$};
    }
    \foreach \x in {5, ..., 8}
    {
      \fill[gray] (\x * 4.0 - 5.0 * 4.0 + 1.0, 0.0) rectangle (\x * 4.0 - 5.0 * 4.0 + 2.0, 1.0);
      \draw (\x * 4.0 - 5.0 * 4.0 + 1.5, 0.5) node {$\x$};
    }
    \foreach \x in {9, 10, 11}
    {
      \fill[lightgray] (\x * 4.0 - 9.0 * 4.0 + 2.0, 0.0) rectangle (\x * 4.0 - 9.0 * 4.0 + 3.0, 1.0);
      \draw (\x * 4.0 - 9.0 * 4.0 + 2.5, 0.5) node {$\x$};
    }
    \foreach \x in {12, 13, 14}
    {
      \fill[white] (\x * 4.0 - 12.0 * 4.0 + 3.0, 0.0) rectangle (\x * 4.0 - 12.0 * 4.0 + 4.0, 1.0);
      \draw (\x * 4.0 - 12.0 * 4.0 + 3.5, 0.5) node {$\x$};
    }
  	\draw[step=1cm] (0,0) grid (14,1);
    \foreach \x in {1,2,3}
    {
      \draw[very thick] (\x * 4.0, -0.3) -- (\x * 4.0, 1.3);
      \draw[very thin, snake=brace, mirror snake] (\x * 4.0 - 4.0, -0.3) -- (\x * 4.0, -0.3);
      \draw (\x * 4.0 - 2.0, -0.8) node {$b+1$};
    }

\end{tikzpicture}
}}
\caption{Color order of positions for $n=14$ and $b=3$.}
\label{pierwszyrysunek}
\end{center}
\end{figure}

Following lemma is the key observation in our algorithm.

\begin{lemma}\label{lemat-najwazniejsze}
Ordering $\pi$
is a $b$-ordering iff for every edge $uv$ such that $\ensuremath{\mathtt{segment}}(\pi(u)) < \ensuremath{\mathtt{segment}}(\pi(v))$ we have $\ensuremath{\mathtt{segment}}(\pi(u)) + 1 = \ensuremath{\mathtt{segment}}(\pi(v))$ and $\ensuremath{\mathtt{color}}(\pi(u)) > \ensuremath{\mathtt{color}}(\pi(v))$.
\end{lemma}

\begin{proof}
Since $\pi$ is a $b$-ordering, for every edge $uv$ we have $|\ensuremath{\mathtt{segment}}(\pi(u)) - \ensuremath{\mathtt{segment}}(\pi(v))| \leq 1$.
If $\ensuremath{\mathtt{segment}}(\pi(u)) = \ensuremath{\mathtt{segment}}(\pi(v))$ then $uv$ is not longer than $b$. Otherwise, suppose w.l.o.g. that
$\ensuremath{\mathtt{segment}}(\pi(u)) + 1 = \ensuremath{\mathtt{segment}}(\pi(v))$. Note that the distance between positions with the same color
in the neighboring segments is exactly $b+1$, so $uv$ is not longer than $b$ iff $u$ has greater color than $v$ (see Figure \ref{drugirysunek}).
\end{proof}

\begin{figure}[htbp]
\begin{center}
{\small{
\begin{tikzpicture}[scale=0.7]

    \foreach \x in {0, ..., 3}
    {
      \fill[darkgray] (\x * 4.0 + 0.0, 0.0) rectangle (\x * 4.0 + 1.0, 1.0);
    }
    \foreach \x in {4, ..., 7}
    {
      \fill[gray] (\x * 4.0 - 4.0 * 4.0 + 1.0, 0.0) rectangle (\x * 4.0 - 4.0 * 4.0 + 2.0, 1.0);
    }
    \foreach \x in {8, 9, 10}
    {
      \fill[lightgray] (\x * 4.0 - 8.0 * 4.0 + 2.0, 0.0) rectangle (\x * 4.0 - 8.0 * 4.0 + 3.0, 1.0);
    }
    \foreach \x in {11, 12, 13}
    {
      \fill[white] (\x * 4.0 - 11.0 * 4.0 + 3.0, 0.0) rectangle (\x * 4.0 - 11.0 * 4.0 + 4.0, 1.0);
    }
  	\draw[step=1cm] (0,0) grid (14,1);
    \foreach \x in {1,2,3}
    {
      \draw[very thick] (\x * 4.0, -0.3) -- (\x * 4.0, 1.3);
      \draw[very thin, snake=brace, mirror snake] (\x * 4.0 - 4.0, -0.3) -- (\x * 4.0, -0.3);
      \draw (\x * 4.0 - 2.0, -0.8) node {$b+1$};
    }
    \draw[dashed, very thin] (3.5,1.0) -- (3.5,1.5);
    \draw[dashed, very thin] (6.5,1.0) -- (6.5,1.5);
    \draw[stealth-stealth, <->, very thin, dashed] (3.5, 1.5) -- (6.5, 1.5);
    \draw (5.0, 2) node {$b$};
    \draw[dashed, very thin] (9.5,1.0) -- (9.5,1.5);
    \draw[stealth-stealth, <->, very thin, dashed] (6.5, 1.5) -- (9.5, 1.5);
    \draw (8.0, 2) node {$b$};
\end{tikzpicture}
}}
\caption{Picturable proof of the Lemma \ref{lemat-najwazniejsze}\label{drugirysunek}}
\end{center}
\end{figure}

\begin{corollary}\label{najwazniejsze}
Ordering $\pi$, is a $b$-ordering iff for every edge $uv$ satisfying
$\ensuremath{\mathtt{segment}}(\pi(u)) + 1 = \ensuremath{\mathtt{segment}}(\pi(v))$ vertex
$u$ is assigned to greater position in the color order than vertex $v$ and for every edge
$uv$ values $\ensuremath{\mathtt{segment}}(\pi(u))$ and $\ensuremath{\mathtt{segment}}(\pi(v))$ differ by at most one.
\end{corollary}

\begin{definition}
By {\em{state candidate}} we denote partial function $s:V \to \{\Theta_i: 0 \leq i < \lceil \frac{n}{b+1} \rceil\}$, i.e.,
   assigning base segments to some of the vertices, such that for every vertex $v$ if $s(v)$ is defined then $s(v) \subset \phi(v)$ (where $\phi$ is the fixed assignment for this phase).
\end{definition}

Note, that for an inner vertex $v$ of the tree $\ensuremath{\mathcal{D}}$ there are two possible values for the state candidate $s$ and there are four possible values
for $v$ being a leaf.

\begin{definition}
A {\em{state}} is a state candidate $s$ satisfying the following conditions:
\begin{enumerate}
\item Vertices from $\ensuremath{\mathrm{dom}}(s)$ can be assigned to the first $|\ensuremath{\mathrm{dom}}(s)|$ positions in the color order,
  in a way compatible with $s$.\label{state:1}
\item For every edge $uv$ one of the following holds.\label{state:2}
  \begin{enumerate}
    \item Both $s(u)$ and $s(v)$ are undefined.
    \item Exactly one value among $\{s(u), s(v)\}$ is defined; say $s(v) = \Theta_i$ and $s(u)$ is undefined. Then if $\phi(u) = \Theta_{(k, l)}$ we have $k \leq i$.
    \item Both $s(u)$ and $s(v)$ are defined; say $s(v) = \Theta_i$ and $s(v) = \Theta_k$. Then $|i-k| \le 1$. \label{state:2c}
  \end{enumerate}
\end{enumerate}
\end{definition}

\begin{definition}
We say that a state $s'$ is an {\em{extension}} of some state $s$ if there exists vertex $v$ such that:
\begin{enumerate}
\item $s(v)$ is undefined and $s'(v)$ is defined;\label{extension:1}
\item $\ensuremath{\mathrm{dom}}(s') = \ensuremath{\mathrm{dom}}(s) \cup \{v\}$ and $s = s'|_{\ensuremath{\mathrm{dom}}(s)}$;\label{extension:2}
\item If $uv$ is an edge in the {\bf whole graph $G$} and $s'(u) = \Theta_k$ and $s'(v) = \Theta_i$, then $k-1 \leq i \leq k$.\label{extension:3}
\end{enumerate}
\end{definition}

\begin{lemma}\label{l:iff1}
Let $\pi$ be a $b$-ordering compatible with the given segment assignment $\phi$. By $s_k$ for $0 \leq k \leq n$ we denote the state candidate, assigning
$\ensuremath{\mathtt{segment}}(\pi(v))$ to every vertex $v$ assigned to one of the first $k$ positions in the color order (i.e., $|\ensuremath{\mathrm{dom}}(s_k)| = k$). Then every $s_k$ is a state
and for every $0 \leq k < n$ state $s_{k+1}$ is an extension of state $s_k$.
\end{lemma}

\begin{proof}
This is a straightforward corollary from Corollary \ref{najwazniejsze}. Point \ref{state:1} of the state definition is obviously satisfied by the definition of $s_k$.
Now lets look at any $s_k$ and edge $uv$. If both $s_k(u)$ and $s_k(v)$ are undefined, Point \ref{state:2} of the state definition is satisfied.
If both are defined, Corollary \ref{najwazniejsze} clearly implies Point \ref{state:2}. Assume then that $s_k(v) = \Theta_i$ and $s_k(u)$ is undefined. By construction
of state candidates $s_k$, $\pi(u)$ is later in the color order than $\pi(v)$. Therefore, by Corollary \ref{najwazniejsze}, $\ensuremath{\mathtt{segment}}(\pi(u)) \leq \ensuremath{\mathtt{segment}}(\pi(v))$. Since
$\pi(u) \in \ensuremath{\mathtt{segment}}(\pi(u)) \subset \phi(u)$, it implies the condition in Point \ref{state:2} of the state definition.

Let us now prove that  $s_{k+1}$ is an extension of the state $s_k$. Points \ref{extension:1} and \ref{extension:2} of the extension definition follow directly from the construction of $s_k$.
Note that if $v$ is the vertex defined by $s_{k+1}$ but not by $s_k$, then $\pi(v)$ is later in the color order than any $u$ defined by $s_k$.
Therefore, if we take any $uv$ as in Point \ref{extension:3} of the extension definition, by Corollary \ref{najwazniejsze} the condition in Point \ref{extension:3} is satisfied.
\end{proof}

\begin{lemma}\label{l:iff2}
Let $s_k$ for $0 \leq k \leq n$ be a set of states such that $s_{k+1}$ is an extension of state $s_k$ for every $0 \leq k < n$. Let $v_k$ be the vertex defined
by $s_k$, but not by $s_{k-1}$. Then ordering $\pi$ which assigns position $k$ in the color order to the vertex $v_k$ is a $b$-ordering.
\end{lemma}

\begin{proof}
We use Corollary \ref{najwazniejsze} once again. Let us prove the thesis by contradiction. Assume that edge $uv$ is longer than $b$ and w.l.o.g. $\pi(u) < \pi(v)$.
Since $\pi(u) + b < \pi(v)$, $\ensuremath{\mathtt{segment}}(\pi(u)) < \ensuremath{\mathtt{segment}}(\pi(v))$. If $\ensuremath{\mathtt{segment}}(\pi(u)) + 1 < \ensuremath{\mathtt{segment}}(\pi(v))$, then $s_n$ is not a state due to Point \ref{state:2c} of
the state definition. Therefore $\ensuremath{\mathtt{segment}}(\pi(u)) + 1 = \ensuremath{\mathtt{segment}}(\pi(v))$ and, by Corollary \ref{najwazniejsze}, $\pi(u)$ is earlier in the color order than $\pi(v)$.
But then there exists $s_k$ for which $s_k(u)$ is defined, $s_k(v)$ is undefined but $s_{k+1}(v)$ is defined. Thus we have contradiction
with Point \ref{extension:3} of the extension definition for the edge $uv$.
\end{proof}

The algorithm for second phase is now quite clear. Given segment assignment $\phi$, we do depth-first search over states, looking for sequence
of states and extensions as in Lemmas \ref{l:iff1} and \ref{l:iff2}. More precisely:
\begin{enumerate}
\item Start with a state $\emptyset$.
\item Having state $s$ with $|\ensuremath{\mathrm{dom}}(s)| = k < n$, try to create extension $s'$ of $s$ in every possible way, i.e., try to assign every undefined vertex in $s$
to the base segment, where $(k+1)$-th position in the color order lies.
\item If we reach state $s$ with $|\ensuremath{\mathrm{dom}}(s)| = n$, construct the $b$-ordering $\pi$ using the DFS stack (that is, states $\emptyset = s_0$, $s_1$, \ldots, $s_n=s$) and return it.\label{alg2:true}
\item If no state $s$ with $|\ensuremath{\mathrm{dom}}(s)| = n$ is reached, return that there is no $b$-ordering compatible with $\phi$.\label{alg2:false}
\end{enumerate}

Note that Lemmas \ref{l:iff1} and \ref{l:iff2} imply that this algorithm returns correct $b$-ordering in Step \ref{alg2:true} and there are no correct $b$-orderings
if the algorithm reaches Step \ref{alg2:false}. Therefore we proved that this algorithm is correct. In Section \ref{s:analiza} we prove the $O(4.83^n)$ time bound and $O^*(4^n)$ space bound.

\section{Time and space analysis}\label{s:analiza}

\subsection{Memory bound}

This analysis is fairly easy. Note that only non-polynomial space used in the algorithm is the space used to track visited states in the depth-first search.
We try to bound the number of visited states for one fixed segment assignment $\phi$ by $4^n$. This implies $O^*(4^n)$ space bound.

\begin{lemma}\label{lemma:mem}
In one run of the second phase, the algorithm visits at most $3^{n-L} 4^L \leq 4^n$ states, where $L$ is the number of leaves in the tree $\ensuremath{\mathcal{D}}$.
\end{lemma}

\begin{proof}
Let $s$ be a visited state and $v$ be a vertex.

If $v$ is an inner vertex of $\ensuremath{\mathcal{D}}$ with $\phi(v) = \Theta_{(i, i+2)}$, then $s(v)$ is either undefined or $s(v) = \Theta_i$ or $s(v) = \Theta_{i+1}$ - three possibilities.

If $v$ is a leaf of $\ensuremath{\mathcal{D}}$ with $\phi(v) = \Theta_{(i-1, i+3)}$, then $s(v)$ is either undefined or one of the base segments $\Theta_{i-1}$, $\Theta_i$, $\Theta_{i+1}$, $\Theta_{i+2}$ - five possibilities, too many.
However, lets look at the parent $u$ of the vertex $v$ in the tree $\ensuremath{\mathcal{D}}$. By construction, $\phi(u) = \Theta_{(i, i+2)}$. Note that, by the definition of the state, since
$uv$ is an edge of $G$:
\begin{enumerate}
\item If $s(u)$ is undefined, then $s(v)$ cannot be $\Theta_{i-1}$.
\item If $s(u) = \Theta_i$, then $s(v)$ cannot be $\Theta_{i+2}$.
\item If $s(u) = \Theta_{i+1}$, then $s(v)$ cannot be $\Theta_{i-1}$.
\end{enumerate}
Therefore, in every case, there are only four possibilities for a leaf $v$. This leads to $3^{n-L}4^L$ bound for the number of valid states.
\end{proof}

\subsection{Time bound}

Note that search for possible expansions of given state and checking if a state candidate is a state can be done in polynomial time.
Therefore the time used by the whole algorithm is bounded by $O((s(G) + 2^n)n^\gamma)$ where $\gamma$ is a constant and $s(G)$ is the number of visited states.
The $2^n$ factor is due to the first phase of the algorithm. Now we focus on bounding $s(G)$, i.e., the number of visited states in the run of the whole algorithm.

\begin{lemma}\label{lemma:time}
The algorithm in the whole run visits at most $3(n+1)\kappa^n$ states for some constant $\kappa < 4.83$.
\end{lemma}

\begin{proof}
Let us do indepth analysis of the number of possible states similar, but more broad, to that in the proof of Lemma \ref{lemma:mem}.
Let $s$ be a state visited while considering segment assignment $\phi$.

Let $u_0$ be the root of $\ensuremath{\mathcal{D}}$. Then $\phi(u_0)$ has at most $\lceil \frac{n}{b+1} \rceil + 1$ possible values.
If $\phi(u_0) = \Theta_{(i, i+2)}$, then $s(u_0)$ is either undefined or equal to $\Theta_i$ or $\Theta_{i+1}$.
In total, $3(n+1)$ possibilities for $u_0$.

Let $u$ be a parent of a leaf $v$. Then there are at most four possibilities to choose for $\phi(v)$ and $s(v)$,
knowing $\phi(u)$ and $s(u)$. The analysis is the same as in the proof of Lemma \ref{lemma:mem}.

Let now $v$ be an inner vertex with parent $u$. Let $\phi(u) = \Theta_{(i, i+2)}$. Then
$\phi(v)$ is either $\Theta_{(i-1, i+1)}$ or $\Theta_{(i+1, i+3)}$, by the way we construct $\phi$.
The following restrictions are implied by the fact that $uv$ is an edge of $G$ and
by the state definition.

\renewcommand{\theenumi}{\Alph{enumi}}
\renewcommand{\labelenumi}{\Alph{enumi}.}

\begin{enumerate}
\item If $s(u)$ is undefined:\label{a:opt:u}
  \begin{enumerate}
    \item If $\phi(v) = \Theta_{(i-1, i+1)}$, then $s(v)$ is undefined or $s(v) = \Theta_i$. State definition forbids possibility that $s(v) = \Theta_{i-1}$.
    \item If $\phi(v) = \Theta_{(i+1, i+3)}$, then $s(v)$ is undefined, $s(v) = \Theta_{i+1}$ or $s(v) = \Theta_{i+2}$.
  \end{enumerate}
  In total, {\bf{five}} possibilities for $\phi(v)$ and $s(v)$.
\item If $s(u) = \Theta_i$:\label{a:opt:l}
  \begin{enumerate}
    \item If $\phi(v) = \Theta_{(i-1, i+1)}$, then $s(v)$ is undefined, $s(v) = \Theta_{i-1}$ or $s(v) = \Theta_i$.
    \item If $\phi(v) = \Theta_{(i+1, i+3)}$, then $s(v) = \Theta_{i+1}$. State definitions forbids possibilities that $s(v)$ is undefined and that $s(v) = \Theta_{i+2}$.
  \end{enumerate}
  In total, {\bf{four}} possibilities for $\phi(v)$ and $s(v)$.
\item If $s(u) = \Theta_{i+1}$:\label{a:opt:r}
  \begin{enumerate}
    \item If $\phi(v) = \Theta_{(i-1, i+1)}$, then $s(v)$ is undefined or $s(v) = \Theta_i$. State definition forbids possibility that $s(v) = \Theta_{i-1}$.
    \item If $\phi(v) = \Theta_{(i+1, i+3)}$, then $s(v)$ is undefined, $s(v) = \Theta_{i+1}$ or $s(v) = \Theta_{i+2}$.
  \end{enumerate}
  In total, {\bf{five}} possibilities for $\phi(v)$ and $s(v)$.
\end{enumerate}

In every option we got at most five options for $\phi(v)$ and $s(v)$ values for every inner, non-root vertex $v$. Together with
four possibilities for leaves and $3(n+1)$ for root, this proves that the algorithm visits at most $3(n+1)5^n$ states.

However, there are few places where we have four, not five possibilities for $\phi(v)$ and $s(v)$: when $v$ is a leaf or when
the parent of $u$ is assigned by the state to the left (smaller position numbers) half of its segment (Option \ref{a:opt:l}). Moreover, in every moment, when we
have five possibilities for $\phi(v)$ and $s(v)$, vertex $v$ might be assigned to the left (smaller position numbers) half of its segment, which gives us Option \ref{a:opt:l}
for the analysis of children of $v$. This leads us to the conclusion that we can use Measure \&{} Conquer technique to obtain better bound.

The Measure \&{} Conquer method was introduced by Fomin, Grandoni and Kratsch (see \cite{fgk:mis}).
As in the above analysis, we analyze vertices in the root-to-leaf order. We use Measure \&{} Conquer method to estimate
number of possible states.

Let us consider hypothetical state generator that generates possible states (i.e., pairs of functions $\phi$ and $s$), by analyzing
the tree $\ensuremath{\mathcal{D}}$ in the root-to-leaf order. The generator first sets $\phi(u_0)$ and $s(u_0)$ in every possible way (at most $3(n+1)$ ways).
Then, while analyzing vertex $v$ with already set $\phi(u)$ and $s(u)$ for parent $u$ of $v$, it assigns $\phi(v)$ and $s(v)$ in every
possible way, keeping in mind limitations described above, both for $v$ being a leaf and inner vertex. We use Measure \&{} Conquer to estimate
number of generated states by this generator.

At any step of the generator we measure the weight of the current problem instance, i.e., already constructed functions $\phi$ and $s$.
The weight of the instance is the sum of weights of vertices. Let $\alpha, \beta \in [0,1]$ be constants to be defined lated.
The weight of the vertex $v$ is:

\renewcommand{\theenumi}{\Roman{enumi}}
\renewcommand{\labelenumi}{\Roman{enumi}.}
\begin{enumerate}
\item $0$, if $v$ is already analyzed;
\item $1$, if $v$ is not analyzed and parent of $v$ is not analyzed or $v$ is the root of $\ensuremath{\mathcal{D}}$;
\item $1$, if $v$ is not analyzed, parent $u$ of $v$ is analyzed and $s(u)$ is undefined;\label{x:opt:1}
\item $\alpha$, if $v$ is not analyzed, parent $u$ of $v$ is analyzed and $\phi(u) = \Theta_{(i, i+2)}$ and $s(u) = \Theta_i$ for some integer $i$.\label{x:opt:2}
\item $\beta$, if $v$ is not analyzed, parent $u$ of $v$ is analyzed and $\phi(u) = \Theta_{(i, i+2)}$ and $s(u) = \Theta_{i+1}$ for some integer $i$.\label{x:opt:3}
\end{enumerate}

Let $T(w)$ be a bound for number of states generated by the generator from point, where the size of the instance is at most $w$. Now we estimate $T$,
using aforementioned limitations for generator choices.

Let $v$ be a non-root vertex currently analyzed by the generator, where vertex $u$ is its parent.
If $v$ is a leaf, there are always four possibilities for $\phi(v)$ and $s(v)$, independent of whether $v$ falls into Category \ref{x:opt:1}, \ref{x:opt:2}, \ref{x:opt:3}, i.e.,
whether $v$ weights $1$, $\alpha$ or $\beta$. Therefore
$$T(w) \leq \max (4T(w-1), 4T(w-\alpha), 4T(w-\beta)).$$

Now lets look at the case when $v$ is an inner vertex. Let $\phi(u) = \Theta_{(i, i+2)}$.
\renewcommand{\theenumi}{\arabic{enumi}}
\renewcommand{\labelenumi}{\arabic{enumi}.}
\begin{enumerate}
\item If $s(u)$ is undefined, $v$ has got weight $1$ and we have five possibilities for $\phi(v)$ and $s(v)$. In two of them, $v$ falls under Category \ref{x:opt:1} for children of $v$,
  in two --- under Category \ref{x:opt:3}, and in one --- under Category \ref{x:opt:2}. Vertex $v$ becomes analyzed and has weight $0$. Since there is at least one child of vertex $v$,
  the following bound holds in this case:
  $$T(w) \leq 2T(w-1) + 2T(w - 1 - (1 - \beta)) + T(w-1-(1-\alpha)).$$
\item If $s(u) = \Theta_i$, $v$ has got weight $\alpha$ and we have four possibilities for $\phi(v)$ and $s(v)$. In one of them, $v$ falls under Category \ref{x:opt:1} for children of $v$,
  in one --- under Category \ref{x:opt:3}, and in two --- under Category \ref{x:opt:2}. Vertex $v$ becomes analyzed and has weight $0$. Since there is at least one child of vertex $v$,
  the following bound holds in this case:
  $$T(w) \leq T(w-\alpha) + T(w - \alpha - (1-\beta)) + 2T(w - \alpha - (1-\alpha)).$$
\item If $s(u) = \Theta_{i+1}$, $v$ has got weight $\beta$ and we have five possibilities for $\phi(v)$ and $s(v)$. In two of them, $v$ falls under Category \ref{x:opt:1} for children of $v$,
  in two --- under Category \ref{x:opt:3}, and in one --- under Category \ref{x:opt:2}. Vertex $v$ becomes analyzed and has weight $0$. Since there is at least one child of vertex $v$,
  the following bound holds in this case:
  $$T(w) \leq 2T(w-\beta) + 2T(w - \beta - (1 - \beta)) + T(w-\beta-(1-\alpha)).$$
\end{enumerate}

By searching the space of possible values $\alpha$ and $\beta$ and by solving the above equations numerically,
we got that for $\alpha = 0.8805$ and $\beta = 1$ function $T(w)$ is bounded by $\kappa^n$ for $\kappa \sim 4.828485 < 4.83$.
This completes the proof.
\end{proof}

The following theorem is a straightforward corollary from Lemmas \ref{lemma:mem} and \ref{lemma:time}.
\begin{theorem}
There exists an algorithm that solves the {\sc{Bandwidth}} problem in $O(4.83^n)$ time and $O^*(4^n)$ space.
\end{theorem}

\bibliographystyle{plain}


In a court motion filed Tuesday, Kelly's attorney, Steven Greenberg, said he signed a contract for the performances — for three to five shows in the UAE from April 17 to 19 — before his arrest in February. Prosecutors charged Kelly with 10 counts of criminal sexual abuse against for different women.
Read more:She used to be R. Kelly's biggest supporter. Now she could send the R&B icon to prison for life.


Figs (Ficus carica) are a popular fruit for home orchards because they are more tolerant of adverse conditions than most common fruit trees. They can be grown in U.S. Department of Agriculture plant hardiness zones 7 through 11, though many gardeners also grow figs in containers and bring them indoors in the winter. Figs are easier than most fruit trees to propagate, but they have an unusual approach to flowering, pollination, and seed formation that varies depending on the variety.
Figs require specialized pollination by fig wasps to produce seed. The flowers actually are inside the flesh of what appears to be immature fruit, but actually is like a flower turned inside out. A small hole at the base allows the wasp to enter and fertilize the flowers of female trees with pollen from male trees. The fruit-like flowers of male trees, called capri figs, fall off after releasing their pollen. If pollinated successfully, the flowers of female trees enlarge to become soft and delicious ripe figs.
Not all figs require this specialized type of pollination to form ripe fruit. Most of the fresh figs bought in stores or produced by backyard orchardists have never been pollinated and do not have viable seeds. These are from fig varieties that have been selected for their ability to develop ripe fruit without pollination. In this case, the unusual fig flower grows into a fleshy fruit without the assistance of pollinating wasps.
Most of the dried figs found in grocery stores are varieties of the "Smyrna" fig. This type of fig requires pollination by fig wasps, which are not native to North America. Commercial orchardists introduce the wasp to pollinate their trees, but because the practice requires specialized knowledge and supplies, it rarely is done by home gardeners. The fruit from pollinated "Smyrna" figs is used for drying because of its high sugar content and the crunchy texture imparted by the seeds.
Fruiting figs are propagated by cuttings, whether they are pollinated varieties containing viable seeds or the seedless type. This method of clonal propagation ensures that a new tree will be produced with the same fruiting qualities as the parent. The use of a separate rootstock is not required as is common with other fruit trees.


text:
package ipam

import "github.com/giantswarm/azure-operator/v6/flag/service/installation/guest/ipam/network"

type IPAM struct {
	Network network.Network
}

redpajama_set_name: RedPajamaGithub


WHO: Dr. Robert Limoni, an orthopedic surgeon with BayCare Clinic, was interviewed for a WBAY Channel 2 News story about his diverse career passions: Surgeon, inventor, farmer.
WHAT: Limoni works full-time as an orthopedic surgeon, runs an organic farm in Door County and betters the lives of his patients through medical device inventions.
WHERE: The story aired Monday night on WBAY Channel 2.


text:
Dr. Ravi Pillai: The Ravishing Biz Epic
Entrepreneur, Dr. Ravi Pillai's success story is one that inspires every young entrepreneur. From humble beginnings, the son of a farmer from the southern Indian state of Kerala, Dr. Pillai is now Chairman and Managing Director of Middle East based RP Group of Companies.
A global conglomerate, RP Group today is involved in construction & infrastructure, healthcare and wellness, education, real estate, hospitality, Information Technology, travel & tourism and retail & malls, with a combined turnover of more than $5 billion, a workforce of 120,000 plus and an operational footprint that extends across nine countries. The group is famed for its rich resource pool of professional, dedicated personnel and its ability to maintain the highest standards of excellence in all its activities.
Dr. Ravi Pillai
As early as 17 years of age, Dr. Pillai began dabbling in trading in his home state of Kerala, and soon after college graduation, entered into the construction business. Despite no background of business and no one to hold hands, he became successful doing construction work for some of South India's biggest companies. However, unforeseen challenges stifled his blossoming business, though he was not deterred. "You have to trust in yourself first," said RP Group Chairman, Dr. Ravi Pillai. "If there is a will, there is a way. Without self-confidence, you can't win."
Armed with determination and faith, he moved to Saudi Arabia in 1978. At that time oil price was around $9 – $10 per barrel, however there were multiple business opportunities in Saudi Arabia. He started his journey with 150 employees in Saudi Arabia. Though it was extremely difficult for Indian companies to win important projects in those days, his undying determination and hard work resulted in getting him infrastructure and defense related construction contracts. One of his early wins was a construction order for the Royal Air Terminal at Riyadh Airport. During the next five years, as the price of oil increased upwards of $25 per barrel, the company moved into industrial contracting, including oil and gas and petrochemical projects critical for nation development, and grew to 5,000 employees. Forty years later, he is considered one of the largest employers in the Gulf. The group employs over 120, 000 employees of various nationalities, including various GCC nationalities, Americans, Europeans and people from the Indian subcontinent.
"My true satisfaction comes from the happiness of my 120,000 happy employees, and their families which they support, which equates to almost 750, 000 people." says Dr. Pillai.
Dr Pillai's focus is on group's human capital, and he believes rest everything will follow. "Money is not everything, most important is to have a kind heart," said Dr Pillai.
RP Group is known for uncompromised business values of commitment, hard work & dedication, honesty and respect. "My policy is that money cannot be made with someone's tears, and must always be earned the right way." Says Dr. Pillai.
The Oil & Gas industry globally recognized Group's companies for giving safe working environment to their employees, by adhering to highest safety standards in the construction world, best amenities, facilities and care as an employer. Dr. Pillai say, "Our dedicated workforce is my real wealth. If you have money, you can accumulate assets, but you can't buy people's trust and commitment. I am fortunate to have committed people and the company has grown because of them. Only if an employee is happy can he or she give their best for the company."
RP Group Employees
Many employees have worked with him for almost three decades. Dr Pillai is delighted that many of his employee's children are doctors and engineers and a good number of them are working for the company. RP Group has grown phenomenally over the years. It has been one of the most trusted infrastructure development partners in the GCC for four decades and the group has delivered projects worth over $30 billion. These include working on some of the world's best refineries and petrochemical plants all over the GCC.
The group is very closely connected with the oil and gas giants in energy sector like Saudi Aramco, Kuwait Oil Company, Kuwait National Petroleum Company, ExxonMobil, Sabic & Its affiliates, Qatar Petroleum, Bapco, Maaden, Alba and several illustrious International Engineering Companies as well. Group companies have received numerous international awards including the recent ExxonMobil's global "Best Contractor" award and Samsung's "Best Partner" award. Group companies are a trusted partner of choice for their clients. Dr. Pillai says, "The biggest and the most demanding companies in the world who deal with us, know that they are at peace with responsibility on our shoulders, and know that the work we undertake shall be carried out by very skilled manpower, in a professional manner and deliver high quality output on time, even for some of the most complex jobs in the world."
Recently, one of the group companies entered the Guinness World Records – for completing the largest continuous concrete pour (of 19,793 cu m for 42 hours) in the world.
In 2011, the group entered IT sector and since then has been growing steadily in India, Middle East and USA. Over the last 20 years, RP Group operations expanded into real estate, hospitality, education, health care, trading, retail and IT sectors across the GCC as well as India, Korea and Australia and the USA. "It is very important to respect the laws and regulations of the countries we work in," said Dr Pillai. "Only then can we achieve success globally."
Dr. Pillai with Indian Prime Minister Narendra Modi
In the Indian tradition of 'Atithi Devo Bhava', where a guest has to be treated with the utmost respect, Dr Pillai established a chain of 5 star luxury hotels – the flagship property Raviz Kovalam, Ravi z Resort and Spa by the Lake Ashtamudi in his native Kollam, the Raviz Kadavu Resort on the banks of the River Chaliyar and a luxury city hotel, the Raviz Calicut. Dr. Pillai's hospitality portfolio also includes 5-star luxury hotels in Pune and Dubai.
RP Group's reach in Kerala also extends to the healthcare and wellness sector. It owns and manages the 300 bed multi-specialty Upasana Hospital at Kollam, a fully-equipped state-of-the-art facility, manned by a team of medical and para medical professionals. The hospital is also responsible for the nearby Upasana School and College of Nursing which offers graduate and post graduate courses. "We have a good name because of our commitment, quality, honesty and credibility," said Dr Pillai.
RP Group over the years has transformed from a highly efficient construction company into a diversified conglomerate.
Dr. Pillai with H.E Sheik Mohammed bin Rashid Al Maktoum
Besides business, Dr Pillai is also mindful of sharing his wealth with the less fortunate. The Dr. Ravi Pillai Foundation works towards the betterment of humanity, with the lion's share of his earnings disbursed every year to the construction of homes, education, thousands of weddings, medical assistance and charity f or the underprivileged and needy.
Dr. Pillai with his family
Dr Pillai and his wife Geetha have two children – Ganesh and Dr Arathi. Both Ganesh and Dr. Arathi are following their father's footsteps by joining the business after completing their studies.
Dr. Ravi Pillai receives Padmashri Award
Throughout his career, Dr. Pillai has been honored with many awards from various governments and organizations. The government of India honored him with one of its most prestigious civilian awards, in 2010, the Padma Shri, for his outstanding contribution to the fields of business and charity.
Dr. Pillai with the former President Dr. A P J Abdul Kalam
He was also recognized by the Indian government with the Pravasi Bhartiya Samman Award in 2008, and has received numerous other honors and awards from around the globe for both his business and philanthropic contributions.
However, the best accolade for him is the happy faces of his employees. "Their happiness is my greatest reward," he added.
Dr. Pillai with the former US President Barack Obama
It is no secret that Dr. Pillai's 'Good Human Beings build lasting Business' motto is the true secret of his success.
The world is at present struggling under the clutches of Covid-19. As a business leader, how do you see this crisis? How the business community can be supported to surmount this crisis? And, what is your expectation about the post-Covid-19 world?
There are few events in the world that leave their long-lasting mark in the history books across the world.
As an entrepreneur, Covid-19 to me is a Black Swan event wherein there is a simultaneous but massive shock to the supply as well as demand side of the businesses across various industries and regions. Apart from the health-related challenges of this pandemic that is on top of everyone's mind, the lockdown which various countries were forced to impose has stifled the demand as well as disrupted the supply chain for various businesses. It goes without saying, just like countries will need years to recover the loss of GDP, businesses will also take time to recover the lost traction.
We have seen a very successful co-operation between various members of our society for coming together to fight the pandemic on healthcare front. Everyone – the government, law & order machinery, businesses and general public have all helped the healthcare professionals who are our frontline warriors fighting against Covid-19.
Similar to the menace of Covid-19, our next challenge collectively as a society is the stagnant economic activities. On this front, I see business leaders and entrepreneurs as our frontline warriors fighting to help us get back to the normal level of GDP, employment and prosperity. I believe, just like the health front, various members of the society need to bring their best foot forward to support the frontline warriors on the economic front as well. These members are everyone including governments, central banks and financial institutions, employees, customers, landlords, vendors and the general public. They all need to be together with the businesses during these difficult times and extend any and every cooperation they can to various businesses. While a lot is being done in this front as well, we need to look at best global practices and adopt these relief measures to the extent feasible here.
Just to give a few examples, globally various governments are giving tax breaks, economic assistance to maintain the staff strength (not reduce the workforce), special bailout package for the most affected industries, direct benefits transfer to the unemployed and jobless to prevent individual insolvency as well as maintain a basic level of purchasing power in the hands of lower income segment of the society to meet their essentials. Central banks have helped by infusing liquidity, buying the corporate loans and bonds, reducing interest rates as well as relaxing the norms on payment timelines of various bank loans for eligible businesses as well as individuals.
Due to the Covid-19 crisis, several Non-Residential Keralities have been forced to return to the state. As a director of NORKA Roots, what is your say on this issue? Can NORKA emerge as a saviour to address the concerns of those who have been forced to choose their lives over their dreams?
Yes, we are passing through one of the worst times. Kerala with one of the largest diaspora outside India has manifold problems. NORKA has been very proactive. We could set up NORKA help desk in several countries to guide Pravasis in the early days of Covid pandemic itself. Kerala has given the maximum number of approvals for flights to bring back people from Asia and the Middle East. Almost one third of flights operated under Vande Bharat Mission was to Kerala. Also, as many as 800 chartered flights were given permission by Kerala Government. This is the highest ever by any state in India. Hon. Chief Minister convened two video conferences with Pravasis and based on their feedback, Hon'ble Chief Minister has announced several relief measures including financial assistance to Pravasis who have lost jobs. NORKA department also has plans to create a skill pool of returnees, for them to gain employment later in Kerala.
India at present maintains a good relation with the Middle East. The Indo-Middle East business community, in which you are an important member, plays a huge role in the growth of this relation. How these two economic giants can support each other to sail through the present crisis safely?
A strong bond of mutual trust, respect and brotherhood has been the foundation of the strong economic, social and political ties between India and the Middle East for decades. This bond has only grown stronger in recent years. Both India and Middle Eastern Countries have demonstrated that " a friend in need is a friend indeed".
India has exported medicines like hydroxychloroquine and sent teams of medical professionals to countries like Kuwait, UAE, etc. to help them combat Covid-19. Given the sudden lockdown, many Indian citizens – not just those who work here but even visitors and tourists were stranded in Middle-East countries. All these countries have taken care of these stranded Indian citizens and treated the patients of Covid-19 amongst visitors and tourists as well, as if they were their own citizens.
Just like on the healthcare front, on the economic front as well I see many avenues of strong co-operation and mutual support. Middle-eastern sovereign wealth funds are always looking for good investment opportunities. India, given her demographics and fundamentals, is an attractive investment choice. The investment by these countries can infuse capital and help create businesses and jobs in India. We have seen record amount of investment in a single company announced from Saudi Arabia and UAE in the middle of this crisis (e.g., Saudi Wealth Fund investing USD 1.5 Billion and UAE's Mubadala and ADIA investing close to Rs. 14,700 crores in Reliance Jio platform). Similar such investments can be mutually beneficial to India as well as Middle-East.
How much disturbing the lockdown was for you? What are the structural changes you have brought in to the administrative structure of your company to transform the system in the way the time demands?
This has indeed been an ineffable and overwhelming time. One that I have not experienced in the past 42 years of my business career. The customary business culture which demands a lot of travel and interaction with people, this global culture has definitely taken a pitch pole.
During the Gulf war in the 1990s, was the last time I had faced a gruelling situation. It was a time that demanded my employees to reach their homes safely. I had made arrangements for Chartered flights for their safe travel, which I must say wasn't a very tough task at the time, in comparison to our current state of uncertainty.
The present situation makes countries cautious in bringing their citizens home whilst handling the pandemic domestically; as such is the dilemma faced by various governments. Covid -19 pandemic has caused panic and apprehension all over the world.
Health and safety protocols with respect to Covid -19 deterrence has become the highest priority. The same goes for my employees, many of them who do require reach their homes safely, has become my prime responsibility.
We have arranged chartered flights to various states of our country, for which we had ample support and coordination from the MEA (Ministry Of External Affairs). I highly appreciate all the support we received from the MEA (Ministry Of External Affairs), all State Governments, Embassies in the GCC in making this mission possible! We also ensured that our employees of other nationalities, travel home safely to countries like Philippines, Nepal and Bangladesh.
Secondly, as businesses need to continue in spite of the present state of affairs, we need to encourage our employees with good Healthcare and safety measures. This has prompted us to reorganize work conditions and ensure a more sterile environment. Our sites have been equipped with qualified Healthcare officials to tackle medical problems faced by our employees on a daily basis. Beds have been arranged in hospitals to ensure timely medical attention in case of any emergencies.
As of now, the relevant Health authorities have cautioned those above 60 years and the provision for traveling has become speculative to a large extent. Hence videoconferencing has become a routine of business and has become a boon. It has become one of my preferred means for communication and has helped me keep in touch with my team across various offices globally. The same goes for meetings scheduled with our various global clients. Time flies these days with the problems we face on a day to day basis.
Companies normally shift burden over employees when they face difficulties. But, your company was different. We have learned that your company has extended enormous support to your employees during this difficult time. Please share us about those initiatives.
This is the time for supporting one another and at the same time extending best understanding towards each other. I consider my employees as family. The biggest strength of my group has been my employees and I share a very personal bond with them. Most of them have worked with us for 30-40 years and have dedicated most of their time with us than with their families. I feel duty bound to take all possible care for them.
One of my biggest focuses has been the health & safety and well-being of my employees. By means of working with some of the biggest companies in the realm of Oil and Gas; my companies have had to adhere to the highest levels of health & safety standards and have been bestowed with numerous global awards for the same. Maintaining the same in a pandemic scenario like this is a very different kind of a challenge, we are doing everything possible to take care of our employees and their well-being.
Me, my family and my team are fully committed to do whatever we can, towards having the minimum impact of the current situation on our employees and their families, and give them maximum possible care towards their lives.
The tourism sector of the state is one of those sectors which have faced the heat of this crisis the most. Your group holds ownership in highly reputed hotels such as Raviz Kovalam, Raviz Astamudi and Raviz Kadavu and Raviz Calicut – which remain as the backbone of the state's tourism sector. Can the sector regain the impetus it has lost to the present crisis?
Unfortunately, tourism and international travel are the worst-hit sectors world over due to the pandemic. Between these 2 sectors, tourism will definitely take longer compared to other sectors to recover back to the level of activities that were seen in 2019. However, I do see some silver lining in these dark clouds:
* I think tourism will return in other countries including India. The young will venture out first. Travel will involve nearby destinations. Economy travel will recover more quickly. People would want to seek some sort of healing to the lockdown (or as it is called "revenge travel") by travelling outdoor and hence, nature-related destinations will be more popular than congested cities. Here, Kerala, God's own country, is the most obvious choice.
* As per RBI estimates, Indians spent roughly $ 1.6 Billion on international travel during summer holidays last year. Till the time a vaccine is developed, widely available and administered to public at large and is considered effective, International Travel (esp. non-essential like tourism) will remain handicapped as many countries are continuing stringent 14-day quarantine period for international travellers which is not practical for most tourists. Thus, all the tourists, Indians and foreigners alike, will be inclined to travel to explore domestic tourist destinations rather than international ones. Thus, unfortunately on one hand, India (& Kerala) may miss out on foreign visitors but Indian travellers are more than capable of compensating for this loss of business as they re-align their travel aspirations with the new normal.
* Ability to maintain social distancing and avoiding crowded cities as well as availability of good healthcare facilities will be the top factors that will influence the choice of tourist destination. Here again, despite my best efforts, I can't think of any other tourist destination that can even match Kerala on both these factors.
* With the present-day advancements in Research and Healthcare, the probability of a vaccine being discovered is unquestionable and is indeed possible at a faster pace. A recovery phase would demand a lot of healing which includes travel and leisure activities.
* With the Kerala Government doing a mighty task in controlling the spreading of Covid -19, Kerala has yet again proved worthy for safe tourism in the future.
Nestled in the secluded backwaters of the Arabian Sea near Trivandrum in Kerala is the five-star deluxe, Raviz Resort & Ayurveda Centre Kollam, a perfect blend of modern architecture and traditional designs of a bygone era. Clearly taking advantage of its scenic backwaters.
Designed to reflect the architectural splendour of the mystical Malabar region, the Hotel Raviz
Resort & Ayurveda Centre Kadavu, a five-star resort on the banks of the Chalayar river in Calicut.
The Raviz Kovalam Beach, Kerala, is a prized possession of the Group in picturesque Kovalam, a popular beachside destination in Kerala. The well placed 5 star Clifftop Beach property is spread over 65 acres of land Kovalam has also been well known for its most coveted Kovalam Palace.
During the post pandemic times , We at Raviz group intend to offer a safe yet enjoyable, world class tourism experience which will help further crown jewel of Kerala tourism.
RP Group of Companies is now one of the most promising business groups in this part of the world. We all know there is a struggle behind this group. Our readers may be interested to hear about that struggle. Please share.
I was born to a farming family with plenty of paddy fields and coconut plantations. I grew up understanding the limited scope of the agricultural sector at the time.
As I have walked in a farmer's shoes, I could learn of the challenges first hand and noticed the hardships of a farmer did not reward proportionate returns. Modern agricultural technology and practices at the time were not well known or available.
At the age of 17, I started my own business, which was of course on a small scale. As I did not have a business background I had no guidance, and I was my own teacher. There were no right and wrong decisions. Only choices that I bore complete responsibility of and of course destiny as well as God's blessings.
The beginning of my business career was in Kerala, my initial ventures being contracts for FACT and Cochin refineries. Unfortunately, due to labour strike, these businesses did not serve me well. It was in 1978, that I had decided to explore my opportunities in the Gulf. With 150 employees, I had initiated my first construction business in Saudi Arabia. Strong determination, hard work and commitment were the only 3 mantras I held close to my heart. Presently, I have more than 1,20 000 employees working for me globally.
Now we are one of the biggest Industrial Construction Companies for Oil and Gas. We have been fortunate to have some of the World's famous oil companies as our clients. My career has indeed been rewarding and I am eternally thankful to God for this rich experience.
My only advise is, anyone who has good knowledge in business can start a business in today's Era as long as you work hard, bear good will and strong commitment.
I strongly believe that Nothing is impossible as long as you have a strong will.
What is your message to the global business community?
My biggest message to the business community is firstly to stay on course and manage this storm with positivity. Let the compromising situation of the world not make business community compromise on their values and ethical system. After the discovery of a vaccine, we will return to a world of perhaps, more and more opportunities.
In my strong opinion "This Year, to stay alive is equivalent to profit! "
Secondly, Covid-19 will bring big changes in the business community and the way business is being conducted. Being fungible and adaptive to the changes will be required more than ever.
What do you want to say to youngsters who see a true role model in you?
Change is the only static thing that occurs with time. The same goes for the pulse in world business.
However, the mantra to success stays the same. Be honest with any business that you wish to pursue. With goodwill comes good opportunities. Be optimistic. With every downfall comes greater opportunities. You still need strong will, hard work, honestly, commitment and strong values to be successful in the long run. Rather, if anything, the current pandemic has only reinforced the need for these values.
You are the recipient of several prestigious awards including the highest civilian honour of Padma Shri. How do you see these achievements?
Each award has only reinstated and instilled my responsibilities to the community and global
society. These honors constantly drive me to stay on my course.
What are the latest CSR initiatives of your company?
Philanthropy is the ideology that drives Dr. Ravi Pillai Foundation which undertakes all the CSR activities of the Group. The Foundation works towards the betterment of the society by providing the underprivileged and the needy with necessary moral and financial support. This is an earnest mission set in motion to help the humanity by Dr. Ravi Pillai, the Chairman and Managing Director of RP Group.
The main activities of the RP Foundation are: Provide assistance and help in the re‐habilitation of the destitute who are suffering from incurable/perennial diseases and who are physically/mentally crippled. Provide assistance to the victims of natural calamities. Provide poverty eradication schemes for the down trodden people. Establish hospitals, clinics, and sanatoriums, to help the poor. Provide assistance to establishments intended for the welfare of the poor. Start educational institutions that provides technical and job‐oriented education with intention for social development. Provide financial assistance for the marriage of young ladies who fall below the poverty line.
I have been able to provide jobs to many who have aspired to work in the Gulf. The current scenario demands all efforts to be focused on CSR activities related to Covid-19. As mentioned above, we are doing everything possible to give back to the community especially with the Covid-19 crisis and have been able to help many families personally with respect to travel, financial aid and healthcare.
During this unprecedented and challenging times of Covid-19, the most pressing need of the pravasis is to safely travel home. While arranging chartered flights for the safe return of all my employees who wanted to travel home, I have been fortunate enough to offer travel assistance to hundreds of workers and families in the gulf countries to travel back to their home countries. I deem it my responsibility to lend a helping hand to my brethren at this time of need and shall continue to offer my assistance to those who are need.
Continuing our mission to support the needy in society, I am still strongly interested in providing more housing facilities to those in need. This I bear in heart and mind as my duty as well as social commitment.
Tags: Ravi Pillaitimes new

redpajama_set_name: RedPajamaCommonCrawl


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Publishing has always given responsibility early, and whilst many people thrive on being 'thrown in at the deep end', the danger of making expensive mistakes is real. Even newcomers with marketing qualifications struggle to apply them in an industry where new product proliferates, resources are tight, and the remit often very broad. But help is at hand!
Anyone new to a marketing position in publishing. Editors wanting a marketing overview to enhance their understanding of the wider publishing context.
The life cycle of a book, from proposal to launch: how do titles get contracted? How are print and price decisions made? Who gets what of the RRP?
Group planning for real titles brought by participants.


At a recent Innovation Conference in Toronto, a seminar focused on "What is the key characteristic/trait of a Successful Leader/Manager or PM in today's day and age".
It must be noted this continues to change as our business and personal worlds evolve.
I find this so interesting. We can be successful, then this can creep up on you and you realize, you need to adapt because the world around you is different.
Humans have an internal desire to get into a routine, feel comfort and then move forward along a defined path. Then once we get bored, we want a little change (but just a little!!!). Then back to the comfort of a path that we can see for miles and miles ahead.
Unfortunately, this world is gone. Every quarter there is a global crisis, merger, bankruptcy, terrorist threat, new competitive product, disruptive technology, etc. The train has left the station.
Okay, lets get back to the question at hand, what is the top characteristic/trait???
other top qualities have included: quality focus, problem solving, compassion, passion, humour, ethics/honesty, confidence, etc.
In today's economy and market, information is readily available, global interconnectivity is exploding and technology continues to advancements (once you get comfortable with a tool or software, be ready to change it next year).
It is becoming less and less about what is delivered, or how it is delivered. It is becoming more and more about learning from the activity, then continuing to move forward in a more modular, agile approach.
One can do all the risk planning they want, but there will always be road blocks that were unplanned. With a more interconnected world of business, these risks become higher in number and greater in impact.
I align this with project management, This does not mean you are allowed to fail and screw up and just keep going. You must learn and not repeat. You must also understand and balance the impacts of the risk events, if your project cannot handle the impact, then the risk must be mitigated. You must create a project environment so when stuff happens, you must not stop. You must find the positive, create an environment of innovation, trust, calm and collaboration, and then maximize the resources at hand (use what you got) and move forward!!!
· who is leading the project?
The documents developed at initiation may seem insignificant due to the lack of detail but these documents are critical. They set the stage for planning and help identify key points that will be continually referenced back to throughout the life of the project. Always ensure that these early documents involve the appropriate stakeholders to get strong buy in and accurate information that covers all angles of the project. In the initiation stage the most significant document that is developed is the Charter.
· Project Manager – manages the project.
· How is it aligned with the organization.
· May start to also document risks.
· Sponsor at a minimum, may include PM and other senior stakeholders.
The Charter is a one time document, you want to have a stake in the sand from which you can now kick off the project and refer back to.
It is critical that strategic aspects are addressed in the Charter, don't worry about the details, they will come out in the scope statement and the development of the WBS. Once the charter is signed, then you begin the scope statement and it can have multiple sign offs and multiple versions as you get more details. The Charter will not be perfect but it at least will get you moving down the road.
TOOOOO often, we take too long to develop the charter and try to add too many details into it. An analogy is that the Charter is a stake in sand. Then the preliminary scope statement is driven into dirt. Then Scope Statement is driven into Clay. Then Project Plan with initial WBS is driven into compacted clay. Then the baseline project plan is driven into wet concrete, etc. As you move further along the project lifecycle, that is when the details come out and the changes should become less.
1) Ensure a Sponsor is identified and confirm their understanding of the project.
2) Ensure the Sponsor and PM have clear roles and responsibilities.
3) Allocate required SME (Subject Matter Experts) to the project and gain their commitment.
4) Document the charter utilizing the standard headings from the organization's template (or leverage a best practice template).
5) Plan a Kick off meeting to launch aspects of the Charter or the entire charter itself to formally kick off the project.
EVM (earned value management) is often discussed but often not understood.
how much will we be under or over budget?
what it will take to get the project back on track.
Organizations have such confidence in the EVM method that they require it as mandatory trending and reporting starting as soon as the project has resources. The attraction to the use of the earned value is that it provides accurate measurements of performance as early as 15 percent into the project. With an accurate "early warning" signal, actions can be taken by a project manager to avoid final cost and schedule over runs.
· Planned Value (PV) - describes how far along a project is supposed to be at any given point in the project schedule. It is a numeric reflection of the budgeted work that is scheduled to be performed, and it is the established baseline (also known as the performance measurement baseline) against which the actual progress of the project is measured. Once it is established, this baseline may change only to reflect cost and schedule changes necessitated by changes in the scope of work. PV is also known as the Budgeted Cost for Work Scheduled (BCWS).
· Earned Value (EV) is a snapshot of a project at a given point in time that reflects the amount of work that has actually been accomplished, expressed as the planned value for that work. EV is also known as the Budgeted Cost for Work Performed (BCWP).
· Actual Cost (AC) is an indication of the level of resources that have been expended to achieve the actual work performed to date or in a time period. AC is also known as the Actual Cost of Work Performed (ACWP).
IF this all sounds complicated, it is because it can be. So until you start to feel comfortable, do not over do it. Stick to accounting basics.
Understand what your stakeholders need to know (ask them if unsure) and understand what you need to know and report on.
For more information regarding EVM methodology, you can refer to the Practice Standard for Earned Value Management (2nd Ed.), published by the Project Management Institute (PMI).
Also, the two links listed below provide more details and understanding related to the EVM methodologies. You will find a deeper level of detail and examples.


text:
Syaifudin (born July 20, 1978 in Jombang, East Java) is an Indonesian footballer. Currently he plays in Indonesia Super League for Persepam Madura United, as goalkeeper. Before the Persepam He played in Persiwa Wamena, Persela Lamongan, Arema Indonesia and Persebaya Surabaya.

References

Indonesian footballers
1978 births
Living people
People from Jombang Regency
Deltras F.C. players
Persebaya Surabaya players
Persibo Bojonegoro players
Arema F.C. players
Persiwa Wamena players
Persela Lamongan players
Persepam Madura Utama players
Indonesian Premier Division players
Liga 1 (Indonesia) players
Association football goalkeepers
Sportspeople from East Java

redpajama_set_name: RedPajamaWikipedia


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text:
Yazidi mass graves: 'Those who did this are not human'
As Kurdish forces reclaim Yazidi towns from ISIL, newly discovered mass graves are revealing the scope of devastation.
by Cathy Otten
A pit containing 36 uncovered bodies was recently found behind an abandoned chicken factory in Snuny, Iraq [Cathy Otten/Al Jazeera]
Snuny, Iraq - By a deserted road near the base of Mount Sinjar, Jamal Murd searched through the pockets of the remains of the dead for traces of his father.
As Murd, a 35-year-old member of the minority Yazidi religion from the town of Khanasoor, searched through a makeshift dirt grave - the remains of a massacre believed to have been committed by Islamic State of Iraq and the Levant (ISIL) fighters as they attacked Yazidi towns around Sinjar - he uncovered his uncle's diabetes medication. He recognised his uncle's clothes. And then he put his hand into the pocket of a dead man.
"I found a key in his pocket," Murd told Al Jazeera. "I took [the key] to my house in Khanasoor. It worked in the front door. I tried the kitchen door and that worked too."
That is when Murd realised the dead man was his father, who disappeared months earlier while fleeing from ISIL. His body was found just last week, part of a pit of 36 uncovered bodies discovered by a roaming shepherd behind an abandoned chicken factory after Iraqi Kurdish forces regained control of the route to the mountain.
RELATED: Iraqi Yazidis: 'If we move they will kill us'
Denouncing them as devil-worshippers, ISIL massacred hundreds of Yazidis and captured thousands more as the group tore through Iraq last summer. But as Kurdish fighters have started to reclaim some Yazidi towns and homeowners have begun to return, the full scope of the ISIL slaughter is only now being unearthed.
Another mass grave was discovered last week in the town of Zummar, 60km northwest of Mosul, which the Peshmerga reclaimed from ISIL last October. The grave contained the bodies of more than 16 Yazidis.
We still need to classify the bones and see which came from which body and to know the cause of death. We are working on it because there are so many bodies.
Dr Hawar Muhammed, director of the forensic medical centre in Dohuk
Qassim Simo, the head of the Kurdish security forces in the town of Snuny, close to where Murd's father was found, believes more mass graves will likely emerge as more territory is won back. So far, eight mass graves have been discovered, but "we can't be sure [how many more there are] because part of Sinjar is still under the control of ISIL", Simo said.
The massacre in Snuny is believed to have taken place in early August, after family members last spoke to the victims as they tried to flee. Forensic experts say it is too early to confirm the victims' cause of death. The bodies include those of two women and two children, Simo told Al Jazeera.
"We found clothes, skulls and shoes. We couldn't recognise their faces; there were just bones and ID cards," he said.
Doctors from the forensic medical team in Kurdish Dohuk descended on the area to collect the bodies, along with a human rights delegation from the Kurdistan Regional Government, sent to investigate ISIL's crimes against Yazidis.
"We still need to classify the bones and see which came from which body and to know the cause of death," said Dr Hawar Muhammed, the director of the forensic medical centre in Dohuk. "We are working on it because there are so many bodies."
In Snuny, layers of graffiti tell the story of ISIL's rampage. On one wall, the words "Long live the Islamic State, death to the Peshmerga" have been scrawled over, but remain visible.
Most of the residents of this area have yet to return, but some have come back to guard their homes. On February 5, ISIL fired mortars at Hardan village, 32km east of Snuny, but no one was injured.
"It is just to show they are there," Simo said, noting there are enough Kurdish troops to keep the town safe.
At the roundabout in the centre of the town, Khalaf Qijo, 48, sells oranges, lemons and cigarettes to passing Kurdish fighters. He came back to Snuny as soon as it was liberated, but found his home burned to the ground and is staying with a cousin.
"I came back to work and make a living for my family who live in a camp in Zakho [Iraqi Kurdistan]," Qijo said, noting the lack of electricity has prevented many from returning. "I hope to bring my family back."
RELATED: Yazidis tell horror stories about ISIL captivity
After fleeing the ISIL onslaught last August, tens of thousands of Yazidis took shelter in camps in Iraqi Kurdistan. Now, as they are returning home to find the brutal evidence of ISIL's massacres, many have accused their Arab neighbours of siding with the group - a charge many Arabs deny. The tensions have erupted in fights between villagers, Simo said, and there have been reports of reprisal attacks by Yazidis against Arabs in Sinjar.
Al Jazeera recently reported that Yazidi militias had attacked the Arab village of Buhanaya, killing many of its residents, setting houses on fire and terrorising women and children. Another report on January 27 said that the same Yazidi militia had also abducted Arab women.
Last week, Ali Savhan, 25, discovered the body of his father in the mass grave in Snuny. As he spoke to Al Jazeera, the road where he stood was empty and in the field behind him, a farmhouse lay in a heap of rubble.
"First we found bones and clothes," Savhan said, noting his father disappeared in early August while trying to help other Yazidis escape Khanasoor. "We searched his pockets and I found his things: two pictures of him, his ID card, his gold ring and $3,300."
Savhan acknowledges that the events of last summer have sewn distrust within the community, noting: "Those who did this are not human."


❶They are made up of bonds the bonds you share with those you care for the most. I believe that families are building blocks of our own society.
When most people think of the word "family," mother, father, children come to mind. Then you that's when grandparents, aunts, uncles, and cousins get thrown in or however far back that goes in a family tree.
This definition goes along with the common perception of the traditional fashionlosdaeroh.cf traditional family has in the past served three primary social functions: providing a labour force, the transmission of property, and the education of children toprepare them to start their own. A family definition essay must highlight these aspects in an interesting manner and describe each element or constituent of a family. On Getting Family Essay Ideas There are several aspects to a family, such as the intricacies of relations, values, and the love and affection amongst its members.
Family does not have limitations or boundaries (true). As long as the definition of family contains the qualities of unconditional love, and endless support, help, and guidance; all different families (awk) will be able to succeed and achieve a greater feeling of happiness and a sense of belonging and acceptance. Oct 20, · Family Definition- Please Read And Give Opinions! Thanks!! Forums Essay, so after moving things around and changing some aspect of my essay I finished and this is my final draft. Thank you so much everyone for your help and for being truthful!!! I love your definition of family. I think that is very true especially when it comes to.


[The king of Spain captures a Christian noblewoman, who gives birth to a daughter, Blancheflour, on the same day as his queen bears a son, Floris. The children are raised together]. When they are seven, Floris refuses to begin school without Blancheflour. The pair studies together for five years and their love blossoms. Fearing that they will marry, the King decides to execute Blancheflour. His queen persuades him to send Floris away instead, hoping that the separation will dull their love.
While he is away, Floris ignores his studies and becomes lovesick. The King resolves to kill Blancheflour but is again dissuaded by his wife. Instead, they sell her to some merchants in exchange for gold and an ornate cup, once owned by Aeneas and decorated with scenes of Paris and Helen of Troy. The king erects a false tomb, and when Floris returns he is told that Blancheflour has died and been buried. He laments wildly and tries to kill himself, until his mother confesses and the empty tomb is opened. Floris vows to seek Blancheflour and his parents give him the cup and a magic ring for protection. He dresses as a merchant and lodges at an inn, where he learns that Blancheflour has been taken to Babylon. He rewards his hosts richly and sets sail the next day.
When he arrives, Floris stays in another inn and learns that the emir plans to keep Blancheflour in his harem. The innkeeper sends the youth to his friend Dares, who warns him that the emir's women are kept in an impenetrable tower, an architectural marvel. He also explains how the emir uses enchantments and a magical garden to select his brides. Floris swoons, but Dares has an ingenious plan: Floris should pretend to be a mason and befriend the tower's porter, ensuring his loyalty by letting him win his ornate cup in a game of checkers.
The porter places Floris in a basket of flowers and sends him into the tower, where he is discovered by Blancheflour's friend Clarys. She agrees not to betray them and the reunited couple go to bed. Blancheflour misses an appointment with the emir, but Clarys makes excuses for her. When it happens a second time, however, his chamberlain discovers her in bed with Floris. Enraged, the emir imprisons the lovers.
When they are brought before the emir's barons to be judged, Floris tries to give Blancheflour his magic ring, but she refuses and it falls to the floor. The barons are moved by their beauty, but the emir demands they be killed, until a king, who saw them arguing over the ring, persuades him otherwise. Floris is spared in exchange for revealing how he entered the tower; the barons laugh at his story and the delighted Amir knights him. The lovers are married, and the emir marries Clarys. Shortly after, news reaches Floris that his father has died, and he and his bride return to Spain, where they are crowned.
From: Jennifer Fellows, Of Love and Chivalry: An Anthology of Middle English Romance. London: J.M. Dent for Everyman, 1993.
Cambridge University Library, MS Gg.4.27 (folio: 1r-5v)c.1300, South-west Midlands. 824 lines. Lacks first two leaves, about 350 lines.
Edinburgh, National Library of Scotland, MS Advocates 19.2.1 (Auchinleck) (folio: 100r-104v)c. 1330, London. 861 lines. Begins imperfect (lacking about 384 lines).
London, British Library, MS Cotton Vitellius D.iii (folio: 6r-8v)South West. Before 1300. Fragment of 451 lines, many imperfect.
London, British Library, MS Egerton 2862 (olim Trentham-Sutherland) (folio: 98r-111r)Late 14th century, Suffolk. 1083 lines. Lacks beginning but the most complete of the MSS -only about 80 lines missing.
A. B. Taylor, Floris and Blancheflour: A Middle-English Romance (Oxford: Clarendon, 1927)Edited from Auchinleck with additions from Cotton and Cambridge manuscripts.
A. C. Gibbs, ed., Middle English Romances, York Medieval Texts (London: Edward Arnold; Evanston: Northwestern University Press, 1966)Pp. 76-83. Selections only. Based on Egerton and Cambridge manuscripts.
D. B. Sands, ed., Middle English Verse Romances (Exeter: Exeter University Press, 1986)Pp. 279-309. Based on Egerton.
D. Laing, A Penni Worth of Witte, Abbotsford Club (Edinburgh, 1857)Pp. 15-44. Edited from Auchinleck.
Emil Hausknecht, ed., Floris und Blauncheflur (Berlin: Weidman, 1885)Based on all MSS.
F. C. De Vries, ed., Floris and Blauncheflur (Gröningen: Drukkerij V.R.B, 1966)Based on all MSS.
George H. McKnight, ed., King Horn, Floriz & Blauncheflur, The Assumption of Our Lady, EETS o.s. 14 (London: Oxford University Press, 1901; rpt. 1962)Pp. 71-110, 146-47. Based on Cambridge, Egerton and Cotton Vitellius manuscripts.
Jennifer Fellows, ed., Of Love and Chivalry: An Anthology of Middle English Romance (London: Everyman, 1992)PAGE Nos/ base text.
Walter Hoyt French and Charles Brockway Hale, eds., The Middle English Metrical Romances, 2 vols (New York: Prentice-Hall, 1930; rpt. New York: Russell & Russell, 1964)Vol. 2. Pp. 823-856. Based on Egerton.