\chapter{Basics}

\label{Basics}

\noindent
{\small An extended version of this chapter is available as chapter Foundations of \cite{LEDAbook}}

\section{Getting Started}

\label{A First Example}
Please use your favourite text editor to create a file {\em prog.c} with the following
program: 

\#include \<LEDA/d\_array.h\>\\
main()\\
\{

\ \ \ d\_array\<string,int\> $N(0)$;\\
\smallskip
\ \ \ string $s$;\\
\smallskip
\ \ \ {\bf while} (cin \>\> $s$ ) $N[s]${\tt ++};\\
\smallskip
\ \ \ {\bf forall\_defined}($s,N$) 
                cout \<\< $s$ \<\< " " \<\< $N[s]$ \<\< $endl$;\\
\smallskip
$\}$\\

\noindent
If you followed the installation guidelines (see chapter \ref{Installation}),
you can compile and link it
with LEDA's basic library $libL.a$ (cf. section \ref{Libraries}).
For example on a Unix machine where g{\tt ++} is installed you can type

\begin{center}
g{\tt ++} -o prog prog.c -lL -lm
\end{center}

When executed it reads a sequence of strings from the standard input and then 
prints the number of occurrences of each string on the standard output. More 
examples of LEDA programs can be found throughout this manual.


The program above uses the parameterized data type dictionary array 
($d\_array\<I,E\>$) from the library. This is expressed by the include 
statement (cf. section \ref{Header Files} for more details). The specification 
of the 
data type $d\_array$ can be found in section \ref{Dictionary Arrays}.
We use it also as a 
running example to discuss the principles underlying LEDA in the following sections. 


\section{The LEDA Manual Page (the type specification)} \label{LEDA Manual Page}

\label{Specifications}
In general the specification of a LEDA data type consists of four parts:
a definition of the set of objects comprising the (parameterized) abstract
data type, a description of how to create an object of the data type,
the definition of the operations available on the objects of the data 
type, and finally, information about the implementation. The four parts 
appear under the headers definition, creation, operations, and implementation 
respectively.
Sometimes there is also a fifth part showing an example.

\begin{itemize}
\item{\bf Definition}

\smallskip
This part of the specification defines the objects (also called instances or
elements) comprising the data type using standard mathematical concepts and 
notation. 

{\bf Example}

The generic data type dictionary array:

An object $a$ of type $d\_array\<I,E\>$ is an injective function from the
data type $I$ to the set of variables of data type $E$. The types $I$ and
$E$ are called the index and the element type respectively, $a$ is called
a dictionary array from $I$ to $E$.

Note that the types $I$ and $E$ are parameters in the definition above.
Any built-in, pointer, item, or user-defined class type $T$ can be used 
as actual type parameter of a parameterized data type. Class types however 
have to provide several operations listed in chapter \ref{UserDefinedParameterTypes}.

\item {\bf Creation}

\smallskip
A variable of a data type is introduced by a \CC variable declaration. 
For all LEDA data types variables are initialized at the time of declaration. 
In many cases the user has to provide arguments used for the initialization 
of the variable.  In general a declaration

$XYZ\<t_1,\dots,t_k\>$ \ \ \ $y(x_1,\dots,x_\ell)$;

introduces a variable $y$ of the data type ``$XYZ\<t_1,\dots,t_k\>$''
and uses the arguments $x_1,\dots,x_\ell$ to initialize it. For example,

$d\_array\<string,int\>$ $A(0)$

introduces $A$ as a dictionary array from strings to integers, and 
initializes $A$ as follows: an injective function $a$ from $string$
to the set of unused variables of type $int$ is constructed, and is
assigned to $A$. Moreover, all variables in the range of $a$ are 
initialized to 0.
The reader may wonder how LEDA handles an array of infinite size. The
solution is, of course, that only that part of $A$ is explicitly stored
which has been accessed already.

For all data types, the assignment operator ($=$) is available for variables 
of that type. Note however that assignment is in general not a constant time 
operation, e.g., if $L_1$ and $L_2$ are variables of type $list\<T\>$ then the 
assignment $L_1 = L_2$ takes time proportional to the length of the list $L_2$
times the time required for copying an object of type $T$.

{\bf Remark}: For most of the complex data types of LEDA, e.g., dictionaries,
lists, and priority queues, it is convenient to interpret a variable name
as the name for an object of the data type which evolves over time by means
of the operations applied to it. This is appropriate, whenever the operations 
on a data type only ``modify'' the values of variables, e.g., it is more 
natural to say an operation on a dictionary $D$ modifies $D$ than to say that 
it takes the old value of $D$, constructs a new dictionary out of it, and 
assigns the new value to $D$. Of course, both interpretations are equivalent.
From this more object-oriented point of view, a variable declaration, e.g.,
$dictionary\<string,int\>$ $D$, is creating a new dictionary object with 
name $D$ rather than introducing a new variable of type 
$dictionary\<string,int\>$; hence the name ``creation'' for this part of 
a specification.
 
\item{\bf Operations}

\smallskip
In this section the operations of the data types are described. For each 
operation the description consists of two parts

\begin{enumerate}
\item 
The interface of the operation is defined using the \CC function declaration
syntax. In this syntax the result type of the operation ($void$ if there is 
no result) is followed by the operation name and an argument list specifying 
the type of each argument. For example,

\smallskip
$list\_item$ $L$.insert ($E\ x,\ list\_item\ it,\ int\ dir=after$)\\
defines the interface of the insert operation on a list $L$ of elements of type
$E$ (cf. section \ref{Linear Lists}). Insert takes as arguments an element $x$ 
of type $E$, 
a $list\_item$ $it$ and an optional relative position argument $dir$. It returns 
a $list\_item$ as result.  

\smallskip
$E\&$  $A[I\ x]$\\
defines the interface of the access operation on a dictionary array $A$. It 
takes an element $x$ of type $I$ as an argument and returns a variable of type $E$.

\item  
{The effect of the operation is defined.  Often the arguments have to 
fulfill certain preconditions. If such a condition is violated the effect 
of the operation is undefined. Some, but not all, of these cases result 
in error messages and abnormal termination of the program (see also 
section \ref{Error Handling}). For the insert operation on lists this 
definition reads:\\
A new item with contents $x$ is inserted after (if $dir=after$) or before 
(if $dir=before$) item $it$ into $L$. The new item is returned.

{\it Precondition}: item $it$ must be in $L$.}

{For the access operation on dictionary arrays the definition reads:\\
returns the variable $A(x)$.
}
\end{enumerate}

\item {\bf  Implementation}

\smallskip
The implementation section lists the (default) data structures used to 
implement the data type and gives the time bounds for the operations and 
the space requirement. For example,

Dictionary arrays are implemented by randomized search trees (\cite{AS89}). 
Access operations $A[x]$ take time $O(\log dom(A))$.
The space requirement is $O(dom(A))$.
\end{itemize}

\section{User Defined Parameter Types} \label{UserDefinedParameterTypes}

If a user defined class type $T$ shall be used as actual type parameter
in a container class, it has to provide the following operations:

\[
\begin{array}{ll}
\mbox{a) a constructor taking no arguments} &\mbox{T::T()} \\
\mbox{b) a copy constructor}     &\mbox{T::T(const T\&)} \\
\mbox{c) an assignment operator} &\mbox{T\&\ T::operator=(const T\&)} \\
\\
\multicolumn{2}{c}{\mbox{and if required by the parameterized data type}}\\
\\
\mbox{d) an input operator} &\mbox{istream\&\ {\bf operator\>\>}(istream\&,T\&)}\\
\mbox{e) an output operator} &\mbox{ostream\&\ {\bf operator\<\<}(ostream\&, const\ T\&)}\\
\mbox{f) a compare function} &\mbox{int \ {\bf compare}(const T\&, const T\&)}\\
\mbox{g) a hash function}    &\mbox{int \ {\bf Hash}(const T\&)}\\
\end{array}
\]

\bigskip\noindent
In the following two subsections we explain the background of the required
compare and hash function. Section \ref{Implementation Parameters} concerns a very
special parameter type, namely implementation parameters.


\subsection{Linear Orders} \label{Linear Orders}

Many data types, such as dictionaries, priority queues, and sorted sequences
require linearly ordered parameter types. Whenever a type $T$ is used in such a
situation, e.g. in $dictionary\<T,\dots\>$ the function
$$int\ \ compare(const\ T\&,\ const\ T\&)$$ 
must be declared and must define a linear order on the data type $T$.

A binary relation $rel$ on a set $T$ is called a linear order on $T$ if for
all $x,y,z\in T$:

\smallskip
1) $x$ $rel$ $x$\\
2) $x$ $rel$ $y$ and $y$ $rel$ $z$ implies $x$ $rel$ $z$\\
3) $x$ $rel$ $y$ or  $y$ $rel$ $x$\\
4) $x$ $rel$ $y$ and $y$ $rel$ $x$ implies $x = y$

\medskip
A function $int$ $compare(const\ T\&,\ const\ T\&)$ defines the linear order 
$rel$ on $T$ if
$$compare(x,y)\ \ \cases{ < 0, &if $x$ $rel$ $y$  and  $x\ne y$\cr 
                       = 0, &if $x = y$\cr 
                       > 0, &if $y$ $rel$ $x$  and  $x\ne y$\cr} $$


For each of the data types $char$, $short$, $int$, $long$, $float$, 
$double$, $integer$, $rational$, $bigfloat$, $real$, $string$, and 
$point$ a function $compare$ is predefined and defines 
the so-called default ordering on that type. The default ordering is the 
usual $\le$ - order for the built-in numerical types, the lexicographic 
ordering for $string$, and for $point$ the lexicographic ordering of the 
cartesian coordinates.  For all other types $T$ there is no default 
ordering, and the user has to provide a $compare$ function whenever a linear 
order on $T$ is required.



{\bf Example}: Suppose pairs of real numbers shall be used as keys 
in a dictionary with the lexicographic order of their components.
First we declare class $pair$ as the type of pairs of real numbers, 
then we define the I/O operations $operator\>\>$ and $operator\<\<$ and the 
lexicographic order on $pair$ by writing an appropriate $compare$ function.


\bigskip
{\bf class} $pair$ $\{$

\ \ \ $double\ \ x$;

\ \ \ $double\ \ y$;

{\bf public:}

\ \ \ $pair()\ \{\ x = y = 0;\ \}$

\ \ \ $pair($const $pair\&\ p)\ \{\ x = p.x;\ y = p.y;\ \}$

\smallskip
\ \ \ friend $istream\&$ operator\>\>($istream$\& $is$, $pair$\&\ $p$) 
      $\{$ $is$ \>\> $p.x$ \>\> $p.y$; $\}$

\ \ \ friend $ostream\&$ operator\<\<($ostream$\& $os$, const $pair$\&\ $p$) 
      $\{$ $os$ \<\< $p.x$ \<\< " " \<\< $p.y$; $\}$

\ \ \ friend $int$ compare(const $pair$\&, const $pair$\&);\\
$\}$;\\
\\
$int$ compare(const $pair\&\ p$, const $pair\&\ q$)\\
$\{$\\
\smallskip
\ \ \ {\bf if} ($p.x$ \< $q.x$) {\bf return } -1;\\
\smallskip
\ \ \ {\bf if} ($p.x$ \> $q.x$) {\bf return }  1;\\ 
\smallskip
\ \ \ {\bf if} ($p.y$ \< $q.y$) {\bf return } -1;\\
\smallskip
\ \ \ {\bf if} ($p.y$ \> $q.y$) {\bf return }  1;\\
\smallskip
\ \ \ {\bf return} 0; \\
$\}$

\smallskip
Now we can use dictionaries with key type $pair$, e.g.,

dictionary\<$pair$,$int$\> D;

\bigskip
Sometimes, a user may need additional linear orders on a data type $T$ 
which are different from the order defined by $compare$, e.g., he might 
want to order points in the plane by the lexicographic ordering of their 
cartesian coordinates and by their polar coordinates. In this example, the 
former ordering is the default ordering for points. The user can introduce 
an alternative ordering on the data type $point$ (cf. section 
\ref{Basic Data Types for Computational Geometry}) by defining
an appropriate comparing function $int$ $cmp$(const $point$\&, const $point$\&)
and then calling the macro 

\begin{center}
DEFINE\_LINEAR\_ORDER($point$, $cmp$, $point_1$). 
\end{center}

\smallskip
After this call $point_1$ is a new data type which is equivalent to the data 
type $point$, with the only exception that if $point_1$ is used as an actual 
parameter e.g. in $dictionary\<point_1,\dots\>$, the resulting data type 
is based on the linear order defined by $cmp$.

In general the macro call

\begin{center}
DEFINE\_LINEAR\_ORDER($T$, $cmp$, $T_1$)  
\end{center}

introduces a new type $T_1$ equivalent to type $T$ with the linear order
defined by the compare function $cmp$.

In the example, we first declare a function $pol\_cmp$ and derive a new type
$pol\_point$ using the DEFINE\_LINEAR\_ORDER macro.\\
\smallskip
$int$  $pol\_cmp$(const $point$\& $x$, const $point$\&\ $y$)\\
$\{$\ \ \co lexicographic ordering on polar coordinates  $\}$\\
\\
DEFINE\_LINEAR\_ORDER($point$, $pol\_cmp$, $pol\_point$)\\

Now, dictionaries based on either ordering can be used.

\smallskip
$dictionary\<pol\_point,int\>$ $D_1$; \ \co polar ordering\\
\smallskip
$dictionary\<point,int\>$ $D_0$;\ \ \ \ \ \ \co default ordering\\

{\bf Remark}: We have chosen to associate a fixed linear order with most of
the simple types (by predefining the function $compare$). This order is used
whenever operations require a linear order on the type, e.g., the operations
on a dictionary. Alternatively, we could have required the user to specify a
linear order each time he uses a simple type in a situation where an ordering
is needed, e.g., a user could define

\quad\quad\quad $dictionary\<point,lexicographic\_ordering,\dots\>$

This alternative would handle the cases where two or more different orderings
are needed more elegantly. However, we have chosen the first alternative
because of the smaller implementation effort.

\subsection{Hashed Types} \label{Hashed Types}

LEDA also contains parameterized data types requiring a {\em hash function}
and an {\em equality test} (operator==) for the actual type parameters. 
Examples are dictionaries implemented 
by hashing with chaining ($\_dictionary\<K,I,ch\_hashing\>$)
or hashing arrays ($h\_array\<I,E\>$).  Whenever a type $T$ is used in such
a context, e.g., in $h\_array\<T,\dots\>$ there must be defined 
\begin{enumerate}
\item a hash function $int\ \ Hash(const\ T\&)$
\item the equality test $bool operator==(const\ T\&,\ const\ T\&)$
\end{enumerate}
$Hash$ maps the elements of type $T$ to integers. It is not
required that $Hash$ is a perfect hash function, i.e., it has not to be
injective. However, the performance of the underlying implementations
very strongly depends on the ability of the function to keep different
elements of $T$ apart by assigning them different integers.
Typically, a search operation in a hashing implementation takes time
linear in the maximal size of any subset whose elements are assigned the
same hash value.
For each of the simple numerical data types char, short, int, long
there is a predefined $Hash$ function: the identity function.

We demonstrate the use of $Hash$ and a data type based on hashing
by extending the example from the previous section. Suppose we
want to associate information with values of the $pair$ class
by using a hashing array $h\_array\<pair,int\>\ A$. We first
define a hash function that assigns each pair $(x,y)$
the integral part of the first component $x$

$int \ \ Hash(const\ pair\&\ p$) $\{$ return $int(p.x)$; $\}$\\

and then can use a hashing array with index type $pair$\\

$h\_array\<pair,int\>\ A$;


\subsection{Implementation Parameters}
   
\label{Implementation Parameters}
For many of the parameterized data types (in the current version: dictionary, 
priority queue, d\_array, and sortseq) there exist variants taking an additional
data structure parameter for choosing a particular implementation 
(cf. chapter \ref{Dictionaries with Implementation Parameter},
\ref{Sorted Sequences with Implementation Parameter},
\ref{Dictionary Arrays with Implementation Parameter}
and \ref{Priority Queues with Implementation Parameter}). Since \CC
does not
allow to overload templates we had 
to use different names: the variants with an additional implementation 
parameters start with an underscore, e.g., \_d\_array\<I,E,impl\>. 
We can easily modify the example program from section \ref{A First Example} 
to use a dictionary 
array implemented by a particular data structure, e.g., skip lists (\cite{Pu90}), 
instead of the default data structure (cf. section \ref{List of data structures}). 
\medskip

\#include \<LEDA/d\_array.h\>\\
\#include \<LEDA/impl/skiplist.h\>\\
\smallskip
main()\\
$\{$

\ \ \ \_d\_array\<string,int,skiplist\> $N(0)$;\\
\smallskip
\ \ \ string $s$;\\
\smallskip
\ \ \ {\bf while} (cin \>\> $s$) $N[s]${\tt ++};\\
\smallskip
\ \ \ {\bf forall\_defined}($s,N$) 
                cout \<\< $s$ \<\< " " \<\< $N[s]$ \<\< $endl$;\\
\smallskip
$\}$\\

Any type \_XYZ\<$T_1,\dots,T_k,xyz\_impl$\> is derived from the corresponding
``normal" parameterized type XYZ\<$T_1,\dots,T_k$\>, i.e., an instance of type 
\_XYZ\<$T_1,\dots,T_k,xyz\_impl$\> can be passed as argument to functions with
a formal parameter of type XYZ\<$T_1,\dots,T_k$\>\&. 
This provides a mechanism
for choosing implementations of data types in pre-compiled algorithms.
See ``prog/graph/dijkstra.c" for an example.

LEDA offers several implementations for each of the data types. For
instance, skip lists, randomized search trees, and red-black trees
for dictionary arrays. Users can also provide their own implementation. 
A data structure ``xyz\_impl" can be used as actual
implementation parameter for a data type $\_XYZ$ if it provides a 
certain set of operations and uses certain virtual functions for type 
dependent operations (e.g. compare, initialize, copy, \dots).
Chapter \ref{Implementations} lists all data structures contained in the current version and 
gives the exact requirements for implementations of 
dictionaries, priority\_queues, sorted sequences and dictionary arrays.
A detailed description of the mechanism for parameterized data types and 
implementation parameters used in LEDA will be published soon.


\section{Arguments}

\label{Arguments}
\begin{itemize}
\item{\bf Optional Arguments}

\smallskip
The trailing arguments in the argument list of an operation may be optional.
If these trailing arguments are missing in a call of an operation the default 
argument values given in the specification are used. For
example, if the relative position argument in the list insert operation
is missing it is assumed to have the value $after$, i.e.,
$L$.insert($it,y$) will insert the item \<$y$\> after item $it$ into $L$.

\item{\bf Argument Passing}

\smallskip
There are two kinds of argument passing in \CC, by value and by reference.
An argument $x$  of type $type$  specified by ``$type\ x$'' in the argument 
list of an operation or user defined function will be passed by value, i.e.,
the operation or function is provided with a copy of $x$.
The syntax for specifying an argument passed by reference is ``$type\&\ x$''.
In this case the operation or function works directly on $x$ ( the variable
$x$ is passed not its value).

Passing by reference must always be used if the operation is to change the
value of the argument. It should always be used for passing large objects
such as lists, arrays, graphs and other LEDA data types to functions.
Otherwise a complete copy of the actual argument is made, which takes time
proportional to its size, whereas  passing by reference always takes constant
time. 


\item{\bf Functions as Arguments}

\smallskip
Some operations take functions as arguments. For instance the bucket sort 
operation on lists requires a function which maps the elements of the list into
an interval of integers. We use the \CC syntax to define the type of a function
argument $f$:
$$T\ \ (*f)(T_1, T_2,\dots, T_k)$$ declares $f$ to be a function 
taking $k$ arguments of the data types $T_1$, \dots, $T_k$,
respectively, and returning a result of type $T$, i.e,
$f: T_1 \times \dots \times T_k \longrightarrow T$ .
\end{itemize}



\section{Items }

\label{Items}
Many of the advanced data types in LEDA (dictionaries, priority queues, 
graphs, \dots), are defined in terms of so-called items. 
An item is a container which can hold an object
relevant for the data type. For example, in the case of dictionaries a
$dic\_item$ contains a pair consisting of a key and an information.
A general definition of items is given at the end of this section. 

We now discuss the role of items for the dictionary example in some detail. 
A popular specification of dictionaries defines a dictionary as a partial
function from some type $K$ to some other type $I$, or alternatively, as a
set of pairs from $K\times I$, i.e., as the graph of the function. In an
implementation each pair $(k,i)$ in the dictionary is stored in some location
of the memory. Efficiency dictates that the pair $(k,i)$ cannot only be
accessed through the key $k$ but sometimes also through the location where it
is stored, e.g., we might want to lookup the information $i$ associated with
key $k$ (this involves a search in the data structure), then compute with the
value $i$ a new value $i\'$, and finally associate the new value with $k$.
This either involves another search in the data structure or, if the lookup
returned the location where the pair $(k,i)$ is stored, can be done by direct
access. Of course, the second solution is more efficient and we therefore
wanted to provide it in LEDA.

In LEDA items play the role of positions or locations in data structures. Thus
an object of type $dictionary\<K,I\>$, where $K$ and $I$ are types, is 
defined as a collection of items (type $dic\_item$) where each item contains 
a pair in $K\times I$. We use \<$k,i$\> to denote an item with key $k$ and 
information $i$ and require that for each $k\in K$ there is at most one 
$i\in I$ such that \<$k,i$\> is in the dictionary. In mathematical terms this
definition may be rephrased as follows: A dictionary $d$ is a partial
function from the set $dic\_item$ to the set $K\times I$. Moreover, for each
$k\in K$ there is at most one $i\in I$ such that the pair $(k,i)$ is in $d$.

The functionality of the operations 

\smallskip
\begin{tabular}{ll}
$dic\_item$  &$D$.lookup($K\ k$)\\
\\
$I$ &$D$.inf($dic\_item\ it$)\\
\\
$void$ &$D$.change\_inf($dic\_item\ it,\ I\ i\'$)
\end{tabular}

\smallskip
is now as follows:
$D$.lookup($k$) returns an item $it$ with contents $(k,i)$, $D$.inf($it$) 
extracts $i$ from $it$, and a new value $i\'$ can be associated with $k$ by
$D$.change\_inf($it,i\'$).


Let us have a look at the insert operation for dictionaries next:

\smallskip
$dic\_item$ $D$.insert($K\ k,\ I\ i$)

There are two cases to consider. If $D$ contains an item $it$ with contents
$(k,i\')$ then $i\'$ is replaced by $i$ and $it$ is returned. If $D$ contains
no such item, then a new item, i.e., an item which is not contained in any 
dictionary, is added to $D$, this item is made to contain $(k,i)$ and is
returned. In this manual (cf. section \ref{Dictionaries}) all of this 
is abbreviated to\\

\parbox{2cm}{$dic\_item$}
\parbox{4cm}{$D$.insert($K\ k,\  I\ i$)} 
\parbox[t]{9cm}{associates the information $i$ with the key $k$.
             If there is an item \<$k,j$\> in $D$ then $j$ is
             replaced by i, else a new item \<$k,i$\> is added
             to $D$. In both cases the item is returned.}\\

We now turn to a general discussion. With some LEDA types $XYZ$ there is an
associated type $XYZ\_item$ of items. Nothing is known about the objects of
type $XYZ\_item$ except that there are infinitely many of them. The only
operations available on $XYZ\_items$ besides the one defined in the
specification of type $XYZ$ is the equality predicate ``=='' and the assignment
operator ``='' . The objects of type $XYZ$ are defined as sets or sequences of
$XYZ\_items$ containing objects of some other type $Z$. In this situation an
$XYZ\_item$ containing an object $z\in Z$ is denoted by \<$z$\>. A new or unused
$XYZ\_item$ is any $XYZ\_item$ which is not part of any object of type $XYZ$.

{\bf Remark}: For some readers it may be useful to interpret a $dic\_item$ as
a pointer to a variable of type $K\times I$. The differences are that the
assignment to the variable contained in a $dic\_item$ is restricted, e.g., the
$K$-component cannot be changed, and that in return for this restriction the
access to $dic\_items$ is more flexible than for ordinary variables, e.g.,
access through the value of the $K$-component is possible.


\section{Iteration}

\label{Iteration}
For many data types LEDA provides iteration macros. These macros can be
used to iterate over the elements of lists, sets and dictionaries or
the nodes and edges of a graph. 
Iteration macros can be used similarly to 
the \CC {\bf for} statement.\\
\ \\
{\bf PLEASE NOTE:} 
inside the body of a loop the corresponding  object must not be altered. 
For instance, it is not allowed to delete nodes from a graph $G$
inside the body of a {\bf forall\_nodes} loop.
Examples are

for all item based data types:

\smallskip
{\bf forall\_items}($it,D$) 
$\{$ the items of $D$ are successively assigned to variable $it\ \}$

\smallskip
{\bf forall\_rev\_items}($it,D$) 
$\{$ the items of $D$ are assigned to $it$ in reverse order $\ \}$

\smallskip
for lists and sets:

\smallskip
{\bf forall}($x,L$)
$\{$ the elements of $L$ are successively assigned to $x\ \}$

\smallskip
{\bf forall\_rev}($x,L$)
$\{$ the elements of $L$ are assigned to $x$ in reverse order $\ \}$

\smallskip
for graphs:

\smallskip
{\bf forall\_nodes}($v,G$)
$\{$ the nodes of $G$ are successively assigned to $v\ \}$

\smallskip
{\bf forall\_edges}($e,G$)
$\{$ the edges of $G$ are successively assigned to $e\ \}$

\smallskip
{\bf forall\_adj\_edges}($e,v$)
$\{$ all edges adjacent to $v$ are successively assigned to $e\ \}$

