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Addition 
interval 
interval 
element 
element 


















Functions and operators that implement Addition
on icl objects are given in the table
above. operator =
and operator 
are behavioral identical to operator
+=
and operator
+
. This is a redundancy that has
been introduced deliberately, because a set union
semantics is often attached operators
=
and 
.

Description of Addition 


Addition on Sets implements set union 

Addition on Maps implements a map
union function similar to set union.
If, on insertion of an element value pair Find more on addability of maps and related semantic issues following the links. Examples, demonstrating Addition on interval containers are overlap counter, party and party's height average. 
For Sets
addition and insertion
are implemented identically. Functions add
and insert
collapse to
the same function. For Maps
addition and insertion work differently. Function
add
performs aggregations
on collision or overlap, while function insert
only inserts values that do not yet have key values.
The admissible combinations of types for member function T& T::add(const P&)
can be summarized in the overload table
below:
// overload table for T\P e i b p T& T::add(const P&) + T& add(T&, const P&) s  s m  m S  S S M  M M
The next table contains complexity characteristics for add
.
Table 1.21. Time Complexity for member function add on icl containers

domain 
interval 
domain 
interval 

O(log n) 






O(log n) 


O(log n) 
amortized 



O(log n) 
O(n) 





O(log n) 
O(n) 
Function T&
T::add(T::iterator
prior,
const P& addend)
allows for an addition in constant time, if addend
can be inserted right after iterator prior
without collision. If this is not possible the complexity characteristics
are as stated for the non hinted addition above. Hinted addition is available
for these combinations of types:
// overload table for addition with hint T\P e i b p T& T::add(T::iterator prior, const P&) + T& add(T&, T::iterator prior, const P&) s  s m  m S  S M  M
The possible overloads of inplace T& operator
+= (T&, const P&)
are given by two tables, that show
admissible combinations of types. Row types show instantiations of argument
type T
. Columns types show
show instantiations of argument type P
.
If a combination of argument types is possible, the related table cell
contains the result type of the operation. Placeholders
e
i
b
p
s
S
m
M
will be used to denote elements, intervals, element value pairs,
interval value pairs, element sets, interval sets, element maps
and interval maps. The first table shows the overloads
of +=
for element
containers the second table refers to interval containers.
// overload tables for element containers: interval containers: T& operator += (T&, const P&) +=  e b s m +=  e i b p S M + + s  s s S  S S S m  m m M  M M M
For the definition of admissible overloads we separate element containers from interval containers. Within each group all combinations of types are supported for an operation, that are in line with the icl's design and the sets of laws, that establish the icl's semantics.
Overloads between element containers and interval containers could also be defined. But this has not been done for pragmatical reasons: Each additional combination of types for an operation enlarges the space of possible overloads. This makes the overload resolution by compilers more complex, error prone and slows down compilation speed. Error messages for unresolvable or ambiguous overloads are difficult to read and understand. Therefore overloading of namespace global functions in the icl are limited to a reasonable field of combinations, that are described here.
For different combinations of argument types T
and P
different implementations
of the operator +=
are selected. These implementations show different complexity characteristics.
If T
is a container type,
the combination of domain elements (e) or element value pairs (b) is faster than a combination of intervals
(i)
or interval value pairs (p) which in turn is faster than the combination
of element or interval containers. The next table shows time complexities
of addition for icl's element containers.
Sizes n
and m
are in the complexity statements are
sizes of objects T y
and P x
:
n = iterative_size(y); m = iterative_size(x); //if P is a container type
Note, that for an interval container the number of elements T::size
is different from the number of intervals that you can iterate over. Therefore
a function T::iterative_size()
is used that provides the desired kind of size.
Time complexity characteristics of inplace addition for interval containers is given by this table.
Table 1.23. Time Complexity for inplace Addition on interval containers


domain 
interval 
domain 
interval 
interval 
interval 

interval_sets 
O(log n) 
amortized 


O(m log(n+m)) 



O(log n) 
O(n) 


O(m log(n+m)) 


interval_maps 



O(log n) 
O(n) 

O(m log(n+m)) 
Since the implementation of element and interval containers is based on the
link redblack tree implementation
of std::AssociativeContainers, we have a logarithmic complexity for addition
of elements. Addition of intervals or interval value pairs is amortized logarithmic
for interval_sets
and
separate_interval_sets
and linear for split_interval_sets
and interval_maps
.
Addition is linear for element containers and loglinear for interval containers.
The admissible type combinations for infix operator
+
are defined by the overload tables
below.
// overload tables for element containers: interval containers: T operator + (T, const P&) +  e b s m +  e i b p S1 S2 S3 M1 M3 T operator + (const P&, T) + + e  s e  S1 S2 S3 b  m i  S1 S2 S3 s  s s b  M1 M3 m  m m p  M1 M3 S1  S1 S1 S1 S2 S3 S2  S2 S2 S2 S2 S3 S3  S3 S3 S3 S3 S3 M1  M1 M1 M1 M3 M3  M3 M3 M3 M3
See also . . .
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