[/
    Copyright (c) 2008-2010 Joachim Faulhaber

    Distributed under the Boost Software License, Version 1.0.
    (See accompanying file LICENSE_1_0.txt or copy at
    http://www.boost.org/LICENSE_1_0.txt)
]

[/ //= Intersection ============================================================]
[section Intersection][/ Intersection]

[section Synopsis][/ Intersection]

[table
[[Intersection]                             [__ch_itv_t__][__ch_itv_sets__][__ch_itv_maps__][__ch_ele_sets__][__ch_ele_maps__]]

[[`void add_intersection(T&, const T&, const P&)`][ ]   [__eiS][__eiS __bpM][ ]     [ ]       ]
[[`T& operator &=(T&, const P&)`]                 [ ]   [__eiS][__eiS __bpM][__es]  [__bm]    ]
[[`T  operator & (T, const P&)`\n
  `T  operator & (const P&, T)`]                 [__i]  [__eiS][__eiS __bpM][__es]  [__bm]    ]
[[`bool intersects(const T&, const P&)`\n
  `bool disjoint(const T&, const P&)`]           [__i]  [__eiS][__eiS __bpM][__es]  [__bm]    ]
]

Functions and operators that are related to ['*intersection*] on *icl* objects
are given in the table above.


[table
[[]      [Description of Intersection]]
[[`Sets`][Intersection on Sets implements ['*set intersection*]]]
[[`Maps`][Intersection on Maps implements a ['*map intersection*] function similar to /set intersection/.
          If, on intersection, an element value pair `(k,v)` it's key `k` is in the map 
          already, the intersection function is propagated to the associated value,
          if it exists for the Map's codomain_type. 
          
          If the codomain_type has no intersection operation, associated 
          values are combined using addition. For partial map types this
          results in an addition on the intersection of the domains of
          the intersected sets. For total maps intersection and
          addition are identical in this case.
                  
          See also
          [link boost_icl.semantics.quantifiers__maps_of_numbers.intersection_on_quantifiers ['intersection on Maps of numbers]]. 
        
          A Map can be intersected with key types: an element 
          (an interval for interval_maps) and and a Set. This 
          results in the selection of a submap, and can be
          defined as a generalized selection function on Maps.
         ]]
]

[endsect][/ Synopsis Intersection]


[section Functions][/ Intersection]

The overloaded function 

`void add_intersection(T& result, const T& y, const P& x)`

allows to accumulate the intersection of `y` and `x` in the first argument `result`. 
`Result` might already contain data. In this case the intersection of `y` and `x` 
is `added` the the contents of `result`.
``
T s1 = f, s2 = f, y = g; P x = h; // The effect of 
add_intersection(s1, y, x);       // add_intersection 
s2 += (y & x);                    // and & followed by +=
assert(s1==s2);                   // is identical
``

This might be convenient, if intersection is used like a generalized selection function.
Using element or segment types for `P`, we can select small parts of a container
`y` and accumulate them in `section`.

The admissible combinations of types for function 
`void add_intersection(T&, const T&, const P&)` can be summarized in the 
['*overload table*] below. 
Compared to other overload tables, placements of function arguments are
different: Row headers denote type `T` of `*this` object.
Columns headers denote type `P` of the second function argument.
The table cells contain the arguments `T` of the intersections
`result`, which is the functions first argument.

``
/* overload table for */                                T\P| e i b p 
void T::add_intersection(T& result, const P&)const      ---+--------
                                                         s | s
                                                         m | m   m
                                                         S | S S     
                                                         M | M M M M 
``

The next table contains complexity characteristics for function `add_intersection`.

[table Time Complexity for function add_intersection on icl containers
[[`void add_intersection(T&, const T&, const P&)const`] [__ch_dom_t__][__ch_itv_t__][__ch_dom_mp_t__][__ch_itv_mp_t__]]
[[__icl_set__]                                   [__Olgn__]    []            []           []          ]
[[__icl_map__]                                   [__Olgn__]    []            [__Olgn__]   []          ]
[[__itv_sets__]                                  [__Olgn__]    [__On__]      []           []          ]
[[__itv_maps__]                                  [__Olgn__]    [__On__]      [__On__]     [__On__]    ]
]

[endsect][/ Function Intersection]


[section Inplace operators][/ Intersection]

The overload tables below are giving admissible
type combinations for the intersection `operator &=`.
As for the overload patterns of /subtraction/
intersections are possible within Sets and Maps
but also for Maps combined with /key objects/
which are /key elements, intervals/ and /Sets of keys/.

``
// 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 M M M M M    
``

While intersection on maps can be viewed as
a ['*generalisation of set intersection*]. The
combination on Maps and Sets can be interpreted as
a ['*generalized selection function*], because it
allows to select parts of a maps using
/key/ or /selection objects/.
So we have a ['*generalized intersection*] for
these overloads,

``
/* (Generalized) intersection */   &= | e b s m            &= | e i b p S M  
                                   ---+--------            ---+------------  
                                   s  | s   s              S  | S S     S    
                                   m  |   m   m            M  |     M M   M  
``

[*and] a ['*selection by key objects*] here:

``
/* Selection by key objects */     &= | e b s m            &= | e i b p S M 
                                   ---+--------            ---+------------ 
                                   s  | s   s              S  | S S     S   
                                   m  | m   m              M  | M M     M   
``

The differences for the different functionalities
of `operator &=` are on the Map-row of the
tables. Both functionalities fall together
for Sets in the function ['*set intersection*].

Complexity characteristics for inplace intersection operations are 
given by the next tables where 
``
n = iterative_size(y);
m = iterative_size(x); //if P is a container type
``

[table Time Complexity for inplace intersection on element containers
[[`T& operator &= (T& y, const P& x)`][__ch_dom_t__][__ch_dom_mp_t__][__ch_icl_set__][__ch_icl_map__]]
[[__icl_set__]                    [__Olgn__]    []               [__Omlgn__]     []              ]
[[__icl_map__]                    [__Olgn__]    [__Olgn__]       [__Omlgn__]     [__Omlgn__]     ]
]

[table Time Complexity for inplace intersection on interval containers
[[`T& operator &= (T& y, const P& x)`][__ch_dom_t__][__ch_itv_t__][__ch_dom_mp_t__][__ch_itv_mp_t__][__ch_itv_sets__][__ch_itv_maps__]]
[[interval_sets]                  [__Olgn__]    [__On__]      []               []               [__Omlgnpm__]    []               ]
[[interval_maps]                  [__Olgn__]    [__On__]      [__Olgn__]       [__On__]         [__Omlgnpm__]    [__Omlgnpm__]    ]
]

[endsect][/ Inplace operators Intersection]

[section Infix operators][/ Intersection]

For the *icl's* infix intersection the
following overloads are available:

``
// 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 m            e  |             S1 S2 S3 M1 M3
                                   b  |       m            i  |    i        S1 S2 S3 M1 M3
                                   s  | s   s m            b  |                      M1 M3
                                   m  | m m m m            p  |                      M1 M3
                                                           S1 | S1 S1       S1 S2 S3 M1 M3
                                                           S2 | S2 S2       S2 S2 S3 M1 M3
                                                           S3 | S3 S3       S3 S3 S3 M1 M3
                                                           M1 | M1 M1 M1 M1 M1 M1 M1 M1 M3
                                                           M3 | M3 M3 M3 M3 M3 M3 M3 M3 M3
``

To resolve ambiguities among interval containers
the ['*finer*] container type is chosen as result type.

Again, we can split up the overload tables of
`operator &` in a part describing
the ['*generalized intersection] on interval containers
and a second part defining the
['*selection by key object] functionality.

``
/* (Generalized) intersection */   &  | e b s m            &  | e  i  b  p  S1 S2 S3 M1 M3
                                   ---+--------            ---+---------------------------
                                   e  |     s              e  |             S1 S2 S3      
                                   b  |       m            i  |    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
``

``
/* Selection by key objects */     &  | e b s m            &  | e  i  b  p  S1 S2 S3 M1 M3
                                   ---+--------            ---+---------------------------
                                   e  |     s m            e  |             S1 S2 S3 M1 M3
                                   b  |                    i  |    i        S1 S2 S3 M1 M3
                                   s  | s   s m            b  |                           
                                   m  | m   m              p  |                           
                                                           S1 | S1 S1       S1 S2 S3 M1 M3
                                                           S2 | S2 S2       S2 S2 S3 M1 M3
                                                           S3 | S3 S3       S3 S3 S3 M1 M3
                                                           M1 | M1 M1       M1 M1 M1      
                                                           M3 | M3 M3       M3 M3 M3      
``

[endsect][/ Inplace operator Intersection]

[section Intersection tester]

[table
[[Tester]                                          [Desctription]]
[[`bool intersects(const T& left, const P& right)`][Tests, if `left` and `right` intersect.]]
[[`bool disjoint(const T& left, const P& right)`]  [Tests, if `left` and `right` are disjoint.]]
[[]                                                [`intersects(x,y) == !disjoint(x,y)`]]
]

``
bool intersects(const T&, const P&)      T\P| e b s m      T\P| e i b p S M 
bool   disjoint(const T&, const P&)      ---+--------      ---+------------
                                          s | 1   1         S | 1 1     1  
                                          m | 1 1 1 1       M | 1 1 1 1 1 1
``

[endsect][/ Intersection tester]


['*See also . . .*]
[table
[]
[[[link boost_icl.function_reference.symmetric_difference ['*Symmetric difference*]] ]]
[[[link boost_icl.function_reference.subtraction  ['*Subtraction*]]  ]]
[[[link boost_icl.function_reference.addition     ['*Addition*]]     ]]
]
['*Back to section . . .*]
[table
[]
[[[link function_synopsis_table ['*Function Synopsis*]]    ]]
[[[link boost_icl.interface ['*Interface*]]                ]]
]


[endsect][/ Intersection]



