Containedness

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### Containedness

Containedness | intervals | interval sets | interval maps | element sets | element maps
--- | --- | --- | --- | --- | ---
`bool T::empty()const` |  | 1 | 1 | 1 | 1
`bool is_empty(const T&)` | 1 | 1 | 1 | 1 | 1
`bool contains(const T&, const P&)` `bool within(const P&, const T&)` | e i | e i S | e i S b p M | e s | b m

This group of functions refers to containedness which should be fundamental to containers. The function `contains` is overloaded. It covers different kinds of containedness: Containedness of elements, segments, and sub containers.

Containedness | O(...) | Description
--- | --- | ---
`bool T::empty()const` `bool is_empty(const T&)` | O(1) | Returns `true`, if the container is empty, `false` otherwise.
`bool contains(const T&, const P&)` `bool within(const P&, const T&)` | see below | Returns `true`, if `super` container contains object `sub`.
| where | n`= iterative_size(sub)`
|  | m`= iterative_size(super)`

    // overload tables for
    bool contains(const T& super, const P& sub)
    bool   within(const P& sub, const T& super)

    element containers:   interval containers:
    T\P| e b s m          T\P| e i b p S M
    --------+---          --------+-------
     s | 1   1             S | 1 1     1
     m | 1 1 1 1           M | 1 1 1 1 1 1

The overloads of `bool contains(const T& super, const P& sup)` cover various kinds of containedness. We can group them into a part (1) that checks if an element, a segment or a container of same kinds is contained in an element or interval container

    // (1) containedness of elements, segments or containers of same kind
    T\P| e b s m          T\P| e i b p S M
    ---+--------          ---+------------
     s | 1   1             S | 1 1     1
     m |   1   1           M |     1 1   1

and another part (2) that checks the containedness of key objects, which can be elements an intervals or a sets.

    // (2) containedness of key objects.
    T\P| e b s m          T\P| e i b p S M
    ---+--------          ---+------------
     s | 1   1             S | 1 1     1
     m | 1   1             M | 1 1     1

For type m = `icl::map`, a key element (m`::domain_type`) and an `std::set` (m`::set_type`) can be a key object.

For an interval map type M, a key element (M`::domain_type`), an interval (M`::interval_type`) and an interval set, can be key objects.

Complexity characteristics for function `bool contains(const T& super, const P& sub)const` are given by the next tables where

    n = iterative_size(super);
    m = iterative_size(sub); //if P is a container type

Table 1.19. Time Complexity for function contains on element containers

`bool contains(const T& super, const P& sub)` `bool within(const P& sub, const T& super)` | domain type | domain mapping type | std::set | icl::map
--- | --- | --- | --- | ---
`std::set` | O(log n) |  | O(m log n) |
`icl::map` | O(log n) | O(log n) | O(m log n) | O(m log n)

Table 1.20. Time Complexity for functions contains and within on interval containers

`bool contains(const T& super, const P& sub)` `bool within(const P& sub, const T& super)` |  | domain type | interval type | domain mapping type | interval mapping type | interval sets | interval maps
--- | --- | --- | --- | --- | --- | --- | ---
interval_sets | `interval_set` | O(log n) | O(log n) |  |  | O(m log n) |
| `separate_interval_set` `split_interval_set` | O(log n) | O(n) |  |  | O(m log n) |
interval_maps | `interval_map` | O(log n) | O(log n) | O(log n) | O(log n) | O(m log n) | O(m log n)
| `split_interval_map` | O(log n) | O(n) | O(log n) | O(n) | O(m log n) | O(m log n)

All overloads of containedness of containers in containers

    bool contains(const T& super, const P& sub)
    bool   within(const P& sub, const T& super)

are of loglinear time: O(m log n). If both containers have same iterative_sizes so that m = n we have the worst case ( O(n log n) ). There is an alternative implementation that has a linear complexity of O(n+m). The loglinear implementation has been chosen, because it can be faster, if the container argument is small. In this case the loglinear implementation approaches logarithmic behavior, whereas the linear implementation stays linear.

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