Containedness

Containedness	interval	interval sets	interval maps	itl::set	itl::map
`void T::clear()`	1	1	1	1	1
`bool T::empty()const`	1	1	1	1	1
`bool T::contains(const P&)const`	e i	e i S	e i S b p M	e s	b m
`bool T::contained_in(const P&)const`	i	S	M	1	1

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
`void T::clear()`	O(n)	All content is removed from the container.
`bool T::empty()const`	O(1)	Returns `true`, if the container is empty, `false` otherwise.
`bool T::contains(const P& sub)const`	see below	Returns `true`, if `*this` container contains object `sub`.
`bool T::contained_in(const P& super)const`	O(m log n)	Returns `true`, if `*this` is contained in object `super`.
	where	n `= this->iterative_size()`
		m `= super.iterative_size()`

// overload tables for 
bool T::contains(const P& src)

element containers:     interval containers:  
contains| e b s m       contains| 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 T::contains(const P& src) 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
contains| e b s m       contains| 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.
contains| e b s m       contains| e i b p S M    
--------+--------       --------+------------    
  s     | 1   1           S     | 1 1     1       
  m     | 1   1           M     | 1 1     1

For type m = itl::map, a key element (m::domain_type) and an itl::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 T::contains(const P& sub)const are given by the next tables where

n = this->iterative_size();
m = sub.iterative_size(); //if P is a container type

Table 1.16. Time Complexity for function contains on element containers

`bool T::contains(const P& sub)const`	domain type	domain mapping type	interval sets	interval maps
`itl::set`	O(log n)		O(m log n)
`itl::map`	O(log n)	O(log n)	O(m log n)	O(m log n)

Table 1.17. Time Complexity for function contains on interval containers

`bool T::contains(const P& sub)const`		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 tests of containedness of containers in containers

bool T::contains(const P& sub_container)const
bool T::contained_in(const P& super_container)const

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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Function Synopsis
Interface