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Equivalences and Orderings

Synopsis
Less on Intervals
Lexicographical Ordering
Sequential Element Ordering
Distinct Equality

Equivalences and Orderings

intervals

interval
sets

interval
maps

element
sets

element
maps

Segment Ordering

bool operator == (const T&, const T&)

1

1

1

1

1

bool operator != (const T&, const T&)

1

1

1

1

1

bool operator < (const T&, const T&)

1

1

1

1

1

bool operator > (const T&, const T&)

1

1

1

1

1

bool operator <= (const T&, const T&)

1

1

1

1

1

bool operator >= (const T&, const T&)

1

1

1

1

1

Element Ordering

bool is_element_equal(const T&, const P&)

S

M

1

1

bool is_element_less(const T&, const P&)

S

M

1

1

bool is_element_greater(const T&, const P&)

S

M

1

1

Distinct Equality

bool is_distinct_equal(const T&, const P&)

M

1

Types

x < y

i

x begins before y or, for equal beginnings x ends before y

All common equality and compare operators are defined for all objects of the icl. For all icl containers equality and compare operators implement lexicographical equality and lexicographical comparison, that depends on the equality of template parameter Compare. This includes the less ordering on intervals, that can be perceived as the sequence of elements between their lower and upper bound. This generalized lexicogrphical comparison in intervals can also be specified this way:

x < y

:=

x begins before y or, for equal beginnings x ends before y.

The other operators can be deduced in the usual way

x > y

:=

y < x

x <= y

:=

!(y < x)

x >= y

:=

!(x < y)

x == y

:=

!(x < y) && !(y < x) induced equivalence

x != y

:=

!(x == y)

Equality and compare operators are defined for all icl objects but there are no overloads between different types.

Containers of different segmentation are different, even if their elements are the same:

split_interval_set<time> w1, w2; //Pseudocode
w1 = {[Mon       ..       Sun)}; //split_interval_set containing a week
w2 = {[Mon .. Fri)[Sat .. Sun)}; //Same week split in work and week end parts.
w1 == w2;                        //false: Different segmentation
is_element_equal(w1,w2);         //true:  Same elements contained  

Complexity is linear in the iterative_size of the shorter container to compare.

The Sequential Element Ordering abstracts from the way in which elements of interval containers are clustered into intervals: it's segmentation.

So these equality and compare operations can be applied within interval container types. The admissible type combinations are summarized in the next overload table.

// overload tables for
bool is_element_equal  (const T&, const P&)
bool is_element_less   (const T&, const P&)
bool is_element_greater(const T&, const P&)

element containers:     interval containers:  
T\P| s m                T\P| S1 S2 S3 M1 M3    
---+----                ---+---------------    
s  | 1                  S1 | 1  1  1    
m  |   1                S2 | 1  1  1    
                        S3 | 1  1  1
                        M1 |          1  1
                        M3 |          1  1

For element containers lexicographical equality and sequential element equality are identical.

The complexity of sequential element comparison functions is linear in the iterative_size of the larger container.

Distinct Equality is an equality predicate that is available for icl::maps and interval_maps. It yields true, if two maps are sequential element equal except for value pairs whose associated values are identity elements.

Complexity is linear in the iterative_size of the larger container to compare.

See also . . .

Semantics

Back to section . . .

Function Synopsis

Interface


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