Subtraction

Synopsis

Subtraction	intervals	interval sets	interval maps	element sets	element maps
`T& T::subtract(const P&)`		e i	b p		b
`T& subtract(T&, const P&)`		e i	b p	e	b
`T& operator -=(T&, const P&)`		e i S	e i S b p M	e s	b m
`T operator - (T, const P&)`		e i S	e i S b p M	e s	b m
`T left_subtract(T, const T&)`	1
`T right_subtract(T, const T&)`	1

Functions and operators that implement Subtraction on icl objects are given in the table above.

	Description of Subtraction
`Sets`	Subtraction on Sets implements set difference
`Maps`	Subtraction on Maps implements a map difference function similar to set difference. If, on subtraction of an element value pair `(k,v)` it's key `k` is in the map already, the subtraction function is propagated to the associated value. On the associated value an aggregation is performed, that reverses the effect of the corresponding addition function. Find more on subtractability of maps and related semantic issues following the links.

Description of Subtraction

Sets

Subtraction on Sets implements set difference

Maps

Subtraction on Maps implements a map difference function similar to set difference. If, on subtraction of an element value pair (k,v) it's key k is in the map already, the subtraction function is propagated to the associated value. On the associated value an aggregation is performed, that reverses the effect of the corresponding addition function.

Find more on subtractability of maps and related semantic issues following the links.

Functions

The admissible combinations of types for subtraction functions can be summarized in the overload table below:

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

The next table contains complexity characteristics for subtract.

Table 1.24. Time Complexity for function subtract on icl containers

`T& T::subtract(const P&)` `T& subtract(T&, const P&)`	domain type	interval type	domain mapping type	interval mapping type
`std::set`	O(log n)
`icl::map`	O(log n)		O(log n)
`interval_sets`	O(log n)	amortized O(log n)
`interval_maps`	O(log n)	O(n)	O(log n)	O(n)

Inplace operators

As presented in the overload tables for operator -= more type combinations are provided for subtraction than for addition.

// 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

Subtraction provides the reverse operation of an addition for these overloads,

// Reverse addition                -= | 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 you can erase parts of icl::maps or interval_maps using key values, intervals or element or interval sets using these overloads:

// Erasure 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

On Sets both function groups fall together as set difference.

Complexity characteristics for inplace subtraction operations are given by the next tables where

n = iterative_size(y);
m = iterative_size(x); //if P is a container type

Table 1.25. Time Complexity for inplace Subtraction on element containers

`T& operator -= (T&, const P&)`	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 1.26. Time Complexity for inplace Subtraction on interval containers

`T& operator -= (T&, const P&)`	domain type	interval type	domain mapping type	interval mapping type	interval sets	interval maps
interval_sets	O(log n)	amortized O(log n)			O(m log(n+m))
interval_maps	O(log n)	amortized O(log n)	O(log n)	O(n)	O(m log(n+m))	O(m log(n+m))

Infix operators

The admissible overloads for the infix subtraction operator - which is a non commutative operation is given by the next overload table.

// overload tables for         -  | e b s m      -  | e i b p S M   
T operator - (T, const P&)     ---+--------      ---+------------   
                               s  | s   s        S  | S S     S     
                               m  | m m m m      M  | M M M M M M

Subtraction on Intervals

Subtraction	Types	Description
`T left_subtract(T right, const T& left_minuend)`	i	subtract `left_minuend` from the interval `right` on it's left side. right_over = left_subtract(right, left_minuend); ... d) : right ... c) : left_minuend [c d) : right_over
`T right_subtract(T left, const T& right_minuend)`	i	subtract `right_minuend` from the interval `left` on it's right side. left_over = right_subtract(left, right_minuend); [a ... : left [b ... : right_minuend [a b) : left_over

Subtraction

Types

Description

T left_subtract(T right, const T& left_minuend)

i

subtract left_minuend from the interval right on it's left side.

right_over = left_subtract(right, left_minuend);
...      d) : right
... c)      : left_minuend
     [c  d) : right_over

T right_subtract(T left, const T& right_minuend)

i

subtract right_minuend from the interval left on it's right side.

left_over = right_subtract(left, right_minuend);
[a      ...  : left
     [b ...  : right_minuend
[a  b)       : left_over