interval$\_$map is a data structure that associates keys of type K with values of type V$.$ It is designed to be used efficiently in situations where intervals of consecutive keys are associated with the same value. Your task is to implement the assign member function of this data structure, which is outlined below.

interval$\_$map is implemented on top of std::map. For more information on std::map, you may refer to cppreference.com.

Each key$-$value$-$pair $($k$,$v$)$ in interval$\_$map::m$\_$map means that the value v is associated with all keys from k $($including$)$ to the next key $($excluding$)$ in m$\_$map. The member interval$\_$map::m$\_$valBegin holds the value that is associated with all keys less than the first key in m$\_$map.

Example: Let M be an instance of interval$\_$map where

M$.$m$\_$valBegin$==\text{'}$A$\text{'},$

M$.$m$\_$map$==\{(1,\text{'}$B$\text{'}),(3,\text{'}$A$\text{'})\},$

Then M represents the mapping

$...$

$-2->\text{'}$A$\text{'}$

$-1->\text{'}$A$\text{'}$

$0->\text{'}$A$\text{'}$

$1->\text{'}$B$\text{'}$

$2->\text{'}$B$\text{'}$

$3->\text{'}$A$\text{'}$

$4->\text{'}$A$\text{'}$

$5->\text{'}$A$\text{'}$

$...$

The representation in the std::map must be canonical, that is$,$ consecutive map entries must not contain the same value: $...,(3,\text{'}$A$\text{'}),(5,\text{'}$A$\text{'}),...$ is not allowed. Likewise, the first entry in m$\_$map must not contain the same value as m$\_$valBegin. Initially, the whole range of K is associated with a given initial value, passed to the constructor of the interval$\_$map data structure.

Key type K

$$ besides being copyable and assignable, is less$-$than comparable via operator$<,$ and

$$ does not implement any other operations, in particular no equality comparison or arithmetic operators.

Value type V

$$besides being copyable and assignable, is equality$-$comparable via operator$==,$ and

$$does not implement any other operations.

Here is what I have so far!!!

$```$

#include

template

class interval$\_$map $\{$

friend void IntervalMapTest$()$;

V m$\_$valBegin;

std::map m$\_$map;

public:

$//$ constructor associates whole range of K with val

interval$\_$map$($V const& val$)$

: m$\_$valBegin$($val$)$

$\{\}$

$//$ Assign value val to interval $[$keyBegin$,$ keyEnd$).$

$//$ Overwrite previous values in this interval.

$//$ Conforming to the C$++$ Standard Library conventions, the interval

$//$ includes keyBegin, but excludes keyEnd.

$//$ If $!($ keyBegin $<$ keyEnd $),$ this designates an empty interval,

$//$ and assign must do nothing.

void assign$($ K const& keyBegin, K const& keyEnd, V const& val $)\{$

if $(!($keyBegin $<$ keyEnd$))\{$

return;

$\}$

auto itLow $=$ m$\_$map.lower$\_$bound$($keyBegin$)$;

if $($itLow $!=$ m$\_$map.begin$())\{$

auto itPrev $=$ prev$($itLow$)$;

if $($itPrev$->$second $==$ val$)\{$

itLow $=$ m$\_$map.erase$($itLow$)$;

$\}$

$\}$

auto itHigh $=$ m$\_$map.lower$\_$bound$($keyEnd$)$;

if $($itLow $!=$ itHigh$)\{$

m$\_$map.erase$($itLow$,$ itHigh$)$;

$\}$

m$\_$map$[$keyEnd$]=$ itHigh $!=$ m$\_$map.end$()?$ itHigh$->$second : m$\_$valBegin;

if $($m$\_$valBegin $!=$ val $||$ keyBegin $!=$ m$\_$map.begin$()->$first$)\{$

m$\_$map$[$keyBegin$]=$ val;

$\}$

if $($keyBegin $==$ m$\_$map.begin$()->$first && m$\_$valBegin $==$ val$)\{$

m$\_$map.erase$($m$\_$map.begin$())$;

$\}$

$\}$

$//$ look$-$up of the value associated with key

V const& operator$[]($ K const& key $)$ const $\{$

auto it$=$m$\_$map.upper$\_$bound$($key$)$;

if$($it$==$m$\_$map.begin$())\{$

return m$\_$valBegin;

$\}$ else $\{$

return $(--$it$)->$second;

$\}$

$\}$

$\}$;

int main$()\{$

interval$\_$map M$(\text{'}$A$\text{'})$;

M$.$assign$(1,4,\text{'}$B$\text{'})$;

M$.$assign$(8,9,\text{'}$C$\text{'})$;

for $($int i $=-2$; i $<9$; $++$i$)\{$

cout $<<$ M$[$i$]<<$ endl;

$\}$

return $0$;

$\}$

$//$ Many solutions we receive are incorrect. Consider using a randomized test

$//$ to discover the cases that your implementation does not handle correctly.

$//$ We recommend to implement a test function that tests the functionality of

$//$ the interval$\_$map, for example using a map of int intervals to char.

$```$

SEEMS RIGHT BUT I IMPLEMENTED IT WRONG! $->$

Unfortunately, this program failed to meet the criterion marked in $***$:

$***$ Type requirements are met: You must adhere to the specification of the key and value type given above.$***$ THIS FAILED, PLEASE HELP WITH THIS

Correctness: Your program should produce a working interval$\_$map with the behavior described above. In particular, pay attention to the validity of iterators. It is illegal to dereference end iterators. Consider using a checking STL implementation such as the one shipped with Visual C$++$ or GCC$.$

Canonicity: The representation in m$\_$map must be canonical.

Running time: Imagine your implementation is part of a library, so it should be big$-$O optimal. In addition:

Do not make big$-$O more operations on K and V than necessary because you do not know how fast operations on K$/$V are; remember that constructions, destructions and assignments are operations as well.

Do not make more than one operation of amortized O$($log N$),$ in contrast to O$(1),$ running time, where N is the number of elements in m$\_$map.

Otherwise favor simplicity over minor speed improvements.