For each question in this homework, you are tasked with implementing a function that:

exhibits the correct behavior, and

meets the required running time complexity.

You cannot receive full credit on a problem unless BOTH conditions are met.

Problem $1-$ remove$\_$characters.py

Write a function remove$\_$characters$($input$\_$string, to$\_$remove$)$ that takes two inputs:

input$\_$string : a string

to$\_$remove: a string

and returns a string that is exactly the same as the input$\_$string but without any of the characters in to$\_$remove. Your function must have a running time complexity of $O(n),$ where $n$ represents the length of input$\_$string.

For clarity, let $m$ be the length of to$\_$remove. Your code cannot have a running time complexity of $O(mn)$ to receive full credit, it must be $O(n).$

NOTE: you may not use any built$-$in string removal methods$/$functions in your code $($e$.$g$.,$ str$.$strip$()).$

Examples

These examples are intended to be illustrative, not exhaustive. Your code may have bugs even if it behaves as below. Write your own unittests to further test expected behaviors.

$$ new$\_$string $=$ remove$\_$characters$(\text{'}$abcd$\text{'},\text{'}$c$\text{'})$

$$ print $($new$\_$string$)$

$\text{'}$ abd $\text{'}$

Problem $2-$ any$\_$two$\_$sum.py

Write a function any$\_$two$\_$sum that takes two inputs:

numbers : a list of integers

total: an integer

and returns True if any two integers in numbers sum to total and False otherwise. Your function must have a running time complexity of $O(n),$ where $n$ represents the length of numbers.

Examples

These examples are intended to be illustrative, not exhaustive. Your code may have bugs even if it behaves as below. Write your own unittests to further test expected behaviors.

$$ result $=$ any$\_$two$\_$sum $([1,3,4,5],7)$

$$ print $($result$)$

True