Input: Given an array A of integers, and a target integer x

$.$

Process: Write an algorithm two

$\_$

sum

$($

A

$,$

$$x

$)$

$,$

$$which traverses A to find the indices i and j of

two numbers st

$.$

$($

i

$$

$=$

$$j

$)($

A

$[$

i

$]$

$+$

$$A

$[$

j

$]$

$=$

$$x

$)$

Meaning that you need to return the indices of two unique elements in A that add up to x

$.$

Remember, these two elements have to be unique in the sense that they have different indices,

yet they can be the same numbers; for ex:

two

$\_$

sum

$([3$

$,$

$2$

$,$

$1$

$,$

$5]$

$,$

$7)$

$=$

$(1$

$,$

$3)$

two

$\_$

sum

$([3$

$,$

$2$

$,$

$8$

$,$

$3]$

$,$

$6)$

$=$

$(0$

$,$

$3)$

two

$\_$

sum

$([3$

$,$

$2$

$,$

$8]$

$,$

$6)$

$=$

$(0$

$,$

$0)$

Output: A tuple of the form

$($

i

$,$

$$j

$)$

$.$

Note: You may assume that there will always be exactly two numbers in A that will add up to

x

$.$

$$Your algorithm can take any number of operations to compute this, i

$.$

e

$.$

$$any time complexity will be accepted. Derive an O

$($

n log n

$)$

$$solution for this question which includes the algorithms, appropriate proofs of correctness, time

$-$

space

complexity analysis. code the whole soluting in sml coding language.