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.