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1. (16 points) Merge is an important subroutine for mergesort. It is an algorithm that takes two sorted lists and outputs a sorted list of all the elements in the two lists. $MERGE([a_{1},…,a_{k}],[b_{1},…,b_{ℓ}])$ 1. $i=1$ 2. $j=1$ 3. output $=[]$ (empty list) 4. while $i⩽k$ and $j⩽ℓ$ : 5. if $a_{i}⩽b_{j}$ : 6. output $=$ output $∘[a_{i}]$ 7. $i=i+1$ 8. else: 9. output $=$ output $∘[b_{j}]$ 10. $j=j+1$ 11. if $i==k+1$ : 12. return output $∘[b_{j+1},…,b_{ℓ}]$ 13. else: 14. return output $∘[a_{i+1},…,a_{k}]$ (a) (5 points) In order to prove the correctness of this algorithm, prove the following loop invariant: After $t$ iterations, output is sorted (b) (5 points) In order to analyze the runtime, prove the following loop invariant: After $t$ iterations, there are exactly $t$ elements in output (c) (3 points) Using the fact that output will have exactly $k+ℓ$ elements and the loop invariant above, give a justification of why the algorithm is correct. (d) (3 points) Using the fact that output will have exactly $k+ℓ$ elements and the loop invariant above, give a brief justification of why the runtime of this algorithm is $O(k+ℓ)$

The question is about Merge subroutine used in the MergeSort algorithm. Merge takes two sorted lists as input and merges them into a single sorted list

By addressing each part of the question, you will provide a thorough explanition of the correctness and runtime analysis of the Merge algorithm used in mergesort.

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