You are given two sorted lists of size m and n. Implement a function to find the kth smallest element in the union of the two lists with linear complexity.

To find the k th smallest element in the union of two sorted lists with linear complexity , we can use a modified version of the merge sort algorithm . Here ' s how it works : \n \n 1 . Mer ge the two sorted lists into a single sorted list . \n 2 . Div ide the merged list into two hal ves . \n 3 . Rec urs ively find the k th smallest element in each half . \n 4 . If k is even , return the average of the two k th smallest elements . \n 5 . If k is odd , return the k th smallest element in the left half . \n \n Here ' s the Python code for this algorithm : \n \n `` ` python \n def find _ k th _ small est ( m , n , k ): \n # Mer ge the two sorted lists into a single sorted list \n merged _ list = sorted ( list ( range ( 1 , m + 1 )) + list ( range ( 1 , n + 1 ))) \n \n # Div ide the merged list into two hal ves \n mid = len ( mer ged _ list ) // 2 \n left _ half = merged _ list [: mid ] \n right _ half = merged _ list [ mid : ] \n \n # Rec urs ively find the k th smallest element in each half \n left _ k th = find _ k th _ small est ( len ( left _ half ), len ( left _ half ), k ) \n right _ k th = find _ k th _ small est ( len ( right _ half ), len ( right _ half ), k ) \n \n # If k is even , return the average of the two k th smallest elements \n if k % 2 == 0 : \n return ( left _ k th + right _ k th ) / 2 \n \n # If k is odd , return the k th smallest element in the left half \n else : \n return left _ k th \n `` ` \n \n The time complexity of this algorithm is O ( m + n ), which is linear . </s>