can you provide the link of this problem ?

my approach: first, absolute values are hard to work with so let's remove them by calculating the max value of MEX(P,a,b) - MEX(Q,a,b) over all a,b for a query (and similarly MEX(Q,a,b) - MEX(P,a,b)) and answer is the max of these 2

now to maximize MEX(P,a,b) - MEX(Q,a,b), let's think of a naive solution first: let's loop over all values X which MEX(P,a,b) can be, and try to minimize MEX(Q,a,b). Well observe that as you extend a subarray by 1 ([a,b] -> [a-1,b] or [a,b+1]) the mex can only increase, so we want to find the smallest subarray where MEX(P,a,b) equals X

well this can be done with a simple loop:

first calculate index:

for(int i = 0; i < n; i++)
  a_index[a[i]] = i;

then something like:

int l = n, r = -1;
for(int i = 0; i < n; i++) {
  l = min(l, a_index[i]);
  r = max(r, a_index[i]);
}

finally we can observe that these "minimal-mex" subarrays in a can only increase in length, so for a query, we can binary search the largest mex for which that corresponding "minimal-length" subarray is contained in the query. And also store MEX(P,a,b) - MEX(Q,a,b) in an array, prefix-maxed