How does a hash table (or B-tree) determine if the two arrays are the
same?
A Hash table is hardly O(n) unless you do not hash the strings.

After designing these two structures (for each array) you still have to
compare the two to see if they are equal (which if it turns out they
are _always_ takes O(n) just for the compare)

My suggestion would be to sort the two arrays and compare item per
item. You must know that strings are the worst things to sort as they
take much longer than numeric values, but you seem to be confined to
that.

You could try to iterate over each element of the arrays
hashing/adding/manipulating each element along the way and compare the
two results for equality, this leaves out the sorting part but puts on
your shoulders the responsibility of ensuring that duplicate values are
as rare as possible and that the operation at each index of the array
is not too taxing (i.e. < O(logn)) or you defeate the purpose of
avoiding the sort in the first place.


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