Binary search in 2 sorted integer arrays

This problem is impossible to solve in guaranteed time complexity faster than O(n) and not possible to solve at all for certain arrays. Binary search runs in O(log n) for a sorted array, but the big array is not guaranteed to be sorted and will in the worst-case require one or more comparisions per element, which is O(n). The best guaranteed time complexity is O(n) with the trivial algorithm: compare every item with its neighbour until you find the "turning point" with A[i] > A[i+1]. However, if you use a breadth-first search, you may get lucky and find the "turning point" early.

Proof that the problem is unsolvable for some arrays: let the array M = [A B] be our big array. To find the point where the arrays meet we're looking for an index i where M[i] > M[i+1]. Now let A=[1 2 3] and B=[4 5]. There is no index in the array M for which the condition holds true, thus the problem is unsolvable for some arrays.

Informal proof for the former: let M=[A B] and A=[1..x] and B=[(x+1)..y] be two sorted arrays. Then swap the positions of element x and y in M. We have no way of finding the index of x without (in the worst case) checking every index, thus the problem is O(n).

Binary search relies on being able to eliminate half the solution space with each comparision, but in this case we cannot eliminate anything from the array and so we cannot do better than a linear search.

(From a practical standpoint, you should never do this in a program. The two arrays should be separate. If this isn't possible, append the length of either array to the bigger array.)

Edit: changed my answer after question was updated. It's possible to do it faster than linear time for some arrays, but not all possible arrays. Here's my idea for an algorithm using breadth-first search:

Start with the interval [0..n-1] where n is the length of the big array.
Make a list of intervals and put the starting interval in it.
For each interval in the list:
    if the interval is only two elements and the first element is greater than the last
        we found the turning point, return it
    else if the interval is two elements or less
        remove it from the list
    else if the first element of the interval is greater than the last
        turning point is in this interval
        clear the list
        split this interval in two equal parts and add them to the list
    else
        split this interval in two equal parts and replace this interval in the list with the two parts

I think a breadth-first approach will increase the odds of finding an interval where A[first] > A[last] early. Note that this approach will not work if the turning point is between two intervals, but it's something to get you started. I would test this myself, but unfortunately I don't have the time now.