How can I find a number which occurs an odd number of times in a SORTED array in O(n) time?

Question

I have a question and I tried to think over it again and again... but got nothing so posting the question here. Maybe I could get some view-point of others, to try and make it w

Answer 1

We don't have any information about the distribution of lenghts inside the array, and of the array as a whole, right?

So the arraylength might be 1, 11, 101, 1001 or something, 1 at least with no upper bound, and must contain at least 1 type of elements ('number') up to (length-1)/2 + 1 elements, for total sizes of 1, 11, 101: 1, 1 to 6, 1 to 51 elements and so on.

Shall we assume every possible size of equal probability? This would lead to a middle length of subarrays of size/4, wouldn't it?

An array of size 5 could be divided into 1, 2 or 3 sublists.

What seems to be obvious is not that obvious, if we go into details.

An array of size 5 can be 'divided' into one sublist in just one way, with arguable right to call it 'dividing'. It's just a list of 5 elements (aaaaa). To avoid confusion let's assume the elements inside the list to be ordered characters, not numbers (a,b,c, ...).

Divided into two sublist, they might be (1, 4), (2, 3), (3, 2), (4, 1). (abbbb, aabbb, aaabb, aaaab).

Now let's look back at the claim made before: Shall the 'division' (5) be assumed the same probability as those 4 divisions into 2 sublists? Or shall we mix them together, and assume every partition as evenly probable, (1/5)?

Or can we calculate the solution without knowing the probability of the length of the sublists?