Question

Suppose you have two data sets, each of which contain n comparable elements. As an basic operation, you may ask one set to tell you the kth largest element of that set, for a value k you choose. Give an algorithm that, with O(log n) queries, determines the median value of the union of the two sets.

Answer 1

The Algorithm so something like this,

We can first search for m1 and M2 the median of the two list using the basic operation of getting a kth number (n/2,n+1/2th number in case of medium?

1) we can have the medians m1 and m2 of the input lists.
2) If m1 and m2 are the same, we can be consider it out of self from
                       return m1 (or m2)
3) If m1 is greater than m2, then median is present in one 
   of the below two subarrays.
    a)  second half of M1 (M1 to the highest in case of a sorted array but we can just work with median no need for calculating the sorted array)
    b) first half of M2 (starting go M2)
4) If m2 is greater than m1, then median is present in one which is just opposite of what we have said.
5) Repeat the above process until size of both the subarrays becomes 2 that is we reach the size of 2 so that only we have both the median terms.
6) If size of the two arrays is 2 then,
  the median.
    Median = (max(array1[0], array2[0]) + min(array1[1], array2[1]))/2

Now on a avrage case we have N --> N/2 ---> N/4 and so on till size is two that is a complexity = O(logN)

If there is any doubts put in comments, hope it help