Question

This is a homework question, binary search has already been introduced:

Given two arrays, respectively N and M elements in ascending order, not necessarily unique:
What is a time efficient algorithm to find the kth smallest element in the union of both arrays?

They say it takes O(logN + logM) where N and M are the arrays lengths.

Let’s name the arrays a and b. Obviously we can ignore all a[i] and b[i] where i > k.
First let’s compare a[k/2] and b[k/2]. Let b[k/2] > a[k/2]. Therefore we can discard also all b[i], where i > k/2.

Now we have all a[i], where i < k and all b[i], where i < k/2 to find the answer.

What is the next step?

Answer 1

You’ve got it, just keep going! And be careful with the indexes…

To simplify a bit I’ll assume that N and M are > k, so the complexity here is O(log k), which is O(log N + log M).

Pseudo-code:

i = k/2
j = k - i
step = k/4
while step > 0
    if a[i-1] > b[j-1]
        i -= step
        j += step
    else
        i += step
        j -= step
    step /= 2

if a[i-1] > b[j-1]
    return a[i-1]
else
    return b[j-1]

For the demonstration you can use the loop invariant i + j = k, but I won’t do all your homework 🙂