Time complexity for merging two sorted arrays of size n and m

Question

I was just wondering what is the time complexty of merging two sorted arrays of size n and m, given that n is always greater than m.

I was thinking

Answer 1

Attention! This answer contains an error

A more efficient algorithm exists and it is presented in another answer.

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The complexity is O(m log n).

Let the long array be called a and the short array be b then the algorithm you described can be written as

  for each x in b
      insert x into a

There are m iterations of the loop. Each insertion into a sorted array is an O(log n) operation. Therefore the overall complexity is O (m log n).

Since b is sorted the above algorithm can be made more efficient

  for q from 1 to m
      if q == 1 then insert b[q] into a
      else 
         insert b[q] into a starting from the position of b[q-1]

Can this give better asymptotic complexity? Not really.

Suppose elements from b are evenly spread along a. Then each insertion will take O(log (n/m)) and the overall complexity will be O(m log(n/m) ). If there exists a constant k>1 that does not depend on n or m such that n > k * m then O(log(n/m)) = O(log(n)) and we get the same asymptotic complexity as above.