Interview Question: How to efficiently search in an ordered matrix?

Question

I have a x by y matrix, where each row and each column are in ascending order as given below.

1   5   7    9
4   6   10   15
8   11  12   19
14  16  18   21

How to search this matrix for a number in O(x+y)?

I was asked this question for an interview, but could not figure out the way. Curious to know if it could be done.

Answer 1

Start at the last element of the first row(top-right corner).
Compare it with the key. We have 3 cases:

• If they are equal we are done.

• If key is greater than that element then it means key cannot be present in that row so move the search to the element below it.

• If key is less than that element then it means key could be present in that row towards left and cannot be present in the column further down, so move the search to the element left of it.

Keep doing it till you find the element or you cannot further move(key does not exist).

Pseudo code:

Let R be number of rows
Let C be number of columns

Let i = 0
Let j = C-1

found = 0
while( i>=0 && i<R) && (j>=0 && j<C) )
   if (matrix[i][j] == key )
      found = 1;
      break
   else if( matrix[i][j] > key )
       j--
   else if( matrix[i][j] < key )
       i++
end-while

Answer 2

Split the Matrix in 4 submatrices. If bottom right of a sub-matrix is less than key, discard it. If the top left of a sub-matrix is bigger than the key, discard it. Repeat the splitting procedure for the remaining sub-matrices.

[Update] For some pseudo code (and a discussion of complexity) see Jeffrey L Whitledge's answer of this question.