Question

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Given a 2d array sorted in increasing order from left to right and top to bottom, what is the best way to search for a target number?
Search a sorted 2D matrix

A time efficient program to find an element in a two dimensional matrix, the rows and columns of which are increasing monotonically. (Rows and columns are increasing from top to bottom and from left to right).

I can only think of binary search, if the 2D array was sorted.

Answer 1

I posed this problem as homework last semester, and two students, which I had considered to be average, surprised me by coming up with a very elegant, straightforward, and (probably) optimal algorithm:

Find(k, tab, x, y)
  let m = tab[x][y]

  if k = m then return "Found"
  else if k > m then
    return Find(k, tab, x, y + 1)
  else
    return Find(k, tab, x - 1, y)

This algorithm eliminates either one line or one column at every call (note that it is tail recursive, and could be transformed into a loop, thereby avoiding the recursive calls). Thus, if your matrix is n*m, the algorithm performs in O(n+m). This solution is better than the dichotomic search spin off (which the solution I expected when handing out this problem).

EDIT : I fixed a typo (k became x in the recursive calls) and also, as Chris pointed out, this should initially be called with the “upper right” corner, that is Find(k, tab, n, 1), where n is the number of lines.