Question

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Write an efficient algorithm that searches for a value in an n x m table (two-dimensional array). This table is sorted along the rows and columns — that is,

Table[i][j] ≤ Table[i][j + 1],
Table[i][j] ≤ Table[i + 1][j]

Related questions

Search a 2D Matrix.

Searching a 2D Sorted Matrix.

Count negative in a 2D Sorted Matrix.

Analysis

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This question is not to be confused with Search a 2D Matrix.

Solution One: Step-wise Linear Search. Standard solution.

Time = O(n). Worse case 2n steps.

Note that this is the best solution!

begin with the upper right corner, we traverse one step to the left or bottom. For example, the picture below shows the traversed path (the red line) when we search for 13. Essentially, each step we are able to eliminate either a row or a column.

Solution Two: Quad Partition 四分法.

Time = O(n¹.58). Analysis can be found in the original question post.

Basic idea is like binary search - get mid and divide. We can then discard ¼ of the matrix. For example: search for 6, we can discard the bottom right sub-matrix.

Solution Three: Diagonal-based binary partition. This is based on previous solution, but better.

Time = O(nlgn).

This is a good solution, but it would fail in a non-square matrix, so…

Code

step-wise linear search

bool stepWise(int mat[][N_MAX], int N, int target, 
              int &row, int &col) {
  if (target < mat[0][0] || target > mat[N-1][N-1]) return false;
  row = 0;
  col = N-1;
  while (row <= N-1 && col >= 0) {
    if (mat[row][col] < target) 
      row++;
    else if (mat[row][col] > target)
      col--;
    else
      return true;
  }
  return false;
}