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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?

The following was asked in a Google interview:

You are given a 2D array storing integers, sorted vertically and horizontally.

Write a method that takes as input an integer and outputs a bool saying whether or not the integer is in the array.

What is the best way to do this? And what is its time complexity?

Answer 1

I would start by asking details about what it means to be “sorted vertically and horizontally”

If the matrix is sorted in a way that the last element of each row is less than the first element of the next row, you can run a binary search on the first column to find out in what row that number is, and then run another binary search on the row. This algorithm will take O(log C + log R) time, where C and R are, respectively the number of rows and columns. Using a property of the logarithm, one can write that as O(log(C*R)), which is the same as O(log N), if N is the number of elements in the array. This is almost the same as treating the array as 1D and running a binary search on it.

But the matrix could be sorted in a way that the last element of each row is not less than the first element of the next row:

1 2 3 4 5 6 7 8  9
2 3 4 5 6 7 8 9  10
3 4 5 6 7 8 9 10 11

In this case, you could run some sort of horizontal an vertical binary search simultaneously:

1. Test the middle number of the first column. If it’s less than the target, consider the lines above it. If it’s greater, consider those below;
2. Test the middle number of the first considered line. If it’s less, consider the columns left of it. If it’s greater, consider those to the right;
3. Lathe, rinse, repeat until you find one, or you’re left with no more elements to consider;

This method is also logarithmic on the number of elements.