Problem Description

Given a matrix grid which has dimensions m x n, our task is to count all the negative numbers within this grid. The matrix is sorted in a non-increasing (also known as non-decreasing) order across both rows and columns—this means the numbers go from larger on the top-left to smaller on the bottom-right.

To visualize, we might have a matrix like this:

4   3   2  -1
3   2   1  -1
1   1  -1  -2
-1 -1  -2  -3

In the above example, you can see that rows and columns are sorted in non-increasing order and there are 8 negative numbers.

The goal is to count how many negative numbers are present.

Intuition

Since the grid is sorted in non-increasing order row-wise and column-wise, we can use a more efficient approach than checking each cell individually.

Here's the intuition step by step:

1. We start at the bottom-left corner of the matrix (grid[m - 1][0]). The reason for doing this is that if this number is negative, then all numbers to its right are guaranteed to be negative because the row is sorted in non-increasing order. On the other hand, if this number is not negative, we can move one column to the right.

2. We keep track of our current position with indices i for rows and j for columns. We set ans to 0 to keep track of our count of negative numbers.

3. We use a while loop that runs as long as i is within the rows and j is within the columns of the grid.

4. If the current element where we are (grid[i][j]) is negative, we add the count of remaining columns in that row to our answer (i.e., ans += n - j) because all of these will be negative. After doing that, we move one row up (i -= 1) since we're done with the current row.

5. If the element is not negative, we move one column to the right (j += 1) to check the next number in the same row.

6. The process is repeated until we've gone through all rows or columns.

7. We return the count ans which contains the number of negative numbers in the grid at the end of our loop.

The solution leverages the sorted property of the matrix to skip over non-negative elements and only count consecutive negatives, which makes it efficient.

Learn more about Binary Search patterns.

Solution Approach

The solution uses a simple algorithm that takes advantage of the properties of the matrix, namely that it is sorted in non-increasing order in both rows and columns. No additional data structures are needed; we can operate directly on the given grid.

Here is a step-by-step walkthrough of the implementation:

1. Initiate two pointers, i for rows, starting from the last row (m - 1), and j for columns, starting from the first column (0).

2. Initiate a variable ans to store the count of negative numbers, initially set to 0.

3. Enter a while loop with the condition that i is greater than or equal to 0 (to make sure we are within the bounds of the rows) and j is less than n (to make sure we are within the bounds of the columns).

4. In the loop, check if the current element grid[i][j] is negative. If it is, increment ans by the number of columns remaining in the current row (n - j). This is because, since the rows are non-increasing, all elements to the right of grid[i][j] will also be negative. Then, decrease the row pointer i by 1, to move up to the previous row.

5. If grid[i][j] is not negative, just move to the next column by incrementing j by 1.

6. The loop continues until one of the exit conditions of the while loop is met (either all rows have been checked or the end of a row has been reached).

7. Finally, return the ans variable, which now contains the count of negative numbers in the grid.

The time complexity of this algorithm is O(m + n), where m is the number of rows and n is the number of columns. This is because at each step of the algorithm, we're either moving one row up or one column to the right until we exit the matrix bounds. In the worst-case scenario, we'll make m+n moves.

The space complexity is O(1) as we are not using any additional space proportional to the size of the input grid; instead, we're just using a few extra variables for counting and indexing.

No complex patterns are used here; the implementation is straightforward once the insight about the grid's sorted property is understood.

Example Walkthrough

Let's consider a smaller example to clearly illustrate the use of the solution approach in a concrete situation. Suppose we have the following grid matrix with dimensions 3 x 4:

3   2   1  -1
2   1  -1  -2
1  -1  -2  -3

We want to count the number of negative numbers in this grid. Here's how we can apply the solution approach:

1. We initialize our pointers i with the index of the last row which is 2 for this grid (since we are using 0-based indexing), and j with the index of the first column which is 0.

2. We set ans to 0 as our count of negative numbers.

3. We begin our while loop. Our initial element to check is grid[2][0], which is 1. Since it's not negative, we move one column to the right by incrementing j to 1.

4. Now, grid[2][1] is -1, which is negative. We find that there are 4 - 1 = 3 columns remaining in the row, including the current column. So, we add 3 to ans, making it 3, and move up a row by decrementing i to 1.

5. Our new position is grid[1][1], which is 1. It's not negative, so we move one column to the right again by incrementing j to 2.

6. We check grid[1][2], it's -1. There are 4 - 2 = 2 columns remaining, so we add 2 to ans, which becomes 5. We then move up a row by decrementing i to 0.

7. The element grid[0][2] is 1, not negative. We increment j to 3.

8. At grid[0][3], we have -1. With only 1 column at this last index, we add 1 to ans, resulting in a final count of 6. Since there are no more rows above, the loop ends here as i is decremented and falls below 0.

Thus, the grid contains 6 negative numbers.

With this example, it is clear how the algorithm smartly traverses the matrix, leveraging the sorted property to count negative numbers efficiently, without needing to check each element.

Solution Implementation

Time and Space Complexity

The provided code implements an algorithm to count the number of negative numbers in an m x n grid, which is assumed to be non-increasing both row-wise and column-wise. The algorithm starts from the bottom-left corner of the grid and moves upwards or to the right depending on the sign of the current cell value.

Time Complexity:

The time complexity of the code can be analyzed as follows:

• The outer loop runs a maximum of m times if we're moving upwards from the bottom row to the topmost row in the worst case.
• The inner condition can lead to moving rightwards across up to n columns.

However, you will never revisit a row or a column once you've moved on from it (since you move up if you find a negative and move right if you find a non-negative number), meaning each cell is visited at most once. Therefore, the maximum number of steps is m + n.

Thus, the time complexity of the algorithm is O(m + n).

Space Complexity:

The space complexity of the code is:

• Constant extra space is used for the variables m, n, i, j, and ans.
• No additional data structures are being used that grow with the size of the input.

Considering the above points, the overall space complexity of the algorithm is O(1), as it uses a constant amount of extra space regardless of the input size.

Learn more about how to find time and space complexity quickly using problem constraints.