Problem Description

The given LeetCode problem asks us to count the number of distinct "corner rectangles" in a 2D grid. A "corner rectangle" is defined by four '1's that are placed in the grid in such a way that they form the corners of an axis-aligned rectangle. Importantly, the interior of this rectangle can contain any mix of '0's and '1's; the only requirement is that the four corners are '1's. The grid, grid, is composed entirely of '0's and '1's, and its dimensions are m x n where m is the number of rows and n is the number of columns. The task is to return the total count of such distinct corner rectangles.

Intuition

To solve this problem, we leverage the fact that a rectangle is defined by its two opposite corners. Here, we iterate over all potential corner pairs in the grid and try to determine if these pairs can form the upper two or lower two corners of a rectangle. We do this by checking if we picked two '1's in the same row. For each pair of '1's in the row, we check to see if the same pair of columns also have '1's in another row, forming a rectangle. To efficiently track this, we employ a counting strategy that uses a dictionary to remember how many times we've seen each pair of '1's in the same row.

Whenever we encounter a pair of '1's in the same row, we check the dictionary to see if this pair has been seen before. If it has, then for each previous occurrence, there is a distinct rectangle. Therefore, the count of rectangles is incremented by the number of times the pair has been seen before. Afterwards, we increment the counter for that pair, indicating that we've found another potential set of top corners for future rectangles.

The algorithm iterates over all rows and, within each row, all pairs of '1's. By using a counter dictionary that keeps track of pairs of indices where '1's are found, it ensures that the process of finding rectangular corners is efficient, avoiding checking every possible rectangle explicitly, which would be computationally expensive especially for large grids.

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Solution Approach

The solution employs a simple yet efficient approach using a hash table to keep track of the count of '1' pairs encountered so far. Here's a step-by-step walkthrough of how the code works:

1. Initialize Variables:

   • Initialize a variable ans to 0, which will hold the final count of corner rectangles.
   • Create a Counter object named cnt from the Python collections module. This counter will map each pair of column indices (i, j) to the number of times that pair has been observed in grid with '1's.

2. Iterate Over the Grid:

   • The first for loop iterates through each row of the grid.

3. Find Pairs of '1's in the Same Row:

   • Within each row, use a nested for loop with enumerate to get both the index i and value c1 at that index.
   • If c1 is '1', it means we have a potential top/left corner of a rectangle.

4. Check for Potential Top/Right Corners:

   • Another nested for loop is used to check to the right of the current '1' for another '1' at index j, forming a pair of top corners (i, j) of a potential rectangle.

5. Count and Update Rectangles:

   • For each such pair (i, j), if they can form the top two corners of a rectangle, we increment ans by cnt[(i, j)] since each count represents another row where a rectangle with these top corners can be completed.
   • After counting the rectangles for (i, j), update cnt[(i, j)] by incrementing by 1 which signifies that we have found another pair of '1's that can form the top corners of potential future rectangles.

6. Return the Answer:

   • After all rows and valid pairs are processed, return ans as the total count of corner rectangles found in the grid.

This solution utilizes a smart counting approach to avoid checking each potential rectangle directly, which drastically reduces the time complexity. The main algorithmic techniques include iterating over array elements and using hash-based counters for an elegant and efficient solution. By checking only rows for pairs of '1's and then counting and updating the number of potential rectangles as more pairs are found, the solution effectively captures all corner rectangles without unnecessary computations.

Example Walkthrough

Let's walk through a small example to illustrate how the solution approach works. Consider the following grid which is a 4x3 matrix containing both 0's and 1's.

grid = [
    [1, 0, 1],
    [1, 1, 1],
    [1, 0, 0],
    [1, 0, 1]
]

1. Initialize Variables:

   • ans starts at 0.
   • cnt is an empty Counter object.

2. Iterate Over the Grid:

   • Start with the first row [1, 0, 1].

3. Find Pairs of '1's in the Same Row:

   • Identify two '1's in the row at index i = 0 and j = 2.

4. Check for Potential Top/Right Corners:

   • These are the potential top corners of multiple rectangles.

5. Count and Update Rectangles:

   • Since cnt[(0, 2)] is not in cnt, ans remains 0, but cnt[(0, 2)] becomes 1.

Repeat steps 2-5 for remaining rows:

• Second row [1, 1, 1] has '1's at indices 0, 1, and 2.
• Pairs (0, 1), (0, 2), and (1, 2) each have the potential to form corner rectangles.
• cnt[(0, 1)] becomes 1, cnt[(0, 2)] is incremented to 2 (since we already have one from the first row), and cnt[(1, 2)] becomes 1.

As we process this row, ans gets updated because cnt[(0, 2)] was already 1 (one rectangle can now be formed with the previous row as the bottom corners).

Now, ans is incremented by 1.

Continuing on to the third and fourth rows, you'll do the same pair checks and counts, but no updates to ans occur until the fourth row.

• In the fourth row [1, 0, 1], the same pair (0, 2) is found as in the first row, meaning that for every previous occurrence of this pair (which is two times so far), a rectangle can be completed, so ans gets incremented by 2.

Finally, after iterating through all rows, we conclude with:

• ans = 3 (total count of corner rectangles found in the grid).

This example illuminated how only pairs of '1's in the same rows are used to determine the possibility of forming rectangles, and with each row's pairs, we effectively keep a running count of potential rectangles without the need for checking all possible rectangles directly. The Counter efficiently handles this incrementation and checking for previously seen pairs.

Solution Implementation

Time and Space Complexity

Time Complexity

The given code's time complexity primarily comes from the nested loops that it uses to iterate over each pair of columns for each row.

• The first loop (for row in grid) goes through each row in the grid, which occurs m times where m is the number of rows.
• Inside this loop, the second loop (for i, c1 in enumerate(row)) iterates over each element in the row, which occurs n times where n is the number of columns.
• The innermost loop (for j in range(i + 1, n)) iterates from the current column i to the last column, with the average number of iterations being roughly n/2 since it's a triangular iteration overall.

Considering the iterations of the third nested loop, the number of column pairs (i, j) that will be considered for each row follows the pattern of a combination of selecting two out of n columns or nC2.

Therefore, the time complexity is O(m * nC2) or O(m * n * (n - 1) / 2), which simplifies to O(m * n^2).

Space Complexity

The space complexity is determined by the storage used by the cnt Counter, which keeps track of the frequency of each pair of columns that have been seen with '1' at both the ith and jth positions.

• In the worst case scenario, the cnt Counter would have an entry for every possible pair of columns. As there are n columns, the number of column pairs would be the same nC2 we calculated before, or n * (n - 1) / 2 pairs.
• Therefore, the space complexity for the cnt Counter is O(n^2).

Thus, the overall space complexity is O(n^2) since the Counter's size is the dominant term, and this space is additional to the input (grid) since the grid itself is not being modified.

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