Скачать или смотреть Counting squares in a Grid: A Simple Algorithm Explained

Discover an effective algorithm for counting `squares` in a grid filled with dots. Learn about input representation and step-by-step logic to implement the solution!
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Counting Squares in a Grid: A Simple Solution

When faced with the challenge of counting the number of squares in a grid filled with dots, the task may seem overwhelming, especially when you have gaps in the grid. In this guide, we'll break down the problem and provide you with a streamlined solution that incorporates a systematic approach using an algorithm that boasts O(n^3) complexity.

Understanding the Problem

Imagine you have a grid consisting of dots and gaps, represented with a 2D array where:

A dot (1) represents a filled position

A gap (0) represents an unfilled position

Your goal is to determine how many squares can be formed solely from filled positions. Each square must contain dots on each of its sides, meaning that any square must completely conform to the definitions of a square in its entirety, including its border.

Example Scenario

In a 5 x 6 grid, certain dots (1's) are missing at specific coordinates. You need to count how many squares exist, including any larger squares encapsulating smaller ones. For example, given the following input:

[[See Video to Reveal this Text or Code Snippet]]

The resulting count of squares (including the largest outer square) might come out to be 6.

Input Representation

To represent this problem efficiently, we use a 2D array of 1's and 0's. Here’s the breakdown:

1s: These represent the dots in the grid.

0s: These indicate gaps where dots are absent.

Algorithm for Counting Squares

The algorithm we're going to discuss will efficiently compute the number of squares using cumulative sums. We'll break it down in clear steps:

Step 1: Initialize Cumulative Arrays

Create two cumulative arrays: R (rows) and C (columns). Here’s what they will represent:

R[i][j]: This will store the number of 1's in the i-th row from the left up to the j-th index.

C[i][j]: This will store the number of 1's in the j-th column from the top up to the i-th index.

Step 2: Populate Cumulative Arrays

Loop through each position in the grid, populating the R and C arrays as follows:

[[See Video to Reveal this Text or Code Snippet]]

Step 3: Count Squares

Now, we’ll need to iterate through each non-zero position and check for possible square sizes:

[[See Video to Reveal this Text or Code Snippet]]

The function validateSquare checks whether all sides of a square of size k from top left position [i][j] is complete without gaps.

Step 4: The Complete Implementation

Here’s the complete code snippet:

[[See Video to Reveal this Text or Code Snippet]]

Now executing countSquares(grid) would provide the count of squares in the grid.

Conclusion

By following these steps, you can effectively count the number of squares within a grid while handling gaps in a clear and methodical manner. Performance-wise, the O(n^3) complexity is manageable for reasonably sized grids, ensuring that you make the most of efficient cumulative sum calculations to derive results.

With this approach, counting squares no longer feels daunting but instead a solvable puzzle where patterns emerge through structured reasoning.