Tiling Problem

Summary

The Tiling Problem is a classic algorithmic challenge that involves finding the minimum number of squares, rectangles, or other shapes needed to cover a given rectangular area. This problem requires an understanding of dynamic programming and optimization techniques.

Detailed Explanation

The problem can be broken down into smaller sub-problems by considering the smallest possible square tiles that can cover the entire rectangular area. The key insight is that any rectangle can be covered by the minimum number of squares if the smallest possible square tile (with side length 1) is used to cover the top-left corner, followed by the next smallest possible square tile (with side length 2), and so on.

To solve this problem, we can create a dynamic programming table where each cell represents the minimum number of tiles needed to cover a rectangle with a given width and height. The base case is when the width or height is 0, in which case no tiles are needed. For larger rectangles, we need to consider the smallest possible square tile that can cover the top-left corner and recursively calculate the minimum number of tiles needed for the remaining area.

Here's a step-by-step breakdown of the solution:

Initialize a dynamic programming table dp with dimensions (width + 1) x (height + 1) filled with infinity values.
Set the base case values in dp: dp[0][i] = dp[i][0] = 0, where i is the index representing the width or height.
For each cell dp[i][j], consider the smallest possible square tile that can cover the top-left corner with side length min(i, j).
Calculate the minimum number of tiles needed to cover the remaining area by recursively calling the dynamic programming function for the sub-problems (i - min(i, j)), (j - min(i, j)), and add 1 to account for the current tile.
Update the value in dp[i][j] with the minimum number of tiles needed to cover the entire rectangle.

Time complexity: O(width * height) Space complexity: O(width + height)

Optimized Solutions

Java

View GeeksforGeeks Problem

public class TilingProblem {
    public static int minTiling(int width, int height) {
        int[][] dp = new int[width + 1][height + 1];
        for (int i = 0; i <= width; i++) {
            for (int j = 0; j <= height; j++) {
                if (i == 0 || j == 0) {
                    dp[i][j] = 0;
                } else {
                    int minTiles = Integer.MAX_VALUE;
                    for (int k = 1; k <= Math.min(i, j); k++) {
                        minTiles = Math.min(minTiles, 1 + dp[i - k][j - k]);
                    }
                    dp[i][j] = minTiles;
                }
            }
        }
        return dp[width][height];
    }
}

Code Editor (Testing phase)

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893411. Tiling Problem