"Cover ""Manhattan Skyline"" Using The Minimum Number of Rectangles"

The "Stone Wall" problem belongs to the conventional computational issues, the goal of which is to estimate the number of cuboid blocks needed to erect a wall of diverse heights. The wall is described by an array H, the components of which indicate the height of the wall at various segments. The objective is to use an algorithm to minimize the number of blocks required to cover the wall without violating height regulations.

Problem Overview

If there is an array H having N positive variables, where variable H[] of length N signifies the height of the wall between meter i to meter i+1. The wall has to be clad with cuboid blocks, each of which has to have uniform thickness and planar rectangular faces. The question arises of how many such blocks would have to be added in order to paint all the wall's unbroken spans, given that the heights change along the length of the wall.

Solution Approach

To solve this problem, we maintain a stack to store the heights of the blocks which will be termed as 'block stack' while going ahead with the wall's height[] array. The stack maintains the current heights of the blocks and facilitates knowing when new blocks are required or when an existing block can be modified.

The core idea is to:

Compare every height in the array.

Take a block, stack it onto another, and use a merge to manage its blocks based on height modification.

When the height changes count the number of new blocks needed to be added out.

 
import java.util.Stack;
 class StoneWall {
    public static void main(String[] args) {
        // Create an instance of the StoneWall class
        StoneWall wallBuilder = new StoneWall();       
        int[] heights = {1, 2, 1, 2, 1};       
        // Call the minimumBlocks method and print the result
        int result = wallBuilder.minimumBlocks(heights);
        System.out.println("Minimum number of cuboid blocks: " + result);
    }
    public int minimumBlocks(int[] heights) {
        // Stack to keep track of the heights of cuboid blocks
        Stack<Integer> heightStack = new Stack<>();
        // Variable to count the number of additional blocks required
        int blockCount = 0;        
        // Iterate through each height in the array
        for (int currentHeight : heights) {
            // Process the stack while the current height is different from the top height in the stack
            while (!heightStack.isEmpty()) {
                int topHeight = heightStack.pop();               
                if (topHeight == currentHeight) {
                    // If the current height is the same as the top height, push it back to the stack
                    heightStack.push(topHeight);
                    break;
                } else if (currentHeight > topHeight) {
                    // If the current height is greater, push the previous height back and add the new height
                    heightStack.push(topHeight);
                    heightStack.push(currentHeight);
                    break;
                } else {
                    // If the current height is lower, increment the block count
                    blockCount++;
                }
            }           
            // If the stack is empty, push the new height into the stack
            if (heightStack.isEmpty()) {
                heightStack.push(currentHeight);
            }
        }        
        // The total number of blocks is the sum of additional blocks and the heights remaining in the stack
        return blockCount + heightStack.size();
    }
}   

Output:

 
Minimum number of cuboid blocks: 3

Time Complexity: The solution has a time complexity of O(N), where N is the length of the height array. It is due to the single pass through the array and efficient stack operations.

Space Complexity: The space complexity is O(N) in the worst case, as the stack may grow to accommodate all heights if there are many height changes.

Considerations and Edge Cases

Handling Different Heights: From the implementation point of view, the stack-based method is excellent because it always keeps track of just how many extra blocks are needed and automatically merges blocks of the same height.

Empty Arrays: If the height array is still empty, the function should return 0 since no blocks are required.

Single Height: If there is only a height value in an array, then only one block is required. Considerations and Edge Cases.

Handling Different Heights: The stack-based method effectively manages changes in height by counting additional blocks only when necessary and merging blocks of the same height.

Conclusion

The "Stone Wall" problem solution efficiently calculates the minimum number of cuboid blocks required to construct a wall of varying heights. By using a stack-based approach, the solution ensures that block counts are minimized and height changes are managed effectively. This method is optimal in terms of both time and space, making it suitable for significant inputs and varying wall profiles.

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