Regions Cut By Slashes

Problem Statement

In computational geometry, a common problem is to calculate disjoint or contiguous regions within given boundaries. Here, we are presented with an n x n grid, where each cell in this grid contains one of three characters: '/', '\\' (escaped backslash), or a blank space ' '. These symbols subdivide the cell diagonally, either splitting it into two triangular halves or keeping it intact in the case of the blank space. The problem is to determine the total number of contiguous regions formed as a result of these subdivisions. The definition of contiguous here extends both along straight edges and shared corners. Given this grid setup as an array of string representations, our task is to compute the count of these regions.

Examples

Example 1

Input:

grid = [" /","/ "]

Output:

2

Example 2

Input:

grid = [" /"," "]

Output:

1

Example 3

Input:

grid = ["/\","\/"]

Output:

5

Explanation:

Recall that because \ characters are escaped, "\/" refers to \/, and "/\" refers to /\.

Constraints

• n == grid.length == grid[i].length
• 1 <= n <= 30
• grid[i][j] is either '/', '\', or ' '.

Approach and Intuition

To solve the problem, we need to consider how the slashes within each cell might divide the space and how these divisions relate to neighboring cells. Here's a step-by-step breakdown of a possible approach:

1. Model each cell with finer granularity: Consider each cell as divided into four parts (top-left, top-right, bottom-left, bottom-right) to better model the diagonal subdivisions made by slashes.

2. Use a Union-Find structure or DFS for region detection: These tools can help in counting the distinct sets (regions).

   • Union-Find (Disjoint set): A Union-Find with path compression and union by rank can efficiently connect and find unique segments.
   • Depth First Search (DFS): Start from each undetected part and mark all connected parts as one region.

3. Mapping cell contents to partitions:

   • A cell with ' ' does not create any separations; all four parts belong to the same region internally.
   • A cell containing '/' splits the cell into top-left and bottom-right parts as one region, and top-right and bottom-left parts as another.
   • A cell containing '\\' splits the cell into top-right and bottom-left parts as one region, and top-left and bottom-right as another.

4. Connectivity across cells: It's imperative to connect the partitioned sections with adjacent pieces in neighboring cells. This includes:

   • Horizontal connectivity (left-right between cells)
   • Vertical connectivity (top-bottom between cells)
   • Handling boundary conditions notably at the edges of the grid.

5. Counting distinct regions: Once all relevant unions are made using either Union-Find or through connected components via DFS, the number of unique regions can be counted.

By applying this systematic exploration of each cell and its relation to its neighbors, the solution computes the distinct contiguous areas formed within the grid. This approach efficiently scales with the grid size, constrained by the problem up to 30x30, and handles complex adjacency scenarios thanks to the granularity of modeling each cell in parts. The intuition hinges upon understanding spatial partitioning in a discrete grid setup and effectively mapping these partitions into a data structure that supports dynamic connectivity queries and updates.

Solutions

class GridRegions {
public:
    int calculateRegions(vector<string>& gridMatrix) {
        int matrixSize = gridMatrix.size();
        int sideLength = matrixSize + 1;
        int totalVertices = sideLength * sideLength;
            
        // Set up the union-find structure
        vector<int> unionFind(totalVertices, -1);
    
        // Connect edges on the grid border
        for (int i = 0; i < sideLength; i++) {
            for (int j = 0; j < sideLength; j++) {
                if (i == 0 || j == 0 || i == sideLength - 1 || j == sideLength - 1) {
                    int vertexIndex = i * sideLength + j;
                    unionFind[vertexIndex] = 0;
                }
            }
        }
            
        // Set the initial point as its own parent
        unionFind[0] = -1;
        int regionCounter = 1;  // Begin with a single region encompassing the borders
    
        // Iterate through the grid cells
        for (int i = 0; i < matrixSize; i++) {
            for (int j = 0; j < matrixSize; j++) {
                // Handle different types of slashes as edges
                if (gridMatrix[i][j] == '/') {
                    int point1 = i * sideLength + (j + 1);
                    int point2 = (i + 1) * sideLength + j;
                    regionCounter += mergeSets(unionFind, point1, point2);
                } 
                else if (gridMatrix[i][j] == '\\') {
                    int point1 = i * sideLength + j;
                    int point2 = (i + 1) * sideLength + (j + 1);
                    regionCounter += mergeSets(unionFind, point1, point2);
                }
            }
        }
    
        return regionCounter;
    }
    
private:
    int locateParent(vector<int>& unionFind, int vertex) {
        if (unionFind[vertex] == -1) return vertex;
    
        return unionFind[vertex] = locateParent(unionFind, unionFind[vertex]);
    }
    
    int mergeSets(vector<int>& unionFind, int v1, int v2) {
        int root1 = locateParent(unionFind, v1);
        int root2 = locateParent(unionFind, v2);
    
        if (root1 == root2) {
            return 1;  // These vertices are part of the same set; counts as new region
        }
    
        unionFind[root2] = root1;  // Merge the sets
        return 0;                  // No new region formed
    }
};