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].length1 <= n <= 30grid[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:
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.
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.
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.
- A cell with
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.
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
- C++
- Java
- Python
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
}
};
The provided solution in C++ focuses on the problem of determining how many regions are formed by slashes ("/" and "") on a grid. This approach uses the union-find data structure to efficiently manage and merge sets, which helps in tracking the connected components of the grid. Here's a breakdown of the solution:
Grid and Union Setup: The code initializes a union-find data structure to keep track of connected components. It treats the grid's border as a single unified region by connecting all border vertices.
Handling Borders: All vertices on the borders of the grid are initially connected to a single vertex, creating a base region that constitutes the outer boundary.
Processing Slashes: As the code iterates through each cell in the grid:
- If a cell contains "/", it merges the top-right and bottom-left corners of that cell.
- If a cell contains "", it merges the top-left and bottom-right corners of that cell.
Region Counting: Each time a merge operation occurs between two separate sets (indicating distinct connected components), it checks if these sets were previously disconnected and, therefore, forms a new region.
Returning Result: The function returns the number of distinct regions, starting with the predefined border region and incrementing with each newly found isolated area through slash inputs.
This method effectively partitions the grid into distinct areas isolated by slashes, managing the complexity of connectivity through the union-find data structure, thus providing a clear and organized way to handle such segmentation problems.
class Solution {
public int countRegions(String[] matrix) {
int size = matrix.length;
int perSide = size + 1;
int totalNodes = perSide * perSide;
// Prepare Union-Find structure
int[] setParent = new int[totalNodes];
Arrays.fill(setParent, -1);
// Link border nodes
for (int row = 0; row < perSide; row++) {
for (int col = 0; col < perSide; col++) {
if (
row == 0 ||
col == 0 ||
row == perSide - 1 ||
col == perSide - 1
) {
int borderNode = row * perSide + col;
setParent[borderNode] = 0;
}
}
}
// Self parent for the very first corner
setParent[0] = -1;
int areas = 1; // Initialize with one area (the surrounding border)
// Evaluate each square in the matrix
for (int i = 0; i < size; i++) {
for (int j = 0; j < size; j++) {
// Handle each slash to connect points
if (matrix[i].charAt(j) == '/') {
int pointUpRight = i * perSide + (j + 1);
int pointDownLeft = (i + 1) * perSide + j;
areas += merge(setParent, pointUpRight, pointDownLeft);
} else if (matrix[i].charAt(j) == '\\') {
int pointUpLeft = i * perSide + j;
int pointDownRight = (i + 1) * perSide + (j + 1);
areas += merge(setParent, pointUpLeft, pointDownRight);
}
}
}
return areas;
}
private int locate(int[] setParent, int vertex) {
if (setParent[vertex] == -1) return vertex;
return setParent[vertex] = locate(setParent, setParent[vertex]);
}
private int merge(int[] setParent, int nodeX, int nodeY) {
int parentX = locate(setParent, nodeX);
int parentY = locate(setParent, nodeY);
if (parentX == parentY) {
return 1; // Already connected, forms a new area
}
setParent[parentY] = parentX; // Merge the nodes
return 0; // No new area formed
}
}
This Java solution tackles the problem of calculating distinct regions created by slashes on a grid. Here's how the solution is structured:
- Initializes a grid (
perSide) that accounts for each point around each cell in the matrix, hence its size ismatrix.length + 1. - Establishes a Union-Find data structure (
setParent) to manage connected components within the grid. Each element represents a node in the grid, and nodes on the border are connected to a common root (node0).
Key Functions and their Responsibilities:
countRegionsmethod:- Fills the Union-Find with initial values, connecting all border nodes.
- Iterates through the given matrix to process each character. Characters
/and\are treated as barriers that split the space, and the connectivity is modified using the Union-Find structure. - For every slash, connects the appropriate corners of the grid squares based on their position and direction, either incrementing the counter for a new region or merging nodes in the existing structure.
locatemethod: A helper function that employs path compression to find the root of a node, which helps in keeping the tree flat and optimizing the Union-Find operations.mergemethod: Combines two nodes' sets into a single set by linking the parent of one node to another. This method also checks if merging nodes creates a new distinct region and updates the count accordingly.
Execution Flow:
- Initialize the grid side length (
perSide) and the Union-Find structure (setParent). - Connect all border nodes to a common root.
- For each vertex represented by a slash or backslash in the matrix, connect nodes accordingly and use the Union-Find to manage these connections.
- The Union-Find structure ensures that all connected components are managed efficiently, allowing the count of distinct areas to be dynamically updated with each union operation.
This implementation cleverly utilizes the Union-Find data structure to efficiently handle dynamic connectivity queries and updates, making the solution highly efficient for large matrices. The use of path compression and union by rank (implicitly through merging by individual elements directly) helps in minimizing the time complexity associated with each operation in the Union-Find structure. Overall, this approach efficiently determines the number of distinct regions formed by slashes in the grid.
class Solution:
def countRegions(self, grid: List[str]) -> int:
n = len(grid)
dimension = n + 1
total_vertices = dimension * dimension
dsu = [-1] * total_vertices
# Join border vertices
for r in range(dimension):
for c in range(dimension):
if r == 0 or c == 0 or r == dimension - 1 or c == dimension - 1:
vertex = r * dimension + c
dsu[vertex] = 0
dsu[0] = -1
regions = 1
for x in range(n):
for y in range(n):
if grid[x][y] == "/":
t_right = x * dimension + (y + 1)
b_left = (x + 1) * dimension + y
regions += self._merge(dsu, t_right, b_left)
elif grid[x][y] == "\\":
t_left = x * dimension + y
b_right = (x + 1) * dimension + (y + 1)
regions += self._merge(dsu, t_left, b_right)
return regions
def _find_root(self, dsu: List[int], vertex: int) -> int:
if dsu[vertex] == -1:
return vertex
dsu[vertex] = self._find_root(dsu, dsu[vertex])
return dsu[vertex]
def _merge(self, dsu: List[int], v1: int, v2: int) -> int:
root1 = self._find_root(dsu, v1)
root2 = self._find_root(dsu, v2)
if root1 == root2:
return 1
dsu[root2] = root1
return 0
This Python solution defines a method countRegions which counts the number of distinct regions that are separated by slashes in a square grid. The main approach used is a union-find algorithm (also known as disjoint-set union, DSU) to efficiently handle the merging and finding of sections created by slashes.
Here's an overview of how the code works:
- The grid size (
n) determines the dimension of an auxiliary grid which is one unit larger (dimension = n + 1). The total vertices in this grid aredimension * dimension. - A list,
dsu, is used to manage these vertices. Initially, all vertices are set as separate sets (-1). - The boundary vertices of the grid are unified into a single set, with all utmost boundary vertices pointing to
0, except the top-left corner, which remains-1. - A variable
regionsstarts with a value of 1, as the entire grid is initially considered one single region. - The grid is then iterated, and depending on whether a cell contains a "/", separating the top-right and bottom-left, or a "", separating the top-left and bottom-right, appropriate vertices in the
dsuare merged. This merging checks if two components are already connected and increments the region count if they were previously disjoint. - The methods
_find_rootand_mergemanage the union-find operations._find_rootprovides the representative or root of a given set by path compression._mergeconnects two vertices and returns0if they were already connected or1if they were not (meaning a region split by a new slash).
This method is effective for geometric partitioning problems, utilizing path compression and union by rank for efficient union-find operations, thus ensuring optimal performance for larger grids. By using the DSU structure, the algorithm effectively tracks components, adjusting the count of distinct regions as slashes add boundaries.
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