{x}
blog image

Regions Cut By Slashes

An n x n grid is composed of 1 x 1 squares where each 1 x 1 square consists of a '/', '\', or blank space ' '. These characters divide the square into contiguous regions.

Given the grid grid represented as a string array, return the number of regions.

Note that backslash characters are escaped, so a '\' is represented as '\\'.

 

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 ' '.

Solution Explanation:

This problem asks to find the number of regions in an n x n grid divided by slashes ('/') and backslashes (''). Two solutions are presented: Union-Find and Depth-First Search (DFS).

Solution 1: Union-Find

This approach is efficient for this type of problem because it efficiently tracks connected components.

Algorithm:

  1. Representation: Each 1x1 square is divided into 4 triangles. We represent the entire grid using a union-find data structure where each of the 4 triangles in a cell is a node.
  2. Initialization: Create a union-find data structure with 4 * n * n nodes, initially all in separate sets.
  3. Connections: Iterate through the grid:
    • Connect adjacent triangles vertically and horizontally.
    • Based on the character ('/', '', or ' '), union the appropriate triangles within each cell. For example, '/' connects the top-left and bottom-right triangles; '' connects top-right and bottom-left.
  4. Counting Regions: The number of disjoint sets in the union-find data structure represents the number of regions.

Time Complexity: O(n²α(n²)), where n is the size of the grid and α is the inverse Ackermann function (which grows extremely slowly, essentially constant for practical purposes). The dominant operation is the union-find operations.

Space Complexity: O(n²) to store the union-find data structure.

Solution 2: Depth-First Search (DFS)

This approach explores the grid to identify connected regions.

Algorithm:

  1. Graph Creation: Create a graph representing the grid. Each cell is divided into four smaller squares, and connections between these squares are established based on the presence of slashes or backslashes. The graph is represented as an adjacency matrix.
  2. DFS Traversal: Perform a depth-first search (DFS) on the graph starting from unvisited nodes (representing empty spaces). Each DFS call marks a connected region as visited.
  3. Count Regions: The number of DFS calls is equal to the number of regions.

Time Complexity: O(n²). The DFS visits each cell at most once.

Space Complexity: O(n²) to store the graph.

Code Explanations (Python, Java, C++, Go, TypeScript, JavaScript)

The code implementations for both solutions (Union-Find and DFS) are provided in multiple languages. The Union-Find solution is generally more efficient. The comments in the code explain the logic step-by-step. The code structures are similar across all languages, primarily differing in syntax and library calls. For example, the find and union functions in the Union-Find solution are common to all implementations but have slight variations in implementation detail based on language features. The DFS solution similarly adapts the graph construction and traversal to language-specific features. The core algorithm remains the same across all versions.