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regionsBySlashes Problem Solution

The regionsBySlashes problem requires us to determine the number of distinct regions formed by slashes (/ and \) in an n x n grid. Each cell in the grid can be considered as divided into four triangles (quadrants), and the slashes determine which triangles are connected or separated. This solution uses Union-Find (Disjoint-Set) to efficiently manage and count the regions.

Step-by-Step Explanation

1. Initialization

  • C++:
    • Initialize a parent array for Union-Find, treating each cell as having 4 sub-cells (or triangles).
    • Initialize a rank array to optimize the Union operation by keeping the depth of the trees minimal.
  • Java:
    • Create arrays parent and rank for the Union-Find structure, similar to the C++ approach.
  • JavaScript:
    • Set up parent and rank arrays using JavaScript’s array functions.
  • Python:
    • Use list comprehensions to initialize parent and rank arrays, representing each cell's triangles.
  • Go:
    • Allocate parent and rank slices to manage the Union-Find structure.

2. Union-Find Helper Functions

  • Find Function:
    • C++/Java/Python/JavaScript/Go: Implement the find function with path compression, which ensures that all nodes in a path point directly to the root. This optimization speeds up future queries.
  • Union Function:
    • C++/Java/Python/JavaScript/Go: Implement the unite (union) function, which merges two sets (triangles) based on their root's rank. This method minimizes the tree height, keeping operations efficient.

3. Iterate Through the Grid

  • Grid Traversal:
    • C++/Java/Python/JavaScript/Go: Loop through each cell of the grid. For each cell:
      • Determine the base index for the cell's 4 triangles.
      • Retrieve the character (/, \, or ``) representing the division in the current cell.

4. Union Operations Based on Slashes

  • Handle /:
    • C++/Java/Python/JavaScript/Go:
      • If the cell contains /, connect the top-left triangle with the bottom-right triangle, and the top-right triangle with the bottom-left triangle.
  • Handle \:
    • C++/Java/Python/JavaScript/Go:
      • If the cell contains \, connect the top-left triangle with the top-right triangle, and the bottom-left triangle with the bottom-right triangle.
  • Handle Space ``:
    • C++/Java/Python/JavaScript/Go:
      • If the cell is empty (space), connect all four triangles within the cell.

5. Connect Adjacent Cells

  • Right Neighbor:
    • C++/Java/Python/JavaScript/Go: If there's a right neighbor cell, connect the right triangle of the current cell with the left triangle of the right neighbor.
  • Bottom Neighbor:
    • C++/Java/Python/JavaScript/Go: If there's a bottom neighbor cell, connect the bottom triangle of the current cell with the top triangle of the bottom neighbor.

6. Count Distinct Regions

  • Count Regions:
    • C++/Java/Python/JavaScript/Go:
      • Traverse the parent array and count how many elements are their own parents. Each such element represents a distinct region.

7. Return the Result

  • Final Output:
    • C++/Java/Python/JavaScript/Go:
      • Return the total number of distinct regions.