Regions Cut By Slashes

Start by expanding each 1x1 square into 4 subcells to handle '/' and '' divisions. Use DFS or union-find to merge connected subcells, treating empty spaces as freely connected. The result counts all isolated regions formed by slashes and spaces efficiently.

Problem Statement

You are given an n x n grid composed of 1 x 1 squares, where each square contains '/', '', or a blank space ' '. Each character divides the square into contiguous subregions, and your task is to determine how many separate regions exist across the entire grid.

Given the grid represented as a string array, implement a function that returns the total number of distinct regions formed by the slashes and empty spaces. Backslashes are escaped as '\' in the input, and connectivity is defined along the edges of the subdivided squares.

Examples

Example 1

Input: grid = [" /","/ "]

Output: 2

Example details omitted.

Example 2

Input: grid = [" /"," "]

Output: 1

Example details omitted.

Example 3

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

Output: 5

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 Approach

Expand Each Cell Into Subcells

Treat each 1x1 square as 4 smaller triangles or subcells to represent divisions by '/' and ''. Map connections between these subcells so DFS or union-find can track distinct regions.

Apply DFS or Union-Find

Traverse the expanded grid using depth-first search or union-find to merge connected subcells. Each unvisited subcell triggers a new region count, ensuring every isolated area is captured.

Count Regions Efficiently

After traversing all subcells, the number of distinct sets or DFS initiations corresponds to the total number of regions. This approach avoids revisiting cells and handles complex slash intersections correctly.

Complexity Analysis

Metric	Value
Time	O(n^2 \cdot \alpha (n^2))
Space	O(n^2)

Time complexity is O(n^2 \cdot \alpha(n^2)) due to union-find operations over n^2 subcells, and space complexity is O(n^2) to store subcell connectivity and visited states.

What Interviewers Usually Probe

• Clarifying whether '/' and '' create 2 or more subregions per square.
• Asking how DFS differs from union-find in this particular expanded grid approach.
• Testing edge cases like all blanks or fully slashed grids to verify correct region counting.

Common Pitfalls or Variants

Common pitfalls

• Ignoring the need to split each square into 4 subcells, leading to undercounting regions.
• Confusing escaped backslashes in the input string with literal characters.
• Overlooking connections between neighboring squares' subcells, causing disconnected regions.

Follow-up variants

• Counting regions in a hexagonal grid with slashes dividing hex cells.
• Handling dynamic updates where slashes are added or removed and recalculating regions efficiently.
• Determining largest region size instead of total number of regions.

FAQ

What is the core pattern in Regions Cut By Slashes?

The problem pattern is array scanning plus hash lookup, where each cell is subdivided to track connectivity and count regions.

How do you handle backslash characters in the grid input?

Backslashes are escaped as '\' in strings. Treat them as '' when dividing each square into subcells.

Can DFS and union-find be used interchangeably here?

Yes, both can track connected subcells. DFS marks visited areas recursively, while union-find merges subcell sets efficiently.

What is the main reason for undercounting regions?

Failing to subdivide each square into 4 subcells or ignoring connections between neighboring subcells can cause undercounting.

How large can the grid be and what are the constraints?

The grid is n x n with 1 <= n <= 30, and each cell contains '/', '', or ' '. Input is always square and properly formatted.