3071. Minimum Operations to Write the Letter Y on a Grid - Detailed Explanation

Problem Statement

You are given a grid (represented as a list/array of strings) where each cell is either painted (represented by the character '#') or not painted (represented by '.'). Your task is to "write" the letter Y on the grid by painting cells (i.e. converting '.' to '#') using as few operations as possible. In one operation you can paint one cell.

A valid Y shape is defined by:

• A pivot cell ((r, c)) which is the junction of the Y.
• Two diagonal arms extending upward from the pivot:
  • The left arm covers cells ((r-i, c-i)) for (i = 1 \ldots L).
  • The right arm covers cells ((r-i, c+i)) for (i = 1 \ldots L).
• A vertical branch extending downward from the pivot:
  • It covers cells ((r+j, c)) for (j = 1 \ldots D).

Both the arm length (L) and the branch length (D) must be at least 1, and the entire candidate Y must lie within the grid boundaries.

The cost of a candidate Y is the number of cells in the pattern (including the pivot, both arms, and the branch) that are not already painted ('#'). Your goal is to find the candidate Y with the minimum cost. If no valid Y can be placed, return (-1).

Examples

Example 1

Input:

["...",
 "...",
 "..."]

Explanation:

• Consider a 3×3 grid with all cells unpainted.
• One valid candidate Y is with pivot at (1,1), arm length (L = 1), and branch length (D = 1).
  • Pivot: ((1,1))
  • Left arm: ((0,0))
  • Right arm: ((0,2))
  • Vertical branch: ((2,1))
• All four cells are unpainted, so the cost is 4.

Output: 4

Example 2

Input:

[
  "......",
  "...#..",
  "......",
  "......",
  "......"
]

Explanation:

• The grid is 5×6 with one pre-painted cell at (1,3).
• The algorithm will try every possible pivot along with all valid arm and branch lengths.
• It computes the cost for every candidate Y (the number of cells that need to be painted) and returns the minimum cost found.
• (The exact minimum cost depends on the optimal placement found by the algorithm.)

Output: (The algorithm will output the minimum number of operations; for instance, it might output 3 or 4 depending on which candidate Y minimizes the operations.)

Hints

• Brute-force Candidate Generation:
  Iterate over every cell as a potential pivot. For each pivot, try every valid arm length (L) (determined by the distance to the grid’s top, left, and right boundaries) and every valid branch length (D) (ensuring the vertical branch fits below the pivot).

• Boundary Checking:
  Carefully compute the maximum arm length (L) for a pivot by using:
  [ L \leq \min(r, c, \text{m}-1-c) ]
  and ensure that (r + D < \text{n}) for the vertical branch.

• Cost Computation:
  For each candidate Y, count the number of cells that are not painted ('.') in the candidate shape.

• Minimization:
  Keep track of the smallest cost found among all candidate Y shapes.