Problem Statement
Largest Plus Sign Finder
You are an urban planner designing a new city layout. You want to place a central plaza in the shape of a plus sign ("+") in the city. The city is represented as an n×n grid where some cells are buildings (denoted by 0) and others are empty spaces (denoted by 1).
A plus sign of order k can be placed in the grid if there are at least k consecutive 1's in all four directions (up, down, left, and right) from the center cell. The center of the plus sign must be an empty space (1), and all cells that are part of the plus sign must also be empty spaces.
Given n and an array of coordinates representing buildings, your task is to find the largest possible plus sign that can be placed in the city.
Examples
Example 1:
Input: n = 5, mines = [[4, 2]]
Output: 2
Explanation: In the 5×5 grid with a building at (4, 2), the largest plus sign that can be placed has order 2.
Example 2:
Input: n = 3, mines = [[0, 0], [0, 2], [2, 0], [2, 2]]
Output: 1
Explanation: In the 3×3 grid with buildings at the four corners, the largest plus sign that can be placed has order 1.
Example 3:
Input: n = 2, mines = [[0, 0], [0, 1], [1, 0], [1, 1]]
Output: 0
Explanation: In the 2×2 grid with buildings at all cells, no plus sign can be placed.
Constraints
- 1 <= n <= 500
- 1 <= mines.length <= 5000
- 0 <= xi, yi < n
- All coordinates in mines are unique.
Problem Breakdown
To solve this problem, we need to:
- The order of a plus sign is determined by the minimum number of consecutive 1's in all four directions from the center.
- We can precompute the number of consecutive 1's in each direction for each cell to efficiently find the largest plus sign.
- Dynamic programming can be used to compute these values in a single pass through the grid.
- The problem can be solved in O(n²) time complexity, which is optimal since we need to consider all cells in the grid.
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
Largest Plus Sign Finder
You are an urban planner designing a new city layout. You want to place a central plaza in the shape of a plus sign ("+") in the city. The city is represented as an n×n grid where some cells are buildings (denoted by 0) and others are empty spaces (denoted by 1).
A plus sign of order k can be placed in the grid if there are at least k consecutive 1's in all four directions (up, down, left, and right) from the center cell. The center of the plus sign must be an empty space (1), and all cells that are part of the plus sign must also be empty spaces.
Given n and an array of coordinates representing buildings, your task is to find the largest possible plus sign that can be placed in the city.
Examples
Example 1:
In the 5×5 grid with a building at (4, 2), the largest plus sign that can be placed has order 2.
Example 2:
In the 3×3 grid with buildings at the four corners, the largest plus sign that can be placed has order 1.
Example 3:
In the 2×2 grid with buildings at all cells, no plus sign can be placed.
Problem Breakdown
Constraints:
- 1 <= n <= 500
- 1 <= mines.length <= 5000
- 0 <= xi, yi < n
- All coordinates in mines are unique.
Key Insights:
The order of a plus sign is determined by the minimum number of consecutive 1's in all four directions from the center.
We can precompute the number of consecutive 1's in each direction for each cell to efficiently find the largest plus sign.
Dynamic programming can be used to compute these values in a single pass through the grid.
The problem can be solved in O(n²) time complexity, which is optimal since we need to consider all cells in the grid.
Real-World Application
This problem has several practical applications:
Urban Planning
Designing city layouts with specific geometric patterns and constraints.
Image Processing
Finding specific patterns or shapes in images represented as grids.
Pattern Recognition
Identifying and measuring the size of cross-shaped patterns in data.