Problem Statement
Maximal Rectangle
You are an urban planner working on a city development project. You have a grid-based map where each cell represents a plot of land. A value of '1' indicates a plot is available for development, while '0' indicates a plot that cannot be used.
Your task is to find the largest rectangular area of available plots (containing only 1's) that can be developed as a single project. This will maximize the land usage efficiency for the city.
Write a function that takes a binary matrix (grid of 0's and 1's) and returns the area of the largest rectangle containing only 1's.
Examples
Example 1:
Input: [
["1","0","1","0","0"],
["1","0","1","1","1"],
["1","1","1","1","1"],
["1","0","0","1","0"]
]
Output: 6
Explanation: The maximal rectangle is shown in the above grid. The area of this rectangle is 6 (2×3).
Example 2:
Input: [
["0","1"],
["1","0"]
]
Output: 1
Explanation: Each '1' forms a rectangle of area 1, and there are no larger rectangles.
Example 3:
Input: [["0"]]
Output: 0
Explanation: There are no '1's in the grid, so the maximal rectangle has area 0.
Constraints
- The number of rows m is between 0 and 200.
- The number of columns n is between 0 and 200.
- Each cell in the matrix contains either '0' or '1'.
- The matrix is represented as a 2D array of characters, not integers.
Problem Breakdown
To solve this problem, we need to:
- This problem can be reduced to finding the largest rectangle in a histogram for each row.
- Dynamic programming can be used to efficiently calculate the heights for each position.
- A stack-based approach can efficiently find the largest rectangle in a histogram.
- The problem requires considering both horizontal and vertical continuity of 1's.
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
Maximal Rectangle
You are an urban planner working on a city development project. You have a grid-based map where each cell represents a plot of land. A value of '1' indicates a plot is available for development, while '0' indicates a plot that cannot be used.
Your task is to find the largest rectangular area of available plots (containing only 1's) that can be developed as a single project. This will maximize the land usage efficiency for the city.
Write a function that takes a binary matrix (grid of 0's and 1's) and returns the area of the largest rectangle containing only 1's.
Examples
Example 1:
The maximal rectangle is shown in the above grid. The area of this rectangle is 6 (2×3).
Example 2:
Each '1' forms a rectangle of area 1, and there are no larger rectangles.
Example 3:
There are no '1's in the grid, so the maximal rectangle has area 0.
Problem Breakdown
Constraints:
- The number of rows m is between 0 and 200.
- The number of columns n is between 0 and 200.
- Each cell in the matrix contains either '0' or '1'.
- The matrix is represented as a 2D array of characters, not integers.
Key Insights:
This problem can be reduced to finding the largest rectangle in a histogram for each row.
Dynamic programming can be used to efficiently calculate the heights for each position.
A stack-based approach can efficiently find the largest rectangle in a histogram.
The problem requires considering both horizontal and vertical continuity of 1's.
Real-World Application
This problem has several practical applications:
Urban Planning
Finding the largest contiguous area for development projects in city planning.
Image Processing
Detecting rectangular regions in binary images for object recognition.
Data Visualization
Finding the largest rectangle in heatmaps or other grid-based data visualizations.