Problem Statement
Cherry Pickup II
You are given a rows x cols matrix grid representing a field of cherries where grid[i][j] represents the number of cherries that you can collect from the (i, j) cell.
You have two robots that can collect cherries for you:
- Robot #1 is located at the top-left corner
(0, 0), and - Robot #2 is located at the top-right corner
(0, cols - 1).
Return the maximum number of cherries collection using both robots by following the rules below:
- From a cell
(i, j), robots can move to cell(i + 1, j - 1),(i + 1, j), or(i + 1, j + 1). - When any robot passes through a cell, It picks up all cherries, and the cell becomes an empty cell.
- When both robots stay in the same cell, only one takes the cherries.
- Both robots cannot move outside of the grid at any moment.
- Both robots should reach the bottom row in
grid.
Examples
Example 1:
Input: grid = [[3,1,1],[2,5,1],[1,5,5],[2,1,1]]
Output: 24
Explanation: Path of robot #1 and #2 are described in color green and blue respectively.
Cherries taken by Robot #1, (3 + 2 + 5 + 2) = 12.
Cherries taken by Robot #2, (1 + 5 + 5 + 1) = 12.
Total of cherries: 12 + 12 = 24.
Example 2:
Input: grid = [[1,0,0,0,0,0,1],[2,0,0,0,0,3,0],[2,0,9,0,0,0,0],[0,3,0,5,4,0,0],[1,0,2,3,0,0,6]]
Output: 28
Explanation: Path of robot #1 and #2 are described in color green and blue respectively.
Cherries taken by Robot #1, (1 + 2 + 2 + 0 + 1) = 6.
Cherries taken by Robot #2, (1 + 3 + 0 + 4 + 6) = 14.
Cherries taken by both robots, (0 + 0 + 9 + 5 + 3) = 17.
Total of cherries: 6 + 14 - 9 = 28.
Constraints
- rows == grid.length
- cols == grid[i].length
- 2 <= rows, cols <= 70
- 0 <= grid[i][j] <= 100
Problem Breakdown
To solve this problem, we need to:
- This problem can be solved using dynamic programming.
- The key insight is to move both robots simultaneously, row by row.
- For each row, we need to consider all possible positions of both robots.
- We can use a 3D DP array dp[row][col1][col2] to represent the maximum cherries that can be collected when robot 1 is at (row, col1) and robot 2 is at (row, col2).
- For each state, we need to consider 9 possible transitions (3 possible moves for each robot).
- We need to be careful about the case where both robots are in the same cell, as only one robot can collect cherries from that cell.
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
Cherry Pickup II
You are given a rows x cols matrix grid representing a field of cherries where grid[i][j] represents the number of cherries that you can collect from the (i, j) cell.
You have two robots that can collect cherries for you:
- Robot #1 is located at the top-left corner
(0, 0), and - Robot #2 is located at the top-right corner
(0, cols - 1).
Return the maximum number of cherries collection using both robots by following the rules below:
- From a cell
(i, j), robots can move to cell(i + 1, j - 1),(i + 1, j), or(i + 1, j + 1). - When any robot passes through a cell, It picks up all cherries, and the cell becomes an empty cell.
- When both robots stay in the same cell, only one takes the cherries.
- Both robots cannot move outside of the grid at any moment.
- Both robots should reach the bottom row in
grid.
Examples
Example 1:
Path of robot #1 and #2 are described in color green and blue respectively. Cherries taken by Robot #1, (3 + 2 + 5 + 2) = 12. Cherries taken by Robot #2, (1 + 5 + 5 + 1) = 12. Total of cherries: 12 + 12 = 24.
Example 2:
Path of robot #1 and #2 are described in color green and blue respectively. Cherries taken by Robot #1, (1 + 2 + 2 + 0 + 1) = 6. Cherries taken by Robot #2, (1 + 3 + 0 + 4 + 6) = 14. Cherries taken by both robots, (0 + 0 + 9 + 5 + 3) = 17. Total of cherries: 6 + 14 - 9 = 28.
Problem Breakdown
Constraints:
- rows == grid.length
- cols == grid[i].length
- 2 <= rows, cols <= 70
- 0 <= grid[i][j] <= 100
Key Insights:
This problem can be solved using dynamic programming.
The key insight is to move both robots simultaneously, row by row.
For each row, we need to consider all possible positions of both robots.
We can use a 3D DP array dp[row][col1][col2] to represent the maximum cherries that can be collected when robot 1 is at (row, col1) and robot 2 is at (row, col2).
For each state, we need to consider 9 possible transitions (3 possible moves for each robot).
We need to be careful about the case where both robots are in the same cell, as only one robot can collect cherries from that cell.
Real-World Application
This problem has several practical applications:
Resource Collection
Optimizing the collection of resources in a grid-based environment, such as robots collecting items in a warehouse.
Multi-Agent Coordination
Coordinating multiple agents to maximize the total reward in a shared environment.
Path Planning
Planning optimal paths for multiple agents in a shared environment, ensuring they don't interfere with each other.