Problem Description
You are given a 0-indexed 2D array grid of size 2 x n, where grid[r][c] represents the number of points at position (r, c) on the matrix. Two robots are playing a game on this matrix. Both robots initially start at (0, 0) and want to reach (1, n-1). Each robot may only move to the right or down. The first robot aims to minimize the points collected by the second robot, while the second robot wants to maximize its own points. Return the number of points collected by the second robot when both play optimally.
Key Insights
- Both robots start at the same point and can only move right or down.
- The first robot's path influences the second robot's available points since it will zero out the cells it traverses.
- The goal for the first robot is to minimize the maximum points the second robot can collect.
- A dynamic programming or greedy approach can be used to determine the optimal paths for both robots.
Space and Time Complexity
Time Complexity: O(n)
Space Complexity: O(1)
Solution
To solve this problem, we can utilize a greedy approach. The idea is to calculate the total points that the first robot can collect on its optimal path while simultaneously keeping track of the maximum points the second robot can collect after the first robot's path is determined. We will iterate through the grid and calculate the possible paths for both robots, updating the points collected by the second robot accordingly. The solution involves keeping track of the cumulative sums for both rows and calculating the optimal paths based on the first robot's movement.