2017. Grid Game

[MyrinNew]

2017. Grid Game


Difficulty: Medium


Topics: Array, Matrix, Prefix Sum


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 ((r, c) to (r, c + 1)) or down ((r, c) to (r + 1, c)).


At the start of the game, the first robot moves from (0, 0) to (1, n-1), collecting all the points from the cells on its path. For all cells (r, c) traversed on the path, grid[r][c] is set to 0. Then, the second robot moves from (0, 0) to (1, n-1), collecting the points on its path. Note that their paths may intersect with one another.


The first robot wants to minimize the number of points collected by the second robot. In contrast, the second robot wants to maximize the number of points it collects. If both robots play optimally, return the number of points collected by the second robot.


Example 1:

• Input: grid = [[2,5,4],[1,5,1]]
• Output: 4
• Explanation: The optimal path taken by the first robot is shown in red, and the optimal path taken by the second robot is shown in blue.
  • The cells visited by the first robot are set to 0.
  • The second robot will collect 0 + 0 + 4 + 0 = 4 points.

Example 2:

• Input: grid = [[3,3,1],[8,5,2]]
• Output: 4
• Explanation: The optimal path taken by the first robot is shown in red, and the optimal path taken by the second robot is shown in blue.
  • The cells visited by the first robot are set to 0.
  • The second robot will collect 0 + 3 + 1 + 0 = 4 points.

Example 3:

• Input: grid = [[1,3,1,15],[1,3,3,1]]
• Output: 7
• Explanation: The optimal path taken by the first robot is shown in red, and the optimal path taken by the second robot is shown in blue.
  • The cells visited by the first robot are set to 0.
  • The second robot will collect 0 + 1 + 3 + 3 + 0 = 7 points.

Constraints:

• grid.length == 2
• n == grid[r].length
• 1 4
• 1 5

Hint:

1. There are n choices for when the first robot moves to the second row.
2. Can we use prefix sums to help solve this problem?

Solution:


We'll use the following approach:

1. Prefix Sum Calculation: We'll compute the prefix sums for both rows of the grid to efficiently calculate the sum of points for any subarray.
2. Simulating Optimal Movement:
   • The first robot determines its path to minimize the points left for the second robot.
   • At each column transition, the first robot can choose to move down, splitting the grid into two segments:
     • Upper remaining points: Points in the top row after the transition column.
     • Lower remaining points: Points in the bottom row before the transition column.
3. Minimizing the Maximum Points for the Second Robot:
   • At each transition, calculate the maximum points the second robot can collect after the first robot's path.
   • Track the minimum of these maximums across all transitions.

Let's implement this solution in PHP: 2017. Grid Game