#2017

Medium

LeetCode Problem Workspace

Grid Game

Solve the Grid Game problem by optimizing the movement of two robots on a 2D matrix grid.

Problem Statement

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 start at (0, 0) and want to reach (1, n-1). Each robot may only move to the right or down at any point in time.

At the start, the first robot moves from (0, 0) to (1, n-1), collecting points from all the cells on its path, and sets these visited cells' values to 0. Then, the second robot moves from (0, 0) to (1, n-1), collecting points on its path. Their paths may intersect, and the goal is to maximize the points collected by the second robot.

Examples

Example 1

Input: grid = [[2,5,4],[1,5,1]]

Output: 4

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

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

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 <= n <= 5 * 104
1 <= grid[r][c] <= 105

Solution Approach

Dynamic Programming with Prefix Sum

To solve the problem efficiently, a prefix sum array can be used to calculate the total points up to any position. First, compute the path for the first robot, marking the grid with 0s for the cells it visits. Then, calculate the possible paths for the second robot, optimizing its path based on the updated grid.

Greedy Strategy for Second Robot's Path

The second robot must choose the best path at each step to maximize the points collected. By leveraging the updated grid after the first robot's movement, you can make greedy choices by selecting the path that offers the highest value, ensuring that its collected points are maximized.

Time Complexity Optimization

Since the problem has constraints that require processing up to 50,000 columns, the solution must be optimized for linear time complexity O(n). By using a greedy approach combined with prefix sums, the problem can be solved efficiently in one pass through the grid.

Complexity Analysis

Metric	Value
Time	O(n)
Space	O(1)

The time complexity is O(n) because the solution involves a single pass through the grid to calculate the paths of both robots, using prefix sums for efficient point calculation. The space complexity is O(1) since the solution only requires a few variables to track the state of the matrix.

What Interviewers Usually Probe

Focus on the interaction between the first and second robots' paths to optimize the second robot's collection of points.
Examine the trade-offs in choosing paths for the second robot, especially when paths overlap with the first robot's path.
The ability to implement efficient algorithms in linear time will be key to solving this problem within the constraints.

Common Pitfalls or Variants

Common pitfalls

Forgetting to account for the cells visited by the first robot, which are marked as 0 and affect the second robot's path.
Failing to optimize the second robot's path by greedily choosing the best option at each step based on updated matrix values.
Not handling the matrix's large size properly within the time limits, leading to inefficient solutions with higher time complexity.

Follow-up variants

Increase the grid size beyond 2 rows and optimize for larger matrixes.
Implementing additional constraints, like diagonal movements or limited robot movement options.
Modifying the game to include dynamic changes in point values during the game, requiring continuous re-evaluation of optimal paths.

FAQ

What is the main challenge in the Grid Game problem?

The main challenge is optimizing the paths of two robots in a matrix grid while maximizing the second robot's points, especially when the robots' paths may overlap.

How does the prefix sum technique help solve this problem?

The prefix sum technique allows you to quickly calculate the total points up to any position on the grid, which is crucial for efficiently solving the problem in O(n) time.

What is the optimal strategy for the second robot's path?

The optimal strategy is to use a greedy approach, where the second robot always chooses the path that maximizes its points based on the remaining grid after the first robot's path.

How can I optimize my solution to handle large grid sizes?

By implementing a linear time complexity solution using prefix sums and a greedy strategy, you can solve the problem efficiently even with large grid sizes, up to 50,000 columns.

What happens if the paths of the robots overlap?

When the paths overlap, the first robot collects the points from the shared cells, marking them as 0. The second robot will avoid these cells and choose the next available best path.

terminal

Solution

Solution 1: Prefix Sum

We notice that if we determine the position $j$ where the first robot turns down, then the optimal path of the second robot is also determined. The optimal path of the second robot is the prefix sum of the first row from $j+1$ to $n-1$, or the prefix sum of the second row from $0$ to $j-1$, taking the maximum of the two.

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class Solution:
    def gridGame(self, grid: List[List[int]]) -> int:
        ans = inf
        s1, s2 = sum(grid[0]), 0
        for j, v in enumerate(grid[0]):
            s1 -= v
            ans = min(ans, max(s1, s2))
            s2 += grid[1][j]
        return ans

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