Grid Game Leetcode Problem 2017 [Python Solution]

In this guide, we will break down and solve a medium-level challenge of the Arrays & Hashing family; Grid Game LeetCode problem.

This problem is an intriguing game of strategy involving two robots on a 2D grid.

The first robot aims to minimize the points the second robot can collect, while the second robot seeks to maximize its score, given that they both play optimally.

Let’s dive into the problem and work through a Python solution step by step.

Problem Overview

You are given a 2D grid of size 2 x n, where grid[r][c] represents the number of points at position (r, c) on the grid.

Two robots start at (0, 0) and need to reach (1, n-1).

They can only move to the right or down, and they play in the following order:

The first robot moves from (0, 0) to (1, n-1), collecting all the points from the cells on its path.

For each cell traversed, grid[r][c] is set to 0.

Then, the second robot moves from (0, 0) to (1, n-1), collecting points on its path.

Their paths may intersect with each other.

The first robot wants to minimize the number of points collected by the second robot, while the second robot aims to maximize its collected points.

If both robots play optimally, you need to 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.

Understanding Constraints

Before we proceed to the solution, it’s essential to understand the constraints of this problem.

grid.length is always 2, meaning we have two rows in the grid.
n represents the number of columns, and it can vary between 1 and 50,000.
The values in the grid are integers ranging from 1 to 105.

Grid Game LeetCode Problem Solution

Now, let’s get into the solution.

The approach we will take involves calculating prefix sums to determine the optimal paths for both robots.

We will implement this in Python.

Efficient Python Code Solution

View on GitHub ➹

class Solution:
    def gridGame(self, grid: List[List[int]]) -> int:
        result = float("inf")
        left, right = 0, sum(grid[0])

        for a, b in zip(grid[0], grid[1]):
            right -= a
            result = min(result, max(left, right))
            left += b
        return result

This Python solution leverages the concept of prefix sums to find the optimal strategy for both robots.

Here’s a breakdown of the code:

We create a class Solution to encapsulate our solution.
In the gridGame method, we initialize the result variable to infinity.

This will be used to store the minimum points the second robot can collect.

We also initialize left and right variables. left represents the points collected by the first robot, and right represents the sum of points in the top row of the grid (row 0).
We iterate through the rows of the grid using the zip function.

For each pair of values (a, b) in the two rows, we perform the following steps:

Subtract a from right to simulate the first robot’s movement.

This means setting the points collected by the first robot to 0.

Update the result by taking the minimum of the current result and the maximum of left and right.

This step is crucial because we want to minimize the points collected by the second robot while maximizing the points collected by the first robot.

Add b to left to simulate the second robot’s movement.
Finally, we return the result, which holds the minimum points collected by the second robot when both robots play optimally.

Time and Space Complexity

Time Complexity: The time complexity of this solution is O(n), where n represents the number of columns in the grid.

We iterate through the columns once to calculate the result.

Space Complexity: The space complexity is O(1) because we use a constant amount of additional space for variables result, left, and right.

Reasoning Behind Our Approach

The key to solving the Grid Game problem efficiently lies in understanding that both robots aim to optimize their points.

The first robot attempts to minimize the points the second robot can collect, while the second robot wants to maximize its score.

To achieve this, we utilize the concept of prefix sums to track the cumulative points along the grid’s rows.

By calculating the prefix sums, we can easily determine the remaining points for each robot’s path, which is essential for making informed decisions.

The first robot explores various options, changing the point at which it crosses rows (indicated by i in the code).

In each case, we calculate the points remaining for the second robot on both the top and bottom rows.

The second robot then selects the maximum of these two options to maximize its collected points.

By iterating through all possible crossing points for the first robot and considering the second robot’s optimal strategy, we find the solution that minimizes the points collected by the second robot.

In conclusion, the Grid Game problem demonstrates the importance of strategic thinking and the use of prefix sums to optimize the outcome for both robots.

View Grid Game on LeetCode

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