Skip to content

Latest commit

 

History

History
121 lines (78 loc) · 4.65 KB

explanation.md

File metadata and controls

121 lines (78 loc) · 4.65 KB

Grid Game Solution: Step-by-Step Explanation

This repository contains solutions for the Grid Game problem implemented in five different programming languages: C++, Java, JavaScript, Python, and Go. Below is a step-by-step explanation of the approach taken for each language without revealing the code itself.

Problem Overview

The problem involves a grid with two rows. The goal is to minimize the maximum sum collected by the second player while moving from the top-left corner to the bottom-right corner of the grid. Both players can only move right or down, and their paths cannot overlap.

Explanation of Solutions

1. C++ Code

Steps

  1. Understanding the Grid Setup: Analyze the sum of elements above (top row) and below (bottom row) for every column.
  2. Prefix and Suffix Sums: Utilize prefix sums to track the total sum collected by Player 1 and suffix sums to estimate Player 2's potential moves.
  3. Minimizing Player 2's Maximum: For each column split, calculate Player 2's maximum score if Player 1 transitions to the bottom row.
  4. Final Decision: Select the column where Player 2's maximum score is minimized.

2. Java Code

Steps

  1. Grid Analysis: Start by splitting the grid into top and bottom rows.
  2. Tracking Scores: Use two cumulative arrays to track sums for both rows:
    • Prefix sums for the top row.
    • Suffix sums for the bottom row.
  3. Simulating Player Transitions: For each column, simulate the scenario where Player 1 moves to the bottom row and compute Player 2's possible scores.
  4. Optimization: Return the minimum score among all possible transitions.

3. JavaScript Code

Steps

  1. Initial Setup: Parse the 2D grid array and calculate cumulative sums for both rows.
  2. Precomputing Scores:
    • Compute the prefix sum for the first row (top).
    • Compute the suffix sum for the second row (bottom).
  3. Iterative Comparison: Iterate over every column and calculate the maximum score Player 2 can achieve for that column split.
  4. Result: Return the minimum of these maximum scores.

4. Python Code

Steps

  1. Grid Representation: Treat the input as a list of two subarrays representing the rows of the grid.
  2. Prefix and Suffix Computation:
    • Calculate cumulative sums from left to right for the top row.
    • Calculate cumulative sums from right to left for the bottom row.
  3. Simulate Player Movement: For each column, evaluate the impact of Player 1 moving down to the bottom row and Player 2's subsequent score.
  4. Optimal Path Selection: Choose the column that minimizes Player 2's maximum score.

5. Go Code

Steps

  1. Grid Input Handling: Read the grid as a slice of slices representing the two rows.
  2. Prefix and Suffix Sums:
    • Use arrays to calculate prefix sums for the top row.
    • Use arrays to calculate suffix sums for the bottom row.
  3. Iterate and Optimize: Loop through each column and calculate the worst-case score for Player 2 if Player 1 transitions at that column.
  4. Determine the Result: Return the column split that yields the least maximum score for Player 2.

Key Concepts in All Solutions

  1. Prefix and Suffix Sums:

    • Prefix sums are used to keep track of Player 1's potential scores.
    • Suffix sums are used to simulate Player 2's possible scores.

  2. Iterative Simulation:

    • For every column, calculate the maximum score Player 2 can achieve.
    • The goal is to minimize this maximum score.

  3. Optimization:

    • The solutions aim to efficiently compute results by avoiding overlapping paths and reducing unnecessary calculations.

Complexity Analysis

Time Complexity

  • All solutions run in O(n), where (n) is the number of columns in the grid. This is because we only compute prefix and suffix sums once and iterate over the columns linearly.

Space Complexity

  • The space complexity is O(n) for maintaining prefix and suffix arrays.

Languages and Code Files

Language File
C++ solution.cpp
Java Solution.java
JavaScript solution.js
Python solution.py
Go solution.go

For the full code implementation, refer to the respective files in the repository.

Final Note

Each solution leverages the same core intuition and approach but adapts it to the syntax and features of the respective language. This ensures the solution is both efficient and easy to understand.