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Longest Path in Grid with Increasing Moves

This README provides a step-by-step explanation for solving the problem of finding the longest increasing path in a grid using multiple programming languages. We start with the problem's approach and then dive into a language-specific breakdown.

Problem Description

Given a 2D grid, each cell contains an integer. You are allowed to start from any cell in the first column and can move to the right, up-right, or down-right, but only to cells with a strictly greater value. The goal is to determine the maximum number of moves that can be made.

Approach

The solution uses dynamic programming (DP) to efficiently find the longest path by storing intermediate results and building upon them.

Steps in the Approach

  1. Initialize a DP Table:

    • Create a dp table where dp[row][col] will hold the maximum number of moves possible starting from grid[row][col].

  2. Backwards Processing:

    • Start filling the dp table from the last column and move leftward to ensure that each cell's result relies on already-calculated values to the right.

  3. Transition Check:

    • For each cell, calculate potential moves:
      • Move directly right.
      • Move up-right.
      • Move down-right.
    • Check if the destination cell has a strictly greater value and update the dp table accordingly.

  4. Extract Result:

    • The final answer is the maximum value in the first column, representing the longest path from any cell in the first column.

Language-Specific Breakdown

Each language follows a similar logic but is tailored to the syntax and nuances of the respective language.

C++ Code Explanation

  1. Set up the DP Table:

    • Initialize a 2D vector, dp, to store results for each cell.

  2. Loop Backward from Last Column:

    • Use a loop to process each column from right to left.

  3. Check Possible Moves:

    • For each cell, check moves to:
      • Right cell (same row).
      • Up-right cell (previous row).
      • Down-right cell (next row).
    • Update dp[row][col] with the maximum moves possible from each option.

  4. Find Maximum Moves:

    • After filling the DP table, loop through the first column to find the maximum number of moves starting from any cell.

Java Code Explanation

  1. Initialize DP Array:

    • Create a 2D integer array dp to store the longest increasing path lengths for each cell.

  2. Iterate Backwards Through Columns:

    • Process each column from the second last to the first, allowing each cell to refer to values in the following column.

  3. Calculate Valid Moves:

    • For each cell:
      • Check possible moves to the right, up-right, and down-right.
      • Only move if the destination cell has a greater value.
    • Update the dp table with the maximum moves from each possible move.

  4. Result Calculation:

    • The answer is the maximum value in the first column after processing all cells.

JavaScript Code Explanation

  1. Create DP Array:

    • Use a 2D array dp to store the maximum moves for each cell.

  2. Process Each Column Backwards:

    • Loop from the second last column back to the first, calculating possible moves for each cell.

  3. Evaluate Moves:

    • For each cell, calculate moves to:
      • Right, up-right, and down-right cells.
    • Only consider moves to cells with greater values and update dp with the maximum path found.

  4. Get Final Result:

    • Find the highest value in the first column of dp, representing the maximum moves starting from any cell in that column.

Python Code Explanation

  1. Set Up DP Table:

    • Initialize a 2D list dp to hold the maximum moves possible from each cell.

  2. Backward Processing Through Columns:

    • Starting from the last column and moving leftward, compute possible moves for each cell.

  3. Check Valid Moves:

    • For each cell, check moves to:
      • Right, up-right, and down-right cells.
    • If the destination cell has a greater value, update dp with the maximum moves possible from that path.

  4. Extract Maximum Moves:

    • The result is the maximum value in the first column, representing the longest path possible starting from any cell in that column.

Go Code Explanation

  1. Initialize DP Table:

    • Create a 2D slice dp to store the longest paths for each cell.

  2. Backwards Processing:

    • Process each column from right to left, calculating the longest path for each cell.

  3. Calculate Valid Moves:

    • For each cell:
      • Check moves to the right, up-right, and down-right cells.
    • Only update if the destination cell is greater and yields a longer path.

  4. Retrieve Result:

    • Find the maximum value in the first column to determine the longest possible path.