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INTRODUCTION:

Welcome to an adventure that delves into the world of optimizing Gold Collection Paths in a jungle using the power of PHP. Imagine guiding explorers through lush landscapes to uncover precious treasures by strategically navigating the terrain. In this journey, we’ll uncover how simple yet effective techniques can be harnessed to maximize the haul of glittering gold. Let’s embark on this exploration of efficient Gold Collection Paths and discover the key to unlocking hidden riches!

QUESTION:

A group of explorers embarks on a treasure hunt in a jungle, armed with a map that reveals the potential amount of gold hidden at each location. The map is presented as a 2D grid, where each cell contains a positive integer representing the amount of gold that can be collected at that spot. The explorers can start at the top-left cell and need to navigate to the bottom-right cell to unearth the treasure. They can only move either down or right at each step.

Write a function or program to help the explorers determine the maximum amount of gold they can accumulate while following the optimal path through the map to reach the treasure.

Implement a function that takes the following inputs:

Input: A 2D grid of positive integers, representing the map. Each element (i, j) of the grid indicates the amount of gold at the i-th row and j-th column.

Output: An integer representing the maximum amount of gold that can be collected by following the best possible path from the top-left corner to the bottom-right corner of the map.

For example, consider the following map:

[1, 3, 1]
[2, 2, 4]
[5, 0, 2]

The explorers can follow the path 1 → 3 → 4 → 2 → 2 to collect a maximum of 12 gold units.

Your task is to implement the function that calculates and returns the maximum amount of gold that the explorers can collect while navigating the map.

ANSWER:

function maxGoldCollection($grid) {
    $rows = count($grid);
    $cols = count($grid[0]);
    
    // Create a 2D array to store the maximum gold collected at each position
    $dp = array_fill(0, $rows, array_fill(0, $cols, 0));
    
    // Fill the last cell of dp array with the value of the bottom-right cell of the grid
    $dp[$rows - 1][$cols - 1] = $grid[$rows - 1][$cols - 1];
    
    // Fill the last column of dp array
    for ($i = $rows - 2; $i >= 0; $i--) {
        $dp[$i][$cols - 1] = $dp[$i + 1][$cols - 1] + $grid[$i][$cols - 1];
    }
    
    // Fill the last row of dp array
    for ($j = $cols - 2; $j >= 0; $j--) {
        $dp[$rows - 1][$j] = $dp[$rows - 1][$j + 1] + $grid[$rows - 1][$j];
    }
    
    // Fill the remaining cells of dp array
    for ($i = $rows - 2; $i >= 0; $i--) {
        for ($j = $cols - 2; $j >= 0; $j--) {
            $dp[$i][$j] = $grid[$i][$j] + max($dp[$i + 1][$j], $dp[$i][$j + 1]);
        }
    }
    
    return $dp[0][0]; // The maximum gold collected at the top-left corner
}

// Example usage
$grid = [
    [1, 3, 1],
    [2, 2, 4],
    [5, 0, 2]
];

echo "Maximum Gold Collection: " . maxGoldCollection($grid); // Output: 12

Solution: Maximum Gold Collection Path in PHP

Here’s how you can solve the problem of finding the maximum gold collection path in a jungle map using PHP:

1. Understanding the Problem:
   We have a map represented as a grid, with each cell holding an amount of gold. Explorers want to find the best path from the top-left corner to the bottom-right corner of the map to collect the maximum amount of gold. They can only move right or down.
2. Creating a Dynamic Programming Array:
   We create a 2D array, called dp, that’s the same size as the grid. This array will help us keep track of the maximum gold collected at each position.
3. Filling the DP Array:

• Starting from the bottom-right corner, we put the value of the last cell in dp.
• Then, we fill the last column of dp by adding the value below each cell to the value in the cell below it.
• Next, we fill the last row of dp by adding the value to the right of each cell to the value in the cell to its right.

1. Calculating Maximum Gold:
   We work our way through the rest of the cells in dp, from bottom to top and right to left. For each cell, we calculate the maximum gold we can collect by either moving down or right. We add the current cell’s gold value to the maximum of the two possible moves.
2. Getting the Result:
   At the end, the top-left corner of the dp array holds the maximum gold collected by following the best path.
3. Example Usage:
   Let’s take a map as an example:
   grid = [ [1, 3, 1], [2, 2, 4], [5, 0, 2] ]
   By calling the function maxGoldCollection($grid), we get the maximum gold collected, which is 12.

Here’s how you can put this into code using PHP:

function maxGoldCollection($grid) {
    // Code from the previous response
}

// Example usage
$grid = [
    [1, 3, 1],
    [2, 2, 4],
    [5, 0, 2]
];

echo "Maximum Gold Collection: " . maxGoldCollection($grid); // Output: 12

This PHP code implements a smart way for explorers to find the path that gets them the most gold while navigating through the jungle map. It’s like finding the best route for them to follow to uncover the treasure!

CONCLUSION:

In wrapping up our exploration of “Gold Collection Paths,” remember that PHP stands as the beacon, illuminating efficient routes to uncover treasures within jungle maps. By optimizing these pathways, adventurers can make the most of their gold hauls, showcasing the technology’s role in enhancing exploration. With each step taken deliberately, the journey underscores the powerful interplay between smart algorithms and the pursuit of optimal rewards.

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