Squirrel Simulation

Problem Statement

In this scenario, you are provided with a garden modelled as a grid defined by its dimensions, height and width. In the garden, there is a tree and a squirrel, each positioned at specific grid coordinates. Additionally, scattered across the garden are multiple nuts, each also positioned at specific grid coordinates. The main focus of this challenge is on the paths the squirrel must take:

1. To fetch each nut one by one.
2. To bring them back to the position of the tree.

Given the constraints of the garden and the characters within it, your objective is to determine the minimal total distance the squirrel must travel to collect all the nuts and deposit them under the tree. The distance measure here is the number of direct moves up, down, left, or right to an adjacent cell the squirrel must make to accomplish this task.

Examples

Example 1

Input:

height = 5, width = 7, tree = [2,2], squirrel = [4,4], nuts = [[3,0], [2,5]]

Output:

12

Explanation:

The squirrel should go to the nut at [2, 5] first to achieve a minimal distance.

Example 2

Input:

height = 1, width = 3, tree = [0,1], squirrel = [0,0], nuts = [[0,2]]

Output:

3

Constraints

• 1 <= height, width <= 100
• tree.length == 2
• squirrel.length == 2
• 1 <= nuts.length <= 5000
• nuts[i].length == 2
• 0 <= treer, squirrelr, nutir <= height
• 0 <= treec, squirrelc, nutic <= width

Approach and Intuition

To solve the problem of finding the minimal distance the squirrel needs to make to collect all nuts and place them under the tree, we need to carefully plan the order and strategy used by the squirrel. Here are some logical deductions and a potential approach based on the examples provided:

1. Identifying Closest Nut Initially:

   • From the examples, particularly the first one, it appears that choosing the closest nut to the squirrel initially can potentially reduce the overall distance required. This could be because the first move doesn't need to consider returning immediately to the tree.

2. Standard Nut to Tree Movement:

   • For each nut, after the first one is collected, the cycle involves collecting a nut and taking it back to the tree. Therefore, the direct distance from each nut to the tree is a constant factor in calculating the total distance.

3. Calculate Distances:

   • Use Manhattan distance as the grid allows only four movement directions (up, down, left, right). For any two points (x1, y1) and (x2, y2) the Manhattan distance is |x1 - x2| + |y1 - y2|.

4. Summing Up the Approach:

   • Compute the direct distance for the squirrel to move from its start point to each nut.
   • Compute the distance from each nut to the tree.
   • Start by sending the squirrel to the nut which gives the minimal initial travel distance (factoring in the journey back to the tree which might allow us to offset the higher initial distance for strategic benefits).
   • For the rest of the nuts, prioritize based on minimizing the looped travel distance (nut to the tree).

This simplistic initial strategy may need complexity tuning considering there could be up to 5000 nuts, and we need to make sure the computation remains efficient within the given constraints.

Solutions

public class Solution {
    public int minimumDistance(int h, int w, int[] treePosition, int[] squirrelPosition, int[][] nutPositions) {
        int totalDistance = 0, maxDiff = Integer.MIN_VALUE;
        for (int[] nut: nutPositions) {
            totalDistance += (calculateDistance(nut, treePosition) * 2);
            maxDiff = Math.max(maxDiff, calculateDistance(nut, treePosition) - calculateDistance(nut, squirrelPosition));
        }
        return totalDistance - maxDiff;
    }
    public int calculateDistance(int[] point1, int[] point2) {
        return Math.abs(point1[0] - point2[0]) + Math.abs(point1[1] - point2[1]);
    }
}