Assign Mice to Holes

The Assign Mice to Holes problem is a classic problem in computer science and mathematics. It involves a scenario where there are 'n' mice and 'n' holes placed in a line. Each mouse must be assigned to a hole, and the goal is to minimize the time taken for all the mice to reach their respective holes. Each mouse moves at a certain speed, and the time it takes for a mouse to reach its hole is proportional to the distance between the mouse and the hole.

Problem Statement and Example

Given the positions of mice and holes along a line, and the speed of each mouse, the task is to find the arrangement of mice in the holes that minimizes the maximum time taken for any mouse to reach its hole. Each mouse should be assigned to a unique hole, and the order of mice and holes should be preserved.

For example, consider the following scenario:

• Mice Positions: -3, 7, 5, 9, 16
• Hole Positions: 1, 9, 15, 4, 14

The goal is to arrange the mice in the holes in a way that minimizes the maximum time taken for any mouse to reach its hole.

Idea to Solve the Problem

To solve this problem, we can follow these steps:

1. Sort the positions of mice and holes in ascending order.
2. Iterate through the sorted arrays and calculate the absolute difference between each mouse's position and its corresponding hole's position.
3. Track the maximum absolute difference during the iteration.
4. The maximum absolute difference represents the time it takes for the mouse to reach its hole in the optimal arrangement.

Algorithm Explanation

1. Sort both the mices and holes arrays in ascending order.
2. Initialize a variable result to store the maximum absolute difference (time taken).
3. Iterate through the arrays from 0 to n-1 (where n is the length of the arrays): a. Calculate the absolute difference between mices[i] and holes[i]. b. Update result if the calculated difference is greater than the current result.
4. Print the result, which represents the minimum time taken for the mice to reach their holes in an optimal arrangement.

Code Solution

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Time Complexity

The time complexity of this algorithm is O(n log n), where 'n' is the number of mice (or holes). This is because the algorithm involves sorting the mice and holes arrays, which takes O(n log n) time using a standard sorting algorithm. The subsequent iteration through the arrays takes linear time O(n), resulting in a total complexity of O(n log n).