Solution Explanation

This problem asks to assign bikes to workers based on the shortest Manhattan distance. The solution utilizes a greedy approach combined with sorting to efficiently find the optimal assignments.

Approach:

1. Calculate Distances: For each worker-bike pair, the Manhattan distance is calculated. This distance is given by |worker_x - bike_x| + |worker_y - bike_y|.

2. Create a Sorted List: A list of tuples (distance, worker_index, bike_index) is created. This list is then sorted based on these three criteria, prioritizing shortest distance, then worker index, then bike index. This ensures that the closest pairings are considered first, resolving ties appropriately.

3. Greedy Assignment: The sorted list is iterated. For each entry, if both the worker and the bike are unassigned, the bike is assigned to the worker. This greedy approach guarantees the optimal solution because it always picks the best available option at each step.

4. Return Assignments: Finally, an array containing the assigned bike index for each worker is returned.

Time Complexity Analysis:

• Distance Calculation: Calculating the Manhattan distance for all n * m worker-bike pairs takes O(n * m) time.

• Sorting: Sorting the list of n * m tuples takes O(n * m * log(n * m)) time using a comparison-based sorting algorithm like merge sort or quicksort.

• Greedy Assignment: Iterating through the sorted list takes O(n * m) time in the worst case (although it's likely to be faster because assignments are made early on).

Therefore, the overall time complexity is dominated by the sorting step, resulting in O(n * m * log(n * m)).

Space Complexity Analysis:

• Distance List: The list of tuples storing distances, worker indices, and bike indices requires O(n * m) space.

• Boolean Arrays: The boolean arrays vis1 and vis2 to track assigned workers and bikes require O(n + m) space.

• Result Array: The result array ans requires O(n) space.

Thus, the overall space complexity is O(n * m), dominated by the distance list.

Code Explanation (Python):

The Python code directly implements the approach described above. The itertools.product function efficiently generates all worker-bike pairs. The sort() method is used to sort the tuples based on the defined criteria. The rest of the code handles the greedy assignment and returns the result.

Code Explanation (Other Languages):

The Java, C++, and Go codes are very similar in structure to the Python code. They follow the same three-step approach: distance calculation, sorting, and greedy assignment. The primary differences lie in the syntax and data structures used for each language. For instance, Java uses Arrays.sort with a custom comparator for sorting, while C++ uses std::sort and Go utilizes sort.Slice. The core logic remains consistent across all implementations.