maximum_compatibility_score_sum.py

"""
There is a survey that consists of n questions where each question's answer is either 0 (no) or 1 (yes).

The survey was given to m students numbered from 0 to m - 1 and m mentors numbered from 0 to m - 1. The answers of the students are represented by a 2D integer array students where students[i] is an integer array that contains the answers of the ith student (0-indexed). The answers of the mentors are represented by a 2D integer array mentors where mentors[j] is an integer array that contains the answers of the jth mentor (0-indexed).

Each student will be assigned to one mentor, and each mentor will have one student assigned to them. The compatibility score of a student-mentor pair is the number of answers that are the same for both the student and the mentor.

For example, if the student's answers were [1, 0, 1] and the mentor's answers were [0, 0, 1], then their compatibility score is 2 because only the second and the third answers are the same.
You are tasked with finding the optimal student-mentor pairings to maximize the sum of the compatibility scores.

Given students and mentors, return the maximum compatibility score sum that can be achieved.

 

Example 1:

Input: students = [[1,1,0],[1,0,1],[0,0,1]], mentors = [[1,0,0],[0,0,1],[1,1,0]]
Output: 8
Explanation: We assign students to mentors in the following way:
- student 0 to mentor 2 with a compatibility score of 3.
- student 1 to mentor 0 with a compatibility score of 2.
- student 2 to mentor 1 with a compatibility score of 3.
The compatibility score sum is 3 + 2 + 3 = 8.
Example 2:

Input: students = [[0,0],[0,0],[0,0]], mentors = [[1,1],[1,1],[1,1]]
Output: 0
Explanation: The compatibility score of any student-mentor pair is 0.
 

Constraints:

m == students.length == mentors.length
n == students[i].length == mentors[j].length
1 <= m, n <= 8
students[i][k] is either 0 or 1.
mentors[j][k] is either 0 or 1.
"""

class Solution:
    def maxCompatibilitySum(self, students: List[List[int]], mentors: List[List[int]]) -> int:
        num_pairs = len(students)
        visited = [False] * num_pairs
        compatibility = [[0] * num_pairs for _ in range(num_pairs)]

        def dfs(index, total_score):
            nonlocal max_score
            if index == num_pairs:
                max_score = max(max_score, total_score)
                return
            for j in range(num_pairs):
                if not visited[j]:
                    visited[j] = True
                    dfs(index + 1, total_score + compatibility[index][j])
                    visited[j] = False
        
        #calculate compatibility scores for each student-mentor pair
        for i in range(num_pairs):
            for j in range(num_pairs):
                compatibility[i][j] = sum(x == y for x, y in zip(students[i], mentors[j]))
        
        max_score = 0
        dfs(0, 0)
        return max_score