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Maximum Compatibility Score Sum

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.

Solution Explanation: Maximum Compatibility Score Sum

This problem involves finding the optimal pairing between students and mentors to maximize the total compatibility score. The compatibility score between a student and a mentor is the number of matching answers in their respective surveys. The solution presented uses a backtracking approach with preprocessing for efficiency.

Approach: Preprocessing + Backtracking

  1. Preprocessing:

    • The solution starts by calculating and storing the compatibility score between each student and each mentor. This is done using nested loops, iterating through all student-mentor pairs and comparing their answers question by question. The result is a 2D array g where g[i][j] represents the compatibility score between student i and mentor j. This preprocessing step avoids redundant calculations during the backtracking phase.

  2. Backtracking (DFS):

    • A recursive Depth-First Search (DFS) function dfs(i, s) is used to explore all possible pairings.
    • i represents the index of the student currently being assigned a mentor.
    • s represents the accumulated compatibility score up to the current assignment.
    • The base case is when i reaches the total number of students (m). This indicates that all students have been assigned, and the current accumulated score s is compared to the global maximum ans, updating ans if s is greater.
    • In each recursive call, the function iterates through all mentors. A boolean array vis is used to track which mentors have already been assigned. If a mentor j is available (!vis[j]), it's temporarily assigned to the current student, the compatibility score g[i][j] is added to the total score s, and the function recursively calls itself to assign the next student (dfs(i + 1, s + g[i][j])). After the recursive call, the mentor j is marked as unassigned (vis[j] = false) to explore other possible assignments.

  3. Result:

    • The function dfs(0, 0) is called to initiate the backtracking process, starting with the first student and an initial score of 0. The final value of ans after all possible pairings have been explored represents the maximum compatibility score sum.

Time and Space Complexity Analysis

  • Time Complexity: The dominant factor in the time complexity is the backtracking process. In the worst case, the algorithm explores all possible permutations of assigning mentors to students, which is O(m!), where 'm' is the number of students (and mentors).

  • Space Complexity: The space complexity is determined by the following:

    • g: A 2D array of size m x m to store precomputed compatibility scores. O(m^2).
    • vis: A boolean array of size m to keep track of assigned mentors. O(m).
    • Recursive call stack: In the worst case, the depth of the recursion is m, leading to O(m) space usage on the call stack.

Therefore, the overall space complexity is O(m^2), dominated by the precomputed compatibility scores array.

Code Examples (Python3, Java, C++, Go, TypeScript, Rust, JavaScript)

The code examples provided in the original response are accurate representations of the solution in various programming languages. They effectively implement the preprocessing and backtracking steps as described above. The slight variations in syntax are due to differences between the languages but the underlying algorithms remain consistent across all implementations.