Best Team With No Conflicts

Best Team With No Conflicts

What is this problem about?

The "Best Team With No Conflicts interview question" is an optimization problem where you must select a subset of players to form a team. Each player has a score and an age. The "conflict" rule is: you cannot have a younger player with a strictly higher score than an older player. Your goal is to maximize the total sum of scores of the players in your chosen team.

Why is this asked in interviews?

Morgan Stanley asks the "Best Team With No Conflicts coding problem" to test a candidate's ability to combine Sorting and Dynamic Programming. It is essentially a variation of the "Longest Increasing Subsequence" (LIS) problem but applied to a 2D constraint. It evaluates whether a candidate can transform a complex problem into a standard algorithmic pattern.

Algorithmic pattern used

This problem is a variation of Longest Increasing Subsequence (LIS).

1. Sort: First, sort the players. The primary sort key should be Age. If ages are equal, sort by Score. Sorting by age simplifies the problem because we only need to worry about the "Score" condition as we move through the list.
2. DP State: dp[i] represents the maximum team score including player $i$ as the last (highest-scoring or oldest) player.
3. Transition: For each player $i$, check all previous players $j < i$. If scores[j] <= scores[i], then player $i$ can be added to the team ending at $j$.
   • dp[i] = max(dp[i], dp[j] + scores[i]).
4. Result: The maximum value in the dp array is the answer.

Example explanation

Players: (Age: 10, Score: 5), (Age: 10, Score: 10), (Age: 11, Score: 8)

1. Sorted: (10, 5), (10, 10), (11, 8)
2. dp[0] = 5.
3. dp[1] (Score 10): Can follow (10, 5) because $5 \leq 10$. `dp[1] = 5 + 10 = 15$.
4. dp[2] (Score 8):
   • Can follow (10, 5) because $5 \leq 8$. Total: $5+8=13$.
   • CANNOT follow (10, 10) because $10 > 8$ and age 10 is younger than age 11. Max Score: 15.

Common mistakes candidates make

• Wrong Sorting Order: Forgetting that if ages are the same, you must sort by score to ensure the DP transition works correctly for players of the same age.
• $O(N^2)$ DP on large $N$: While LIS is $O(N^2)$, if $N$ were $10^5$, you would need a Fenwick Tree or Segment Tree to optimize the DP to $O(N \log N)$.
• Misinterpreting the Conflict: Thinking the conflict is about older players having higher scores, when it's actually about younger players having strictly higher scores.

Interview preparation tip

Always look for "Subsequence" properties in subset optimization problems. Sorting is almost always the first step for problems involving two independent variables (like age and score). This is a foundational "Dynamic Programming" technique.