#1626

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Best Team With No Conflicts

Find the highest score basketball team by choosing players without conflicts in age and score using dynamic programming.

array dynamic programming sorting

Problem Statement

You manage a basketball team and aim to select a team with the highest total score. Each player's score and age are provided, but there is a conflict if a younger player has a strictly higher score than an older player. A conflict does not exist between players of the same age.

Given two lists of integers, scores and ages, representing the score and age of each player, return the maximum possible total score of a valid team where no conflicts arise between the selected players.

Examples

Example 1

Input: scores = [1,3,5,10,15], ages = [1,2,3,4,5]

Output: 34

You can choose all the players.

Example 2

Input: scores = [4,5,6,5], ages = [2,1,2,1]

Output: 16

It is best to choose the last 3 players. Notice that you are allowed to choose multiple people of the same age.

Example 3

Input: scores = [1,2,3,5], ages = [8,9,10,1]

Output: 6

It is best to choose the first 3 players.

Constraints

1 <= scores.length, ages.length <= 1000
scores.length == ages.length
1 <= scores[i] <= 106
1 <= ages[i] <= 1000

Solution Approach

Sort the Players by Age and Score

Start by sorting the players by age in ascending order. If two players have the same age, sort them by their score in descending order. This sorting step ensures that younger players are considered first, and ties in age are broken by score.

Dynamic Programming to Maximize Score

Use dynamic programming to compute the maximum possible score. Initialize a dp array where each element represents the maximum score achievable by selecting players up to that point. Iterate through the sorted list and calculate the maximum score by considering each player while maintaining the no-conflict rule.

Track the Best Team Combination

As you iterate through the sorted players, for each player, check all previous players to see if they can be included without conflict (i.e., the older players should have a score equal to or greater than the younger players). Update the dp array accordingly to track the best team combination.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity is O(n^2) due to the nested loop in dynamic programming, where n is the number of players. Sorting the players takes O(n log n), which is the dominant factor in this approach. The space complexity is O(n), where n is the number of players, due to the dp array used for storing intermediate results.

What Interviewers Usually Probe

Assess if the candidate recognizes the need to sort players to avoid conflicts.
Check if the candidate can explain dynamic programming in the context of maximizing the score.
Evaluate how the candidate handles edge cases, such as when all players are of the same age or score.

Common Pitfalls or Variants

Common pitfalls

Forgetting to sort the players correctly, which could lead to conflicts and incorrect results.
Using a greedy approach instead of dynamic programming, missing the optimal team combination.
Not considering edge cases like players of the same age or when no valid team can be formed.

Follow-up variants

Consider additional constraints where the number of players is limited or when there is an upper bound on the team size.
Change the problem to allow conflicts if the difference in scores is small.
Introduce other constraints, such as a fixed number of players that must be selected.

FAQ

What is the main approach to solving the Best Team With No Conflicts problem?

The problem is solved using dynamic programming. First, sort the players by age and score, then use dynamic programming to find the maximum score by considering each player in order.

How do I avoid conflicts between players when selecting a team?

To avoid conflicts, sort players by age, and if two players have the same age, sort them by their score. Then, use dynamic programming to ensure the team selection adheres to the no-conflict rule.

What are the time and space complexities of the solution?

The time complexity is O(n^2), where n is the number of players, due to the nested loops in dynamic programming. The space complexity is O(n) due to the dp array.

Can the problem be solved with a greedy approach?

A greedy approach is not optimal for this problem, as it may miss the best combination of players. Dynamic programming ensures the optimal solution is found.

What edge cases should I consider when solving this problem?

Consider edge cases like all players having the same age or score, or when no valid team can be formed due to conflicts between all players.

terminal

Solution

Solution 1

#### Python3

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class Solution:
    def bestTeamScore(self, scores: List[int], ages: List[int]) -> int:
        arr = sorted(zip(scores, ages))
        n = len(arr)
        f = [0] * n
        for i, (score, age) in enumerate(arr):
            for j in range(i):
                if age >= arr[j][1]:
                    f[i] = max(f[i], f[j])
            f[i] += score
        return max(f)

Solution 2

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class Solution:
    def bestTeamScore(self, scores: List[int], ages: List[int]) -> int:
        arr = sorted(zip(scores, ages))
        n = len(arr)
        f = [0] * n
        for i, (score, age) in enumerate(arr):
            for j in range(i):
                if age >= arr[j][1]:
                    f[i] = max(f[i], f[j])
            f[i] += score
        return max(f)

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