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Best Team With No Conflicts

You are the manager of a basketball team. For the upcoming tournament, you want to choose the team with the highest overall score. The score of the team is the sum of scores of all the players in the team.

However, the basketball team is not allowed to have conflicts. A conflict exists if a younger player has a strictly higher score than an older player. A conflict does not occur between players of the same age.

Given two lists, scores and ages, where each scores[i] and ages[i] represents the score and age of the ith player, respectively, return the highest overall score of all possible basketball teams.

 

Example 1:

Input: scores = [1,3,5,10,15], ages = [1,2,3,4,5]
Output: 34
Explanation: You can choose all the players.

Example 2:

Input: scores = [4,5,6,5], ages = [2,1,2,1]
Output: 16
Explanation: 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
Explanation: 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

1626. Best Team With No Conflicts

This problem asks to find the highest overall score of a basketball team, given the scores and ages of players, with the constraint that no younger player can have a strictly higher score than an older player.

Approach 1: Dynamic Programming

This approach uses dynamic programming to efficiently compute the maximum score.

  1. Sort: We first sort the players by score in ascending order. If two players have the same score, we sort them by age in ascending order. This sorting helps to ensure that we consider players in a way that respects the age constraint.

  2. DP Array: A DP array f is created, where f[i] stores the maximum team score achievable considering players up to index i.

  3. Iteration: The algorithm iterates through the sorted players. For each player i:

    • It considers including the current player in the team. To do this it checks previous players (j from 0 to i-1). If the current player's age is greater than or equal to the previous player's age, and the current player's score is greater than or equal to the previous player's score (to avoid conflicts), it updates f[i] with the maximum between its current value and the maximum score achievable up to player j plus the current player's score (f[j] + score[i]). This ensures that we are building a team with no conflicts.
    • Otherwise, it means that there was a player j that is younger than i but had a higher score, violating the condition. Then we will simply choose to not include that player i.

  4. Result: The maximum value in the f array represents the highest overall score attainable.

Time Complexity: O(N^2), where N is the number of players. This is due to the nested loops in the dynamic programming step.

Space Complexity: O(N) to store the DP array f.

Code: (Python, Java, C++, Go, TypeScript, JavaScript) [See the code in the previous response]

Approach 2: Binary Indexed Tree (Fenwick Tree)

This approach uses a Binary Indexed Tree (BIT) to optimize the dynamic programming step from O(N^2) to O(N log N).

  1. Sort: Same as Approach 1, sort players by score (ascending) and then age (ascending).

  2. BIT: A Binary Indexed Tree is used to efficiently query the maximum score achievable for players with ages up to a certain value.

  3. Iteration: Iterate through the sorted players. For each player, update the BIT with the maximum score achievable including this player and the players before it with no conflicts. This update involves a query to find the best score among previous players and an update to the BIT for the player's age.

  4. Result: After iterating through all players, the maximum value in the BIT represents the best team score.

Time Complexity: O(N log M), where N is the number of players and M is the maximum age. The sorting takes O(N log N), and the BIT operations take O(N log M).

Space Complexity: O(M) for the Binary Indexed Tree.

Code: (Python, Java, C++, Go) [See the code in the previous response]

Note: Approach 2 provides a performance improvement over Approach 1, especially when the number of players is large and the range of ages is relatively small. However, Approach 1 is often simpler to understand and implement. The choice of approach depends on the specific constraints and performance requirements.