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Maximize Value of Function in a Ball Passing Game

Difficulty: Hard

Problem Description

You are given an integer array receiver of length n and an integer k. n players are playing a ball-passing game. You choose the starting player, i. The game proceeds as follows: player i passes the ball to player receiver[i], who then passes it to receiver[receiver[i]], and so on, for k passes in total. The game's score is the sum of the indices of the players who touched the ball, including repetitions. Return the maximum possible score.

Key Insights

  • The ball passing sequence can form cycles, making it essential to identify these cycles to optimize the score.
  • For each possible starting player, we can compute the score for k passes, but due to the constraints, we need to avoid direct simulation for large k.
  • Using properties of cycles and prefix sums allows us to compute the maximum score efficiently.

Space and Time Complexity

Time Complexity: O(n + m), where n is the number of players and m is the length of the cycle. Space Complexity: O(n), used for storing visited players and cycle information.

Solution

To solve the problem, we can use a combination of cycle detection and prefix sums:

  1. Iterate through each player to find the passing sequence and identify cycles.
  2. For each starting player, simulate the passing while keeping track of the score.
  3. If we encounter a cycle, calculate the contribution of the cycle based on how many more passes remain.
  4. Keep track of the maximum score encountered during the simulations.

Code Solutions

def maxScore(receiver, k):
    n = len(receiver)
    max_score = 0

    for start in range(n):
        visited = {}
        current = start
        score = 0
        passes = 0
        
        while passes < k and current not in visited:
            visited[current] = passes
            score += current
            current = receiver[current]
            passes += 1

        if passes < k:  # Cycle detected
            cycle_length = passes - visited[current]
            cycle_sum = sum(receiver[i] for i in visited if visited[i] >= visited[current])
            remaining_passes = k - passes

            # Calculate contributions from the cycle
            score += (remaining_passes // cycle_length) * cycle_sum
            score += sum(receiver[i] for i in visited if visited[i] >= visited[current]) * (remaining_passes % cycle_length)

        max_score = max(max_score, score)

    return max_score
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