Maximize Value of Function in a Ball Passing Game

Maximize Value of Function in a Ball Passing Game - Problem

You are given an integer array receiver of length n and an integer k. There are n players 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.

The score formula is: i + receiver[i] + receiver[receiver[i]] + ... + receiver^(k)[i]

Return the maximum possible score.

Note: receiver may contain duplicates and receiver[i] may be equal to i.

Input & Output

Example 1 — Basic Case

$ Input: receiver = [2,0,1], k = 4

› Output: 6

💡 Note: Starting from player 1: 1→0→2→1→0. Score = 1+0+2+1+0 = 4. Starting from player 2: 2→1→0→2→1. Score = 2+1+0+2+1 = 6. Maximum is 6.

Example 2 — Self-Loop

$ Input: receiver = [1,1,1,2,3], k = 3

› Output: 10

💡 Note: Starting from player 4: 4→3→2→1. Score = 4+3+2+1 = 10. This gives the maximum score since we start from the highest index and move to progressively smaller indices.

Example 3 — Small k

$ Input: receiver = [1,1], k = 1000

› Output: 1001

💡 Note: Starting from player 1: stays at 1 for all 1001 positions (including start). Score = 1 × 1001 = 1001.

Constraints

1 ≤ receiver.length ≤ 10⁵
0 ≤ receiver[i] ≤ n - 1
1 ≤ k ≤ 10¹⁰

Visualization

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Understanding the Visualization

Input

Array receiver=[2,0,1] and k=4 passes

Process

Try different starting positions and simulate ball passing

Output

Return maximum score from all possible starting positions

Key Takeaway

🎯 Key Insight: Binary lifting allows us to handle k up to 10^10 by precomputing jumps in powers of 2

Asked in

G Google 25 M Meta 18 a Amazon 15

The key insight is to use binary lifting to handle very large k values efficiently. For small k, simulation works, but for k up to 10^10, we need binary lifting with prefix sums. Best approach is binary lifting with Time: O(n log k), Space: O(n log k).

Common Approaches

Approach	Time	Space	Notes
✓ Brute Force Simulation	O(n × k)	O(1)	Try every starting position and simulate k passes step by step
Binary Lifting with Prefix Sums	O(n log k)	O(n log k)	Use binary lifting to handle large k efficiently by precomputing jumps

Brute Force Simulation — Algorithm Steps

Try each starting player i from 0 to n-1
For each start, simulate k passes following receiver[current]
Sum all player indices during simulation
Track maximum score across all starting positions

Visualization

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Step-by-Step Walkthrough

Try Start 0

Simulate k passes from position 0

Try Start 1

Simulate k passes from position 1

Maximum

Return the highest score found

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>
#include <string.h>

long long solution(int* receiver, int n, int k) {
    long long maxScore = 0;
    
    for (int start = 0; start < n; start++) {
        long long score = 0;
        int current = start;
        
        for (int i = 0; i <= k; i++) {  // k+1 because we include starting player
            score += current;
            current = receiver[current];
        }
        
        if (score > maxScore) {
            maxScore = score;
        }
    }
    
    return maxScore;
}

int main() {
    char line[10000];
    fgets(line, sizeof(line), stdin);
    
    // Parse array manually
    int receiver[100000];
    int n = 0;
    char* token = strtok(line + 1, ",]");
    while (token != NULL) {
        receiver[n++] = atoi(token);
        token = strtok(NULL, ",]");
    }
    
    int k;
    scanf("%d", &k);
    
    long long result = solution(receiver, n, k);
    printf("%lld\n", result);
    
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(n × k)

For each of n starting positions, we simulate k passes

✓ Linear Growth

Space Complexity

O(1)

Only using constant extra variables

✓ Linear Space

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Input & Output

Constraints

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Related Problems

Common Approaches

Brute Force Simulation — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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