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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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

✓ Two Pass Hash

⏱️ Time: N/A Space: N/A

Optimized

⏱️ Time: N/A Space: N/A

Binary Lifting with Prefix Sums

⏱️ Time: O(n log k) Space: O(n log k)

Precompute where each player goes after 2^j passes and the sum of indices for that path. Use binary representation of k to combine these precomputed jumps efficiently.

Brute Force Simulation

⏱️ Time: O(n × k) Space: O(1)

For each possible starting player (0 to n-1), simulate the ball passing game for exactly k passes. Follow the receiver chain and sum all player indices encountered.

Algorithm Steps — Algorithm Steps

Code -

solution.c — C

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

long long getMaxScore(int* receiver, int n, int k) {
    long long maxScore = 0;
    
    // Try each starting position
    for (int start = 0; start < n; start++) {
        long long score = 0;
        int current = start;
        
        // Simulate k passes
        for (int pass = 0; pass <= k; pass++) {
            score += current;
            if (pass < k) {
                current = receiver[current];
            }
        }
        
        if (score > maxScore) {
            maxScore = score;
        }
    }
    
    return maxScore;
}

void parseArray(const char* str, int* arr, int* size) {
    *size = 0;
    const char* p = str;
    
    // Find opening bracket
    while (*p && *p != '[') p++;
    if (*p == '[') p++;
    
    // Parse numbers
    while (*p && *p != ']') {
        // Skip whitespace and commas
        while (*p == ' ' || *p == ',') p++;
        if (*p == ']' || *p == '\0') break;
        
        // Parse number
        char* endp;
        arr[(*size)++] = (int)strtol(p, &endp, 10);
        p = endp;
    }
}

int main() {
    char line1[10000], line2[10000];
    
    // Read receiver array
    if (!fgets(line1, sizeof(line1), stdin)) return 1;
    
    // Read k
    if (!fgets(line2, sizeof(line2), stdin)) return 1;
    
    // Parse receiver array
    static int receiver[1000];
    int n = 0;
    parseArray(line1, receiver, &n);
    
    // Parse k
    int k = atoi(line2);
    
    // Calculate result
    long long result = getMaxScore(receiver, n, k);
    printf("%lld\n", result);
    
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

⚡ Linearithmic

Space Complexity

⚡ Linearithmic Space

25.8K Views

Medium Frequency

~35 min Avg. Time

890 Likes

Ln 1, Col 1

Smart Actions

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