What is the time complexity of Find the Child Who Has the Ball After K Seconds?

The optimal solution has a time complexity of O(1).

What data structures are used to solve Find the Child Who Has the Ball After K Seconds?

This problem uses Math, Simulation. Understanding these concepts is essential for solving this Easy problem efficiently.

What are the key insights for solving Find the Child Who Has the Ball After K Seconds?

The pattern of ball passing repeats after a certain period. This period is 2 * (n - 1) because it takes (n-1) seconds to reach the end from the beginning and another (n-1) seconds to return to the beginning. We can use the modulo operator (%) to find the remainder after dividing k by the period. This remainder represents the effective number of seconds within a single cycle of passing.

Which companies ask LeetCode 3178. Find the Child Who Has the Ball After K Seconds in interviews?

This problem is frequently asked at Agoda.

JavaScript is required for the full interactive experience. However, all content is accessible without JavaScript.

Find the Child Who Has the Ball After K Seconds - LeetCode 3178 Solution

Find the Child Who Has the Ball After K Seconds - Complete Solution Guide

Find the Child Who Has the Ball After K Seconds is LeetCode problem 3178, a Easy level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

You are given two positive integers n and k . There are n children numbered from 0 to n - 1 standing in a queue in order from left to right. Initially, child 0 holds a ball and the direction of passing the ball is towards the right direction. After each second, the child holding the ball passes it to the child next to them. Once the ball reaches either end of the line, i.e. child 0 or child n - 1 , the direction of passing is reversed . Return the number of the child who receives the ball after

Detailed Explanation

The problem simulates passing a ball among children standing in a line. Initially, child 0 has the ball. The ball is passed to the right until it reaches the end, then the direction reverses, and it's passed to the left until it reaches the beginning, and so on. The problem asks to determine which child will hold the ball after a given number of seconds (k). The inputs are the number of children (n) and the number of seconds (k). The output is the index (0-based) of the child holding the ball after k seconds.

Solution Approach

The solution uses a mathematical approach to avoid explicit simulation. It calculates the period of the ball-passing pattern and then uses the modulo operator to determine the effective number of seconds within that cycle. Based on the remainder, it directly calculates the index of the child holding the ball without simulating each second.

Step-by-Step Algorithm

Calculate the period of the pattern: `period = 2 * (n - 1)`
Find the remainder after dividing k by the period: `rem = k % period`
If the remainder is less than n, the ball is still in the first half of the cycle (passing to the right), and the child's index is simply the remainder.
If the remainder is greater than or equal to n, the ball is in the second half of the cycle (passing to the left). The calculation `n - 1 - (rem - (n - 1))` determines the child's index in this case.

Key Insights

The pattern of ball passing repeats after a certain period. This period is 2 * (n - 1) because it takes (n-1) seconds to reach the end from the beginning and another (n-1) seconds to return to the beginning.
We can use the modulo operator (%) to find the remainder after dividing k by the period. This remainder represents the effective number of seconds within a single cycle of passing.
Handling the direction change efficiently is key. Once we understand the cycle, a simple calculation can determine the child's index.

Complexity Analysis

Time Complexity: O(1)

Space Complexity: O(1)

Topics

This problem involves: Math, Simulation.

Companies

Asked at: Agoda.

Find the Child Who Has the Ball After K Seconds | LeetCode 3178

Easy

Math Simulation

Time: O(1)Space: O(1)

Solution Code

Available in:

class Solution {
    public int numberOfChild(int n, int k) {
        int period = (n - 1) * 2;
        k %= period;
        if (k < n) {
            return k;
        } else {
            return n - (k - (n - 1)) - 1;
        }
    }
}

View Full Solution on GitHubGitHub Solution View on LeetCodeLeetCode

Problem Info

Description

Problem 3178: Find the Child Who Has the Ball After K Seconds - Detailed explanation and solution approaches.

Topics

Math Simulation

Companies

Agoda

Stats

Difficulty:Easy

Languages:4

Companies:1

Problem #:3178

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

4 steps

1
Calculate the period of the pattern: `period = 2 * (n - 1)`
2
Find the remainder after dividing k by the period: `rem = k % period`
3
If the remainder is less than n, the ball is still in the first half of the cycle (passing to the right), and the child's index is simply the remainder.
4
If the remainder is greater than or equal to n, the ball is in the second half of the cycle (passing to the left). The calculation `n - 1 - (rem - (n - 1))` determines the child's index in this case.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(1)

⚡ Constant time - Super fast! Same speed regardless of input size.

💾

Space Complexity

O(1)

💪 Constant space - Uses same memory regardless of input!

📝

Why?

The solution involves only a few arithmetic operations; the number of operations is constant and does not depend on the input values n and k. Therefore, both time and space complexity are constant.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Simulation

Simulate the ball passing for each second until k seconds are elapsed. Keep track of the current child and direction.

Time: O(k)Space: O(1)

Trade-offs: Simpler to understand but less efficient than the mathematical approach, especially for large values of k.

✨

Key Insights

1The pattern of ball passing repeats after a certain period. This period is 2 * (n - 1) because it takes (n-1) seconds to reach the end from the beginning and another (n-1) seconds to return to the beginning.
2We can use the modulo operator (%) to find the remainder after dividing k by the period. This remainder represents the effective number of seconds within a single cycle of passing.
3Handling the direction change efficiently is key. Once we understand the cycle, a simple calculation can determine the child's index.

📚

Key Terms Explained

For Students

Edge Cases

k = 0: The solution correctly returns 0 as child 0 starts with the ball.
k = 1: The solution correctly returns 1, assuming that k=1 means one second is passed.
k = 2*(n-1): The solution correctly returns 0 because the cycle completes, and the ball returns to child 0
n = 2, k = any number: The solution correctly handles the case where there are only two children, oscillating between them.

⚠️

Common Mistakes to Avoid

✗Incorrectly handling the direction reversal: Forgetting to reverse the direction when the ball reaches either end.

✗Not considering the cyclical nature of the ball passing: Trying to solve the problem using a linear approach without noticing the pattern repetition.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Modeling a simple process in a distributed system, where a task is passed between nodes in a linear fashion.

🏦Simulating simple queuing systems with a limited number of resources or processes.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

Use this problem to demonstrate clean code practices. Variable naming and code organization matter.

⚠️ Watch Out

Always test your solution with edge cases: empty input, single element, and boundary values.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: September 27, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

Try solving this without using extra space first. It's a great exercise for understanding the underlying pattern.

@CodeNinjaJul 14

In a real interview, always clarify constraints first. Ask about input size and value ranges.

@TechLeadJul 7

❓

Frequently Asked Questions

How should I approach easy-level algorithm problems?

Easy problems build foundational skills. Focus on: 1) Understanding the problem completely before coding, 2) Writing clean, readable code, 3) Handling all edge cases (empty input, single element, etc.), 4) Analyzing time and space complexity. Don't skip complexity analysis even for simple solutions.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "Find the Child Who Has the Ball After K Seconds"?

For this easy-level Math problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether simulation techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

🤖 Need help with this problem?