2582. Pass the Pillow

There are n people standing in a line labeled from 1 to n. The first person in the line is holding a pillow initially. Every second, the person holding the pillow passes it to the next person standing in the line. Once the pillow reaches the end of the line, the direction changes, and people continue passing the pillow in the opposite direction.

• For example, once the pillow reaches the nth person they pass it to the n - 1th person, then to the n - 2th person and so on.

Given the two positive integers n and time, return the index of the person holding the pillow after time seconds.

Example 1:

Input: n = 4, time = 5
Output: 2
Explanation: People pass the pillow in the following way: 1 -> 2 -> 3 -> 4 -> 3 -> 2.
After five seconds, the 2nd person is holding the pillow.

Example 2:

Input: n = 3, time = 2
Output: 3
Explanation: People pass the pillow in the following way: 1 -> 2 -> 3.
After two seconds, the 3rd person is holding the pillow.

Constraints:

• 2 <= n <= 1000
• 1 <= time <= 1000

Approach 01:

#include <bits/stdc++.h>

using namespace std;

class Solution {
public:
    int passThePillow(int num_people, int time_elapsed) {
        // Calculate the effective time within one complete cycle of passing the pillow
        int cycle_time = (num_people - 1) * 2;
        int effective_time = time_elapsed % cycle_time;

        // If the effective time is less than the number of people, the pillow is moving forward
        if (effective_time < num_people) {
            return 1 + effective_time;
        }

        // If the effective time is greater than or equal to the number of people, the pillow is moving backward
        return num_people - (effective_time - (num_people - 1));
    }
};

class Solution:
    def passThePillow(self, num_people: int, time_elapsed: int) -> int:
        # Calculate the effective time within one complete cycle of passing the pillow
        cycle_time = (num_people - 1) * 2
        effective_time = time_elapsed % cycle_time

        # If the effective time is less than the number of people, the pillow is moving forward
        if effective_time < num_people:
            return 1 + effective_time

        # If the effective time is greater than or equal to the number of people, the pillow is moving backward
        return num_people - (effective_time - (num_people - 1))

Time Complexity

• Modulo Operation:

  Calculating effective_time using modulo operation takes \( O(1) \) time.

• Conditional Checks:

  Checking the condition if effective_time < num_people and the subsequent operations inside if-else block take \( O(1) \) time.

• Overall Time Complexity:

  The overall time complexity is \( O(1) \) since all operations are constant time operations.

Space Complexity

• Space Usage:

  The algorithm uses a constant amount of extra space for variables cycle_time and effective_time, which takes \( O(1) \) space.

• Overall Space Complexity:

  The overall space complexity is \( O(1) \) as there are no additional data structures used that grow with input size.