Understanding the Josephus Problem: A Comprehensive Guide

Josephus Problem Explained 🎯

There are N people standing in a circle waiting to be executed. The counting out begins at some point in the circle and proceeds around the circle in a fixed direction. In each step, a certain number of people are skipped and the next person is executed. The elimination proceeds around the circle (which is becoming smaller and smaller as the executed people are removed), until only the last person remains, who is given freedom.

Given the total number of persons N and a number k which indicates that k-1 persons are skipped and the kth person is killed in a circle. The task is to choose the person in the initial circle that survives.

Examples:

Input: N = 5 and k = 2
Output: 3
Explanation: Firstly, the person at position 2 is killed,
then the person at position 4 is killed, then the person at position 1 is killed.
Finally, the person at position 5 is killed. So the person at position 3 survives.

Input: N = 7 and k = 3
Output: 4
Explanations: The persons at positions 3, 6, 2, 7, 5, and 1 are killed in order,
and the person at position 4 survives.

Brute Force Approach

To solve this problem using a brute-force approach, we can simulate the game step by step. This involves maintaining a list of friends and eliminating every (k)-th friend until only one friend remains. Here’s how we can implement this:

1. Initialize the List: Create a list of friends numbered from 1 to (n).
2. Simulate the Elimination: Start from the first friend and count (k) friends in the clockwise direction, wrapping around the list if necessary.
3. Eliminate the Friend: Remove the (k)-th friend from the list.
4. Repeat: Continue the process with the next friend immediately clockwise of the eliminated friend.
5. Stop: When only one friend is left, they are the winner.

Explanation:

1. Initialization:

   • friends is a list of integers from 1 to (n) representing the friends.
   • index keeps track of the current position in the circle (0-based index).

2. Elimination Loop:

   • The loop runs until only one friend remains in the list.
   • In each iteration:
     • Calculate the index of the friend to eliminate: (index + k - 1) % len(friends). This ensures that the counting wraps around the list correctly.
     • Remove the friend at the calculated index from the list.
     • The index automatically points to the next friend (because removing an element shifts all elements to the left).

3. Winner:

   • After the loop, the only remaining element in the friends list is the winner.

Example Walkthrough ((n = 5), (k = 2)):

1. Initial list: [1, 2, 3, 4, 5], start at index = 0.
2. Eliminate friend at (0 + 2 - 1) % 5 = 1, list becomes [1, 3, 4, 5], next start at index = 1.
3. Eliminate friend at (1 + 2 - 1) % 4 = 2, list becomes [1, 3, 5], next start at index = 2.
4. Eliminate friend at (2 + 2 - 1) % 3 = 0, list becomes [3, 5], next start at index = 0.
5. Eliminate friend at (0 + 2 - 1) % 2 = 1, list becomes [3].

Thus, the winner is friend number 3.

Time Complexity (TC)

The brute-force approach has a time complexity that can be analyzed as follows:

1. Initial List Creation: Creating the list of friends takes (O(n)) time.

2. Elimination Process:

   • In each iteration, we need to find the (k)-th friend to eliminate. This involves calculating the next index and removing the element from the list.
   • Calculating the next index is an (O(1)) operation.
   • Removing an element from the list takes (O(n)) time in the worst case because it involves shifting elements in the list.
   • Since we perform the removal operation (n-1) times (once for each eliminated friend), the total time complexity for the elimination process is (O(n \times n) = O(n^2)).

Therefore, the overall time complexity of the brute-force approach is (O(n + n^2) = O(n^2)).

Space Complexity (SC)

The space complexity can be analyzed as follows:

1. Space for the List: We maintain a list of friends, which requires (O(n)) space.
2. Auxiliary Space: No additional auxiliary space is required beyond the input and the list of friends.

Therefore, the overall space complexity is (O(n)).

Summary

• Time Complexity: (O(n^2))
• Space Complexity: (O(n))

These complexities indicate that the brute-force approach is feasible for smaller values of (n), but may become inefficient for larger values due to the quadratic time complexity.

Code

// package dailychallenge;

import java.util.ArrayList;
import java.util.stream.Collectors;
import java.util.stream.IntStream;

class Solution {
    public static int findTheWinner(int n, int k) {
        ArrayList<Integer> friends = IntStream.range(1, n + 1).boxed().collect(Collectors.toCollection(ArrayList::new));
        int idx = 0;
        while (friends.size() > 1) {
            idx = (idx + k - 1) % friends.size();
            friends.remove(idx);
        }
        return friends.get(0);
    }

    public static void main(String[] args) {
        int n = 6, k = 5;
        int theWinner = findTheWinner(n, k);
        System.out.println(theWinner);

    }
}

---

Optimized Approach

To solve this problem optimally, we can use the mathematical approach known as the Josephus problem, which has a well-known efficient solution. The optimal approach leverages the recursive formula of the Josephus problem to find the winner in (O(n)) time and (O(1)) space.

Josephus Problem Recurrence Relation

The Josephus problem can be defined recursively:
[
J(n, k) = (J(n-1, k) + k) \% n
]
where (J(n, k)) is the position of the winner in a circle of size (n) with every (k)-th person being eliminated, and the base case is:
[
J(1, k) = 0
]

Conversion to Iterative Approach

We can convert the recursive relation to an iterative approach to avoid the overhead of recursion and achieve an (O(n)) time complexity.

1. Initialize the Winner's Position: Start with (J(1, k) = 0) for one person.
2. Iterate and Update: Use the recurrence relation iteratively to update the winner's position for increasing sizes of the circle.
3. Adjust for 1-Based Index: The problem requires a 1-based index, so we convert the 0-based result to a 1-based result by adding 1 at the end.

Python Code for the Optimal Solution

def find_the_winner(n, k):
    winner = 0  # Base case: when there's only one person, they are the winner (0-based index).
    for i in range(2, n + 1):  # Iterate from 2 to n
        winner = (winner + k) % i  # Update the winner's position
    return winner + 1  # Convert from 0-based index to 1-based index

# Example usage:
n = 5
k = 2
print(find_the_winner(n, k))  # Output: 3

Explanation

1. Initialization: Start with winner = 0 (0-based index for (n = 1)).
2. Iterate: For each (i) from 2 to (n):
   • Update the winner's position using the recurrence relation: winner = (winner + k) % i.
   • This step ensures that the position of the winner is correctly computed as the circle grows.
3. Convert to 1-Based Index: After the loop, convert the result to a 1-based index by returning winner + 1.

Complexity Analysis

• Time Complexity: The loop runs (n-1) times (from 2 to (n)), and each iteration involves a constant-time operation. Therefore, the time complexity is (O(n)).
• Space Complexity: The space complexity is (O(1)) because we only use a few variables regardless of the input size.

Example Walkthrough ((n = 5), (k = 2))

1. Initial winner: winner = 0 (for (n = 1)).
2. (i = 2): winner = (0 + 2) % 2 = 0
3. (i = 3): winner = (0 + 2) % 3 = 2
4. (i = 4): winner = (2 + 2) % 4 = 0
5. (i = 5): winner = (0 + 2) % 5 = 2

Convert to 1-based index: 2 + 1 = 3. Thus, the winner is friend number 3.

The loop starts from 2 because we are building up the solution incrementally from the smallest case. Here’s a detailed explanation:

Josephus Problem Explanation

The Josephus problem can be thought of as eliminating every ( k )-th person in a circle until only one person remains. The position of the winner for ( n ) people can be derived from the position of the winner for ( n-1 ) people.

Recursive Relation

The recursive relation for the Josephus problem is:
[
J(n, k) = (J(n-1, k) + k) \% n
]
This means:

• ( J(n, k) ) is the position of the winner in a circle of size ( n ).
• ( J(n-1, k) ) is the position of the winner in a circle of size ( n-1 ).

Base Case

For ( n = 1 ) (a single person), the position of the winner is trivially 0 (0-based index):
[
J(1, k) = 0
]

Iterative Approach

The iterative approach uses the recursive relation to build up the solution from the base case. Starting from the base case for ( n = 1 ), we iteratively compute the position of the winner for larger values of ( n ).

Why Start the Loop from 2?

When ( n = 1 ), the winner is known to be at position 0. We use this as our starting point. The loop then iterates from 2 to ( n ) to compute the winner for larger circles:

1. Base Case: Initialize winner to 0 for ( n = 1 ).
2. Iterate from 2 to ( n ): Update the winner's position using the recurrence relation.

Here’s the code with detailed comments:

def find_the_winner(n, k):
    winner = 0  # Base case: for n = 1, the winner is at position 0 (0-based index).
    for i in range(2, n + 1):  # Iterate from 2 to n to build up the solution
        winner = (winner + k) % i  # Update the winner's position using the recurrence relation
    return winner + 1  # Convert from 0-based index to 1-based index

# Example usage:
n = 5
k = 2
print(find_the_winner(n, k))  # Output: 3

Detailed Example Walkthrough

For ( n = 5 ), ( k = 2 ):

1. Initialization: Start with winner = 0 (for ( n = 1 )).

2. Iteration:

   • For ( i = 2 ): [ \text{winner} = (0 + 2) \% 2 = 0 ]
   • For ( i = 3 ): [ \text{winner} = (0 + 2) \% 3 = 2 ]
   • For ( i = 4 ): [ \text{winner} = (2 + 2) \% 4 = 0 ]
   • For ( i = 5 ): [ \text{winner} = (0 + 2) \% 5 = 2 ]

3. Convert to 1-Based Index: 2 + 1 = 3.

Thus, the winner is friend number 3.

Summary

• Loop Starting from 2: The loop starts from 2 because we already know the base case for ( n = 1 ). We build up the solution from this base case to the desired ( n ) using the recurrence relation.
• Efficiency: This approach efficiently computes the winner in ( O(n) ) time and ( O(1) ) space.

Dryrun :

Let's perform a dry run of the recursive solution for ( n = 5 ) and ( k = 2 ).

Dry Run of the Recursive Solution

We use the function josephus_recursive(n, k) to determine the position of the winner in a circle of size ( n ) with every ( k )-th person being eliminated.

Here's the code again for reference:

def josephus_recursive(n, k):
    if n == 1:
        return 0
    else:
        return (josephus_recursive(n - 1, k) + k) % n

def find_the_winner(n, k):
    return josephus_recursive(n, k) + 1

Steps and Recursive Calls

1. Initial Call:

   find_the_winner(5, 2)

This calls josephus_recursive(5, 2).

1. Recursive Calls:

   • josephus_recursive(5, 2) calls josephus_recursive(4, 2).
   • josephus_recursive(4, 2) calls josephus_recursive(3, 2).
   • josephus_recursive(3, 2) calls josephus_recursive(2, 2).
   • josephus_recursive(2, 2) calls josephus_recursive(1, 2).

2. Base Case:

   • josephus_recursive(1, 2) returns 0 because the base case is reached (( n = 1 )).

3. Unwinding the Recursion:

   • josephus_recursive(2, 2): [ (josephus_recursive(1, 2) + 2) \% 2 = (0 + 2) \% 2 = 0 ] Returns 0.
   • josephus_recursive(3, 2): [ (josephus_recursive(2, 2) + 2) \% 3 = (0 + 2) \% 3 = 2 ] Returns 2.
   • josephus_recursive(4, 2): [ (josephus_recursive(3, 2) + 2) \% 4 = (2 + 2) \% 4 = 0 ] Returns 0.
   • josephus_recursive(5, 2): [ (josephus_recursive(4, 2) + 2) \% 5 = (0 + 2) \% 5 = 2 ] Returns 2.

4. Convert to 1-Based Index:

   • find_the_winner(5, 2): [ josephus_recursive(5, 2) + 1 = 2 + 1 = 3 ] Returns 3.

Summary of Recursive Calls

• josephus_recursive(5, 2) returns 2.
• Convert the 0-based index to a 1-based index: ( 2 + 1 = 3 ).

The winner is friend number 3.

Step-by-Step Visualization

1. ( n = 1 ):

   • Returns 0.

2. ( n = 2 ):

   • ( (0 + 2) \% 2 = 0 )
   • Returns 0.

3. ( n = 3 ):

   • ( (0 + 2) \% 3 = 2 )
   • Returns 2.

4. ( n = 4 ):

   • ( (2 + 2) \% 4 = 0 )
   • Returns 0.

5. ( n = 5 ):

   • ( (0 + 2) \% 5 = 2 )
   • Returns 2.

Convert the result to 1-based index: ( 2 + 1 = 3 ).

Thus, the winner is friend number 3.