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In this problem, you are tasked with finding the losers of a circular game. The game is played in a circular arrangement, and each round, we eliminate participants in a specific manner.

Given the total number of participants n, and a step value k which determines how participants are eliminated, continue eliminating participants every k-th turn until only one participant remains. You need to return a list of participants who were eliminated in the order they were eliminated.

Clarifying Questions

1. Input Constraints: Can the number of participants (n) and the step value (k) be large numbers?
2. Type of Data: Are the inputs guaranteed to be integers?
3. Output Order: Should the output list the eliminated participants in the order they were eliminated?
4. Special Cases: What should be returned if n is 1? (Since there’s only one participant, no one should be eliminated).

Assumptions

• n is a positive integer greater than 0.
• k is a positive integer greater than 0.
• The participants are numbered from 1 to n.

Strategy

1. Circular Elimination: Use a list to simulate the circle of participants.
2. Index Calculation: Use modular arithmetic to efficiently handle the circular nature of the game.
3. Elimination Simulation: Use a loop to go through and eliminate participants, keeping track of the current index and removing participants until only one remains.
4. Track and Store: Store the eliminated participants in a list in the order they were eliminated.

Algorithm

1. Create a list of participants from 1 to n.
2. Use a variable to keep track of the current index in the participant list.
3. Iterate until the list of participants is reduced to one element:
   • Calculate the index of the participant to be eliminated using the step value k.
   • Remove the participant from the list and append them to the result list.
4. Return the result list containing eliminated participants in order.

Code Implementation

def findLosers(n, k):
    # Step 1: Initialize the participants list
    participants = list(range(1, n + 1))
    eliminated = []
    index = 0

    # Step 2: Eliminate participants until only one remains
    while len(participants) > 1:
        # Calculate the index of the participant to eliminate
        index = (index + k - 1) % len(participants)
        # Remove the participant and add to eliminated list
        eliminated.append(participants.pop(index))

    # Return the list of eliminated participants
    return eliminated

# Example usage:
n = 5
k = 2
print(findLosers(n, k))  # Outputs the order in which participants are eliminated

Time Complexity

• Initialization: Creating the initial list of participants takes O(n).
• Loop for Elimination:
  • Each iteration removes one participant: O(1) for each removal in average case but O(n) worst case if implemented using a naive list.
  • Total iterations: n - 1 to eliminate participants.
  • So, total time in naive list removal is O(n^2) in the worst case.

The algorithm is effective and clear for smaller values of n. For larger values, an optimized structure (like a linked list) could be used to improve complexity but isn’t shown here for simplicity.

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