Prison Escape States: Algorithm, Correctness, and Complexity

Prison Escape States: An O(n) Algorithm

This article explores an algorithm to count the number of possible escape states in a prison scenario. The prison has 'N' rooms, each housing a prisoner who follows one of 'M' religions. Escape is possible if adjacent rooms have prisoners of the same religion.

1. Algorithm Description and Pseudo-code

Natural Language:

• Begin by assuming at least one escape state exists. This is because even a single room can represent a possible escape state.
• Iterate through each room, starting from the second room (room number 2).
• Check if the current room's religion differs from the previous room's religion. If so, increment the escape state count by 1.
• Continue this process until the last room is reached.
• The final count of escape states is the result.

Pseudo-code:

function countEscapeStates(N, M):
    escape = 1
    prevReligion = randomReligion() // Assuming a function to generate a random religion between 1 to M
    
    for i = 2 to N:
        currentReligion = randomReligion()
        if currentReligion != prevReligion:
            escape = escape + 1
        prevReligion = currentReligion
    
    return escape

2. Proof of Correctness

The algorithm's core principle is counting religion changes between adjacent rooms. If the religions are different, an escape state is possible. Therefore, incrementing 'escape' by 1 for each religion change accurately reflects the number of possible escape states.

3. Complexity Analysis

Time Complexity: The algorithm iterates through all 'N' rooms exactly once. The time complexity of generating a random religion ('randomReligion()') is considered constant. Therefore, the overall time complexity is O(n).

Space Complexity: The algorithm only requires a constant amount of extra space to store the 'escape' variable. This makes the space complexity O(1).