Prison Escape States: Counting Escape Possibilities with Different Religions

Calculating Escape States in a Prison with Different Religions

This problem involves a prison with N rooms, each holding a prisoner who follows one of M religions. An escape can occur if two adjacent rooms have prisoners of the same religion. Our goal is to find an efficient algorithm that counts the number of states where an escape is possible.

Algorithm Description

The algorithm iterates through each room, checking if the current room's prisoner's religion differs from the previous room's prisoner's religion. If they are different, it signifies a potential escape state, and the count is incremented. This process continues until all rooms have been examined.

Pseudocode

def countEscapeStates(N, M):
    count = 0
    previousReligion = 0
    for i in range(1, N + 1):
        currentReligion = randomReligion(M)  # Assigns a random religion to the current room
        if currentReligion != previousReligion:
            count += 1
        previousReligion = currentReligion
    return count

Correctness Proof

The algorithm's correctness stems from its logic of identifying escape states. It iterates through each room, comparing the current room's religion to the previous room's. If they differ, an escape is possible from that state, and the count is incremented. This process accurately reflects the problem's definition of an escape state.

Complexity Analysis

The algorithm exhibits a time complexity of O(N) due to the single iteration through all N rooms. This linear relationship with the number of rooms makes it an efficient solution for determining the number of escape states in a prison with a varying number of rooms.