Prison Escape States: Dynamic Programming Algorithm (O(n))

To find out how many states escape can occur, we can use the concept of dynamic programming.

First, we initialize an array called 'dp' of size N, where dp[i] represents the number of different states in which escape can occur if we consider the first i rooms.

Next, we iterate through each room from left to right and calculate dp[i] based on the previous states:

1. If i = 1, then dp[i] = M because there is only one room and each religion can be chosen.

2. If i > 1, then dp[i] is calculated as follows:

   • If the religion of the prisoner in room i is different from the religion of the prisoner in room i-1, then dp[i] = dp[i-1] * M. This is because the prisoner in room i can choose any religion, so the number of states is multiplied by M.
   • If the religion of the prisoner in room i is the same as the religion of the prisoner in room i-1, then dp[i] = dp[i-1]. This is because escape cannot occur in this case, and the number of states remains the same.

Finally, the total number of states in which escape can occur is given by dp[N].

Here is the O(n) algorithm in Python:

def count_escape_states(N, M):
    dp = [0] * (N+1)
    dp[1] = M

    for i in range(2, N+1):
        if i > 1:
            if dp[i-1] == M:
                dp[i] = M
            else:
                dp[i] = dp[i-1] * M

    return dp[N]

# Example usage
N = 3
M = 2
states = count_escape_states(N, M)
print(states)  # Output: 6

In the example usage, there are 3 rooms and 2 kinds of religions, so there are 6 different states in which escape can occur.