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This approach involves simulating each day's changes directly. Given the constraints of the problem, this can be manageable for relatively small values of 'n'. We iterate over each day, applying the rules to determine the state of each cell. Importantly, the edge cells are always vacant as they lack two neighbors. By updating the state iteratively, we can reach the final configuration after 'n' days.
Time complexity: O(1), as the complexity per cycle of states is fixed. During each cycle, we just iterate over the 8 states.
Space complexity: O(1), only a fixed array is used for state computation.
1using System;
2
3public class PrisonCells {
4 public static int[] NextDay(int[] cells) {
5 int[] nextCells = new int[8];
6 for (int i = 1; i < 7; ++i) {
7 nextCells[i] = cells[i - 1] == cells[i + 1] ? 1 : 0;
8 }
9 return nextCells;
10 }
11
12 public static int[] PrisonAfterNDays(int[] cells, int N) {
13 N = N % 14 == 0 ? 14 : N % 14; // Optimize for large N
14 for (int i = 0; i < N; ++i) {
15 cells = NextDay(cells);
16 }
17 return cells;
18 }
19
20 public static void Main(string[] args) {
21 int[] cells = new int[] {0, 1, 0, 1, 1, 0, 0, 1};
22 int N = 7;
23 int[] result = PrisonAfterNDays(cells, N);
24 Console.WriteLine(string.Join(", ", result));
25 }
26}The C# implementation follows a similar pattern of computing each day's change, leveraging the cycle to minimize operations when dealing with larger 'n'. The use of arrays allows for straightforward manipulation of each cell's state.
This alternative approach builds on recognizing patterns within the transformations. Given only 8 cells and binary states, the maximum number of possible unique states is 27 = 128. Through simulation, we can identify that the pattern cycles or repeats after at most 14 days, reducing the need to simulate all given 'n' days. By finding cycles, one can compute the state that would correspond to the final day directly, drastically minimizing operations.
Time complexity: O(128), as we explore states to identify cycles, bounded by maximum unique states of 128.
Space complexity: O(128), accommodates tracking up to 128 states.
The C code leverages a direct approach to identify patterns using the findCycle method, where state sequences are preserved and revisited to determine recurring configurations. This understanding is then used to restrict operations to only necessary repetitions through precise cycle length determination.