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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.
1#include <stdio.h>
2#include <string.h>
3
4void nextDay(int *cells, int *nextCells) {
5 nextCells[0] = 0;
6 nextCells[7] = 0;
7 for (int i = 1; i < 7; ++i) {
8 nextCells[i] = cells[i - 1] == cells[i + 1];
9 }
10}
11
12void prisonAfterNDays(int *cells, int n, int *result) {
13 int nextCells[8];
14 n = n % 14 == 0 ? 14 : n % 14; // Optimization purpose
15 for (int i = 0; i < n; ++i) {
16 nextDay(cells, nextCells);
17 memcpy(cells, nextCells, 8 * sizeof(int));
18 }
19 memcpy(result, cells, 8 * sizeof(int));
20}
21
22int main() {
23 int cells[8] = {0, 1, 0, 1, 1, 0, 0, 1};
24 int n = 7;
25 int result[8];
26 prisonAfterNDays(cells, n, result);
27 for(int i = 0; i < 8; i++) {
28 printf("%d ", result[i]);
29 }
30 return 0;
31}The C program defines a function nextDay to calculate the state of cells for the next day based on the current state using the rules provided. The prisonAfterNDays function uses this helper function to simulate day-by-day changes. The logic accommodates the cyclic nature of the problem by using n % 14 to optimize large iterations, where 14 is the length of the repeating cycle determined by observation and testing.
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.
By using a HashMap, this Java solution efficiently tracks cell states with corresponding cycle iterations, allowing for early detection of repeated sequences. The cycle law of conservation cuts down the computational overhead effectively for large 'n' inputs.