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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.
1function nextDay(cells) {
2 const nextCells = Array(8).fill(0);
3 for (let i = 1; i < 7; i++) {
4 nextCells[i] = cells[i - 1] === cells[i + 1] ? 1 : 0;
5 }
6 return nextCells;
7}
8
9function prisonAfterNDays(cells, N) {
10 N = N % 14 === 0 ? 14 : N % 14; // Cycle detection
11 for (let i = 0; i < N; i++) {
12 cells = nextDay(cells);
13 }
14 return cells;
15}
16
17// Example usage
18const cells = [0, 1, 0, 1, 1, 0, 0, 1];
19const N = 7;
20console.log(prisonAfterNDays(cells, N));In JavaScript, we utilize the power of arrays to handle state transitions. The nextDay function computes the next state for each cell, and with optimized looping in prisonAfterNDays, we acknowledge the repetitive cycles for efficient computation.
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.
JavaScript implementation uses object keys to identify similar configurations previously seen. It allows harnessing these repetitive states effectively by cutting round trips, through concise implementation logic.