Prison Cells After N Days

Start by simulating each day's cell state with careful attention to the transformation rules. Track seen configurations using a hash map to detect cycles, allowing you to skip repeated computations and jump ahead efficiently. This approach combines array manipulation, hash-based lookup, and bit-level reasoning to handle very large N without full iteration.

Problem Statement

You are given a row of 8 prison cells, each either occupied (1) or vacant (0). Every day, a cell changes state based on its two adjacent neighbors: it becomes occupied if both neighbors are in the same state, otherwise it becomes vacant. The first and last cells only have one neighbor each and are always updated to vacant the next day.

Given the initial state of the cells and a number N representing days, compute the state of the prison after N days. Since cell states can repeat, efficiently detecting cycles allows large N values to be processed without simulating every day sequentially.

Examples

Example 1

Input: cells = [0,1,0,1,1,0,0,1], n = 7

Output: [0,0,1,1,0,0,0,0]

The following table summarizes the state of the prison on each day: Day 0: [0, 1, 0, 1, 1, 0, 0, 1] Day 1: [0, 1, 1, 0, 0, 0, 0, 0] Day 2: [0, 0, 0, 0, 1, 1, 1, 0] Day 3: [0, 1, 1, 0, 0, 1, 0, 0] Day 4: [0, 0, 0, 0, 0, 1, 0, 0] Day 5: [0, 1, 1, 1, 0, 1, 0, 0] Day 6: [0, 0, 1, 0, 1, 1, 0, 0] Day 7: [0, 0, 1, 1, 0, 0, 0, 0]

Example 2

Input: cells = [1,0,0,1,0,0,1,0], n = 1000000000

Output: [0,0,1,1,1,1,1,0]

Example details omitted.

Constraints

• cells.length == 8
• cells[i] is either 0 or 1.
• 1 <= n <= 109

Solution Approach

Simulate with Array Scanning

Iterate through the cell array each day, updating intermediate states in a temporary array while applying the rules. This ensures neighbor comparisons are accurate and prevents overwriting values that will be needed for the same day's computation.

Use Hash Lookup to Detect Cycles

Store each cell configuration as a string or integer key in a hash map with the day it occurred. When a configuration repeats, compute the cycle length and jump ahead by N modulo cycle length to avoid redundant simulation steps.

Optimize with Bit Manipulation

Represent the 8 cells as an 8-bit integer to speed up state updates and comparisons. Apply bit shifts and XOR operations to calculate the next day's state, reducing both memory and computation time for large N scenarios.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Without cycle detection, time complexity is O(N) and space O(N) to store states. With cycle detection, time reduces to O(C) where C is the cycle length, and space is O(C) for the hash map.

What Interviewers Usually Probe

• Will ask if the solution handles very large N efficiently without full simulation.
• May probe about how to detect repeating patterns and cycle lengths.
• Could question the bit manipulation optimization for faster state updates.

Common Pitfalls or Variants

Common pitfalls

• Forgetting to treat the first and last cells as always vacant the next day.
• Overwriting the current day's cell values before computing neighbors, causing incorrect results.
• Failing to detect cycles, leading to excessive computation for large N values.

Follow-up variants

• Prison cells of arbitrary length instead of 8, requiring generalized neighbor rules.
• Changing the state transition rule to different neighbor conditions, affecting cycle detection.
• Return the number of days until the cells first repeat instead of final state after N days.

FAQ

What is the best approach for Prison Cells After N Days with very large N?

Use a hash map to detect repeating cell patterns and compute N modulo the cycle length to jump ahead efficiently.

Can bit manipulation really speed up cell updates?

Yes, representing the 8 cells as an 8-bit integer allows state updates with shifts and XOR, reducing computation and memory.

Why do the first and last cells always become vacant?

They have only one neighbor, so the transformation rule for two neighbors cannot apply; they are set to vacant by definition.

How do I detect cycles in cell configurations?

Store each day's configuration in a hash map; when a repeat occurs, the distance between repeats gives the cycle length.

Does this approach generalize to arrays longer than 8 cells?

Yes, the simulation and cycle detection methods work for longer arrays, but bit manipulation optimization will need larger integer representations.