# Solution Alignment The prompt asks for an implementation-level answer. The safest way to present it is to define the state, maintain clear invariants, then walk through complexity and tests. ## Problem Restatement ##### Scenario Coding rounds and infrastructure-style algorithm questions. ##### Question Write an efficient algorithm to return the length of the longest increasing subsequence in an integer array. Design a function that rearranges a classroom so no adjacent students share the same team. How would you sort 50 numbers in a file, and how would your approach change for one million numbers? ##### Hints Discuss O(n log n) solutions, dynamic programming with binary search, and external/parallel sorting for large files. ## Recommended Approach Define a state that captures exactly the remaining decision information. Fill base cases first, then transition from smaller subproblems to larger ones. For games, use score difference or minimax DP; for counting, sum valid predecessor states. ## Correctness The implementation should maintain an invariant after each loop or operation that directly matches the problem statement. At termination, that invariant implies the returned value has considered every valid candidate exactly once, or has preserved the required data-structure state after every API call. ## Complexity Typical DP time is number_of_states times transition_cost. Space can often be reduced when transitions only need the previous layer or diagonal. ## Edge Cases and Tests Empty input, length 1, invalid symbols, negative values where allowed, ties under optimal play, and large counts requiring modulo arithmetic.