# 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 ##### Question Given a large password file, build a program to read the required password(s) while minimizing memory usage. How would you optimize the space so that only the needed content is loaded? Given a mathematical expression as a string, write code to evaluate it using a depth-first-search (DFS) approach. ## Recommended Approach Use the string constraints to choose between two pointers, a stack, frequency counts, prefix/suffix state, or dynamic programming. Maintain the invariant that processed characters have already been normalized, counted, or matched according to the operation. ## 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 Most direct string scans are O(n) time. Space ranges from O(1) for two pointers to O(n) for stacks, maps, or DP tables. ## Edge Cases and Tests Empty string, length 1, repeated characters, invalid characters, case sensitivity, Unicode vs ASCII, and very long input.