Evaluate Mathematical Expressions

Start by clarifying the exact contract for Evaluate Mathematical Expressions: input representation, mutation rules, duplicate behavior, and what should happen on invalid or empty input. The core idea is grammar-aware parsing with precedence and parentheses. Work a tiny example aloud and name the invariant after each step, because that is the proof that the approach stays correct as the input grows.

A dependable implementation plan is to define supported grammar; tokenize numbers and signs; apply higher-or-equal precedence operators; resolve parentheses; validate syntax and final stack. Unary minus and integer division semantics must be specified. Keep the code aligned with that invariant instead of adding special cases that silently change the contract. Test the smallest input, a normal case, duplicates or ties where relevant, an adversarial boundary, and a case that exercises the failure path. The expected complexity is O(n) time and O(n) space. Explain both time and auxiliary memory, including recursion or helper structures; do not hide sorting, copying, or repeated scans. If the title omits critical constraints, say what you need to know before committing to a more specialized solution.