Debug a multithreaded code where race conditions were occurring in shared state

Start by clarifying the exact contract for Debug Race Conditions: input representation, mutation rules, duplicate behavior, and what should happen on invalid or empty input. The core idea is how to find and protect a violated shared-state invariant. 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 name shared state; identify every compound read-modify-write; reproduce under contention; protect the whole invariant with a mutex or matching atomic. Visibility primitives alone do not provide mutual exclusion. 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(1) work per update, with contention-dependent latency. 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.