Make a Graph Algorithm Implementation Thread-Safe

Problem: Given a project that implements a graph algorithm (from a research paper), explain how to make the implementation thread-safe for concurrent execution, and write pseudocode reflecting this.

Approach: Identify the shared mutable state accessed by the algorithm (e.g. visited-node sets, adjacency structures being mutated, shared result accumulators) and guard each with an appropriate synchronization primitive — a mutex/lock around critical sections that mutate shared state, or lock-free structures (atomic flags for visited-marking) where contention is high. Favor partitioning work so each thread owns disjoint subgraphs where possible, minimizing the need for locking altogether.