My first instinct was to just BFS from every node and count reversals each time, which technically works but blows up on large graphs.
Model the problem as finding a root that minimizes the number of edges pointing toward it. Use two DFS traversals: first compute the reversal count for an arbitrary root, then reroot to compute counts for all nodes in O(V+E) time. Return the minimum count.
Pro tip: Clarify that the graph may not be connected; if disconnected, no single root can reach all nodes, so the problem is infeasible. Also, mention that the optimal root is the centroid of the underlying undirected tree if the graph is a tree.
Confirm that the graph is directed, 1-indexed, and may have cycles or be disconnected. Discuss that if disconnected, no valid root exists, so assume connected or handle separately.
Pick an arbitrary root (e.g., node 1) and perform a DFS/BFS to count how many edges need to be reversed to make all edges point away from it. For each edge u->v, if v is the parent of u, it needs reversal.
Use the rerooting technique: when moving the root from u to its child v, the reversal count changes by +1 if the edge u->v exists (since it now points toward the new root), and -1 if the edge v->u exists. Update counts accordingly.
After computing reversal counts for all nodes, return the minimum value. If the graph is disconnected, return -1 or indicate impossibility.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.