The Order Is the Algorithm

Once you see best[v] = f(best of v's predecessors), the graph problem becomes a DP problem — but only if you can process nodes in an order where every predecessor comes first. That order is the algorithm:

• DAG → topological sort, then one linear sweep. O(V + E).
• General graph, reachability/counting → DSU merges components offline, or SCC contraction, then DP on the condensed graph.
• Tree → the traversal order (post-order for upward DP, pre-order for downward) plays the topo role.

The misread that costs a medal

The classic stumble: treating an arbitrary graph as a DAG. If cycles exist, "longest path" is not just hard — it is undefined (infinite) for the reachability version, and NP-hard for the simple-path version. Before running topo DP, ask: is acyclicity actually given, or derived? Deriving it (e.g., events ordered by time, states decreasing by a measure) is itself the key observation of many problems.

Two-pointer on graphs?

When edge weights are 0/1, 0-1 BFS replaces Dijkstra; when all weights are equal, plain BFS is optimal. Recognizing the weakest tool that still works is the mastery skill — heavyweight tools invite heavyweight bugs.