Graphs

The two basic graph traversals; BFS gives hop distances and levels, DFS gives discovery/finish order for cycles, components and topological order.

Order the vertices of a DAG so every edge points forward; the core of build systems, schedulers and dependency resolution, with cycle detection for free.

Single-source shortest paths with non-negative edge weights by settling vertices in order of distance with a priority queue.

Dijkstra ordered by cost-so-far plus an admissible estimate of the remaining cost; optimal paths while exploring far fewer nodes toward the goal.

Rank the nodes, remove them one by one while adding shortcut edges that keep distances intact, then answer queries with two tiny searches that only climb the ranking.

Disjoint-set forest with union by size and path compression; merges sets and answers same-set queries in effectively constant time.

Cheapest set of edges connecting all vertices; Kruskal sorts edges and joins components with union-find, Prim grows one tree, Boruvka merges in parallel rounds.

Maximum s-t flow by BFS level graphs and blocking flows; also gives the min cut, bipartite matching and assignment with capacities.

Optimal one-to-one matching of rows to columns in O(n³) by growing a matching along shortest augmenting paths over dual potentials.

Split a graph into k balanced parts with few edges between them; multilevel coarsen-partition-refine (METIS, KaHIP) for quality, streaming heuristics (LDG, Fennel) for one-pass scale.