Graph Algorithms

Graph traversal forms the backbone of many graph algorithms. Breadth-First Search (BFS) systematically visits nodes in increasing distance from the source, which makes it ideal for finding shortest paths in unweighted graphs and for level-by-level exploration. Depth-First Search (DFS) dives deep into one branch before backtracking, revealing the structure of connected components, detecting cycles, and enabling algorithms that rely on post-order processing. When implementing these traversals, you typically maintain a visited set to avoid reprocessing nodes and a data structure to hold the frontier: a queue for BFS or a stack/recursion for DFS. In weighted graphs, additional data structures like priority queues are used to select the next node to process based on accumulated weights. The choice between BFS and DFS depends on the problem: BFS for shortest path in unweighted graphs and graph reachability with minimal steps, DFS for cycle detection, topological ordering in DAGs, and graph decomposition.

Shortest path problems are central in graph algorithms. In unweighted graphs, BFS naturally yields the shortest path in terms of the number of edges from a source to all reachable nodes. In weighted graphs, Dijkstra's algorithm computes the minimum total weight path by greedily selecting the next closest vertex using a priority queue. The core idea is to maintain a distance estimate for each node and to relax edges, updating the best-known distance when a shorter path is found. A practical insight is to use a min-heap (priority queue) to retrieve the next vertex with the smallest tentative distance efficiently. For graphs with negative weights, Bellman-Ford can handle them but may be slower; when all weights are non-negative, Dijkstra with a priority queue is typically preferred. If you need all-pairs shortest paths, algorithms like Floyd-Warshall or repeated Dijkstra are employed. Mastery here involves understanding the relaxation step, edge relaxation conditions, and when to stop the algorithm.

Spanning trees are subgraphs that connect all vertices with the minimum number of edges. They preserve connectivity without cycles. Two classical algorithms produce spanning trees efficiently: Kruskal's algorithm builds the minimum spanning tree (MST) by repeatedly adding the smallest edge that does not create a cycle, typically using a disjoint-set (union-find) data structure to detect cycles. Prim's algorithm grows a tree from an arbitrary starting vertex by always attaching the smallest edge that connects the tree to a new vertex. Both algorithms assume a connected, undirected, weighted graph and aim to minimize total edge weight. The concept of MST is central in network design, where you want to connect points with the least total wiring or cost while maintaining full reachability. In practice, MSTs underpin clustering, image segmentation, and heuristic approximations for more complex network problems.

Network flow theory studies the maximum amount of a commodity that can be transferred from a source to a sink through a network with capacity constraints. The classic problem is the max-flow, min-cut theorem: the maximum flow equals the minimum cut capacity separating source and sink. Algorithms like Ford-Fulkerson, Edmonds-Karp (which uses BFS for augmenting paths), and Dinic's algorithm solve this by repeatedly finding augmenting paths and augmenting flow along them. Graphs with capacities allow you to model real-world systems: traffic, data routing, supply chains. A residual graph tracks remaining capacity and guides the next augmentation. Understanding flows requires grasping the notions of capacity, residual capacity, augmenting paths, and cuts. In practice, you model the network, initialize zero flow, and iteratively push flow until no more augmenting paths exist, achieving a maximum flow that cannot be improved without increasing capacity.

Planar graphs and graph drawing are practical topics where structure matters. Planarity concerns whether a graph can be drawn on a plane without crossing edges. Some algorithms assume planarity to simplify computations. Graph drawing aims to place vertices and route edges aesthetically and legibly, which improves understanding and visualization in domains like circuit design or social networks. While many graph algorithms focus on combinatorial properties, visual representations can reveal patterns such as communities, clusters, and central nodes. Techniques like BFS layering, radial layouts, and force-directed methods combine algorithmic insights with geometric reasoning to produce informative diagrams. Understanding planarity also links to powerful theorems such as Kuratowski's theorem, which characterizes non-planar graphs via minor containment. In practice, when you model problems, choosing representations that support efficient traversal, matching, or flow computations is as important as the theoretical correctness of the algorithm.