Graph Algorithm
A graph algorithm is a procedure that operates on a graph — a set of nodes (vertices) connected by edges — to solve problems such as finding shortest paths, detecting cycles, or determining connectivity between nodes.
Definition
A graph algorithm is a procedure that operates on a graph — a set of nodes (vertices) connected by edges — to solve problems such as finding shortest paths, detecting cycles, or determining connectivity between nodes.
Overview
Graphs are among the most flexible data structures because they can model almost any relationship between entities: web pages linked by hyperlinks, cities connected by roads, people connected by friendships, or tasks connected by dependencies. Graph algorithms are the tools used to extract useful information from these structures. Traversal algorithms like breadth-first search (BFS) and depth-first search (DFS) form the foundation, visiting nodes systematically to answer questions like 'can I reach node B from node A?' or 'what is the shortest number of hops between two nodes?' Built on top of traversal, shortest-path algorithms like Dijkstra's algorithm and Bellman-Ford compute the cheapest route between nodes in weighted graphs, while algorithms like Kruskal's and Prim's find minimum spanning trees that connect all nodes at minimum total edge cost. Many of these algorithms are, at their core, applications of dynamic programming or greedy strategies layered on top of BFS/DFS traversal, and their runtime is typically analyzed using Big O Notation in terms of the number of vertices and edges. Topological sorting, another key graph algorithm, orders nodes in a directed acyclic graph so that every edge points from an earlier node to a later one — this is how build systems and task schedulers determine a valid order for dependent operations. Cycle detection, connected-components analysis, and network-flow algorithms round out the toolkit used across routing, scheduling, and social-network analysis. Graph algorithms show up constantly in real infrastructure: routing engines, recommendation systems, dependency resolvers in package managers, and social network features like 'people you may know' are all graph problems in disguise, which is why they're a heavily tested topic in technical interviews.
Key Concepts
- Breadth-first search (BFS) and depth-first search (DFS) as core traversal strategies
- Dijkstra's and Bellman-Ford algorithms for shortest-path computation
- Kruskal's and Prim's algorithms for minimum spanning trees
- Topological sorting for ordering tasks with dependencies
- Cycle detection and connected-components analysis
- Directed vs. undirected and weighted vs. unweighted graph variants
- Runtime typically expressed in terms of vertices (V) and edges (E)