Graph Algorithm
Algorithms that operate on graphs of nodes and edges to answer questions about connectivity and paths.
Definition
A graph algorithm operates on a graph, a set of nodes (vertices) connected by edges. Graphs model networks, dependencies, and relationships, and graph algorithms answer questions about paths, connectivity, flow, and structure.
Graph representation matters for performance: an adjacency list is efficient for sparse graphs, while an adjacency matrix suits dense ones or fast edge lookups. Choosing the representation to match the graph's density and the operations needed is a first design decision.
Graphs unify a remarkable range of problems, from social networks and web links to circuit connectivity, task scheduling, and the sparsity patterns of matrices. This generality means that recognizing a problem as a graph problem often unlocks a large toolkit of established algorithms. The main design choices, representation, traversal order, and whether edges carry weights or directions, follow from the structure of the specific graph at hand.
Foundational algorithms
- Breadth-first and depth-first search for traversal.
- Dijkstra and Bellman-Ford for shortest paths.
- Topological sort for dependency ordering.
- Max-flow and minimum spanning tree algorithms.
Why it matters
Graphs are among the most general data models in computing, describing everything from road networks to circuit connectivity to task dependencies. Efficient graph algorithms make analysis of these structures tractable at scale.
Fusion connection
Sparse-matrix solvers for plasma simulation rely on graph algorithms internally, since the matrix structure corresponds to a graph over mesh nodes that must be ordered and partitioned efficiently.