Graph Algorithms

Abstract

Many programming problems can be solved by considering the situation as a graph and using an appropriate graph algorithm. In this chapter, we will learn the basics of graphs and a selection of important graph algorithms. Section 7.1 discusses graph terminology and data structures that can be used to represent graphs in algorithms. Section 7.2 introduces two fundamental graph traversal algorithms. Depth-first search is a simple way to visit all nodes that can be reached from a starting node, and breadth-first search visits the nodes in increasing order of their distance from the starting node. Section 7.3 presents algorithms for finding shortest paths in weighted graphs. The Bellman–Ford algorithm is a simple algorithm that finds shortest paths from a starting node to all other nodes. Dijkstra’s algorithm is a more efficient algorithm which requires that all edge weights are nonnegative. The Floyd–Warshall algorithm determines shortest paths between all node pairs of a graph. Section 7.4 explores special properties of directed acyclic graphs. We will learn how to construct a topological sort and how to use dynamic programming to efficiently process such graphs. Section 7.5 focuses on successor graphs where each node has a unique successor. We will discuss an efficient way to find successors of nodes and Floyd’s algorithm for cycle detection. Section 7.6 presents Kruskal’s and Prim’s algorithms for constructing minimum spanning trees. Kruskal’s algorithm is based on an efficient union-find structure which has also other uses in algorithm design.