Greedy Graph Algorithms

Introduction

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The Minimum Spanning Tree Problem

Theorem (The Spanning Tree Lemma). Suppose $T$ is a spanning tree of a graph $G$ . Let $e$ be an edge in $G \setminus T$ . Then the following are true:

$T \cup \{e\}$ contains a unique cycle $C$ .
Let $e' \in C$ . Then $T^\prime = T - \{e\} \cup \{e'\}$ is a spanning tree of $G$ .

Figure 1: A spanning tree of a graph. Hover over a non-tree edge to reveal the cycle guaranteed by the minimum spanning tree lemma.

The Shortest Path Problem

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Greedy Graph Algorithms

Introduction

The Minimum Spanning Tree Problem

The Shortest Path Problem

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