03 Shortest Paths

Shortest Paths

Having explored spanning trees, we now turn our attention to another fundamental concept in graph theory: paths, specifically shortest paths. In many applications, we are not merely interested in whether a path exists between two vertices, but in finding the most efficient path, often measured in terms of distance, cost, or time. This leads us to the problem of finding shortest paths in graphs.

Problem Definition: Given a connected graph $G=(V,E)$, two vertices $s,t\in V$, and a cost function $c:E\to {\mathbb{R}}_{\ge 0}$ (assigning non-negative costs to edges), the shortest path problem seeks to find an $s-t$-path $P$ in $G$ such that the sum of costs of edges in $P$ is minimized.

This problem arises in numerous contexts, from navigation systems finding the quickest route between locations to network routing protocols seeking the least congested path for data transmission.

Algorithms for Shortest Paths

The script mentions two prominent algorithms for solving the shortest path problem:

• Dijkstra’s Algorithm: Dijkstra’s algorithm is a classic algorithm for finding the shortest paths from a single source vertex $s$ to all other vertices in a graph with non-negative edge weights. It efficiently computes shortest paths by maintaining a set of vertices for which the shortest path from the source is already known and iteratively expanding this set.

• Floyd-Warshall Algorithm: The Floyd-Warshall algorithm, in contrast to Dijkstra’s, solves the all-pairs shortest paths problem. It computes the shortest paths between all pairs of vertices in a graph, even in the presence of negative edge weights (though it cannot handle negative cycles). It employs dynamic programming to systematically improve path lengths by considering intermediate vertices.

These algorithms, while both addressing shortest path problems, differ in their scope, applicability, and underlying techniques. Dijkstra’s algorithm is more efficient for single-source shortest paths in graphs with non-negative edge weights, while Floyd-Warshall is versatile for all-pairs shortest paths and can handle negative edge weights (but not negative cycles).

In subsequent sections, we will delve deeper into these algorithms and explore their implementations, complexities, and applications. For now, it is crucial to recognize the significance of shortest paths as a fundamental graph problem and to appreciate the existence of efficient algorithms designed to solve it.

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