Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Dijkstra's Algorithm and Shortest Path

In course

MATH-261: Discrete optimization

This course is an introduction to linear and discrete optimization. Warning: This is a mathematics course! While much of the course will be algorithmic in nature, you will still need to be able to prove theorems. ...

Description

This lecture covers Dijkstra's algorithm for the ALL-TO-ONE shortest path problem, combining it with Bellman-Ford for an effective ALL-PAIRS shortest path algorithm. The slides detail the steps of Dijkstra's algorithm, the importance of nonnegative edges, and the process of finding the optimal solution. It also explores the concept of ALL-TO-ONE and ALL-PAIRS algorithms, discussing the implications of negative edge costs and the significance of running Dijkstra multiple times. The lecture concludes with a practical problem set involving the application of the algorithms to find optimal solutions for various scenarios.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructors (2)

Rom Pinchasi

Friedrich Eisenbrand

Friedrich Eisenbrand's main research interests lie in the field of discrete optimization, in particular in algorithms and complexity, integer programming, geometry of numbers, and applied optimization. He is best known for his work on efficient algorithms for integer programming in fixed dimension and the theory of cutting planes, which are an important tool to solve large scale industrial optimization problems in practice. Before joining EPFL in March 2008, Friedrich Eisenbrand was a full professor of mathematics at the University of Paderborn. Friedrich received the Heinz Maier-Leibnitz award of the German Research Foundation (DFG) in 2004 and the Otto Hahn medal of the Max Planck Society in 2001.

Official source

Ontological neighbourhood

Mathematics

Discrete mathematics: Graph theory

Related concepts (32)

Dijkstra's algorithm

Dijkstra's algorithm (ˈdaɪkstrəz ) is an algorithm for finding the shortest paths between nodes in a weighted graph, which may represent, for example, road networks. It was conceived by computer scientist Edsger W. Dijkstra in 1956 and published three years later. The algorithm exists in many variants. Dijkstra's original algorithm found the shortest path between two given nodes, but a more common variant fixes a single node as the "source" node and finds shortest paths from the source to all other nodes in the graph, producing a shortest-path tree.

Shortest path problem

In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The problem of finding the shortest path between two intersections on a road map may be modeled as a special case of the shortest path problem in graphs, where the vertices correspond to intersections and the edges correspond to road segments, each weighted by the length of the segment.

Path (graph theory)

In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A directed path (sometimes called dipath) in a directed graph is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction. Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts.

Graph theory

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics.

Vertex (graph theory)

In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices). In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another.

Related lectures (32)

Minimum Spanning Trees: Prim's Algorithm

Explores Prim's algorithm for minimum spanning trees and introduces the Traveling Salesman Problem.

Graph Algorithms II: Traversal and Paths

Explores graph traversal methods, spanning trees, and shortest paths using BFS and DFS.

Dynamic Programming: Pascal's Triangle & Floyd's Algorithm

Explores dynamic programming through Pascal's Triangle and Floyd's Algorithm.

Dynamic Programming: Shortest Paths Algorithms

Explores dynamic programming strategies for finding shortest paths in networks with various algorithms and complexities.

Shortest Paths: Bellman-Ford and Dijkstra