🥑 Graph Algorithms | Data Structures & Algorithms

#DataStructures #Algorithms #GraphTheory #ComputerScience This video explores several advanced Graph Algorithms used to analyze graph properties and solve famous computer science problems. Presented by Hayden Smith, this lecture covers cycle checking, connected components, and the important distinction between Hamiltonian and Euler paths. 🔄 Cycle Checking & Connectivity Beyond simple traversal, algorithms can analyze a graph's structure. Cycle Checking: A modified Depth-First Search (DFS) can be used to determine if a graph contains a cycle (a path that starts and ends at the same vertex). Connected Components: This algorithm identifies how many separate, disjoint subgraphs exist within a larger graph, which is useful for understanding connectivity. 📍 Hamiltonian Paths & Circuits * Definition: A Hamiltonian path is one that visits every vertex in a graph exactly once. A Hamiltonian circuit is a Hamiltonian path that is also a cycle. * Application: This relates to problems like the famous "Traveling Salesman Problem." * Complexity: Finding a Hamiltonian path is an extremely difficult problem. It is NP-hard, meaning there is no known efficient algorithm to solve it. ↔️ Euler Paths & Circuits * Definition: An Euler path is one that travels along every edge in a graph exactly once. An Euler circuit is an Euler path that is also a cycle. * Application: This is useful for route-planning problems, such as designing a route for mail delivery or garbage collection. * Complexity: Unlike Hamiltonian paths, there is a very simple and efficient shortcut to determine if an Euler path exists. The check relies on the degree of the vertices (the number of edges connected to them): * An Euler circuit exists if and only if every vertex in the graph has an even degree. * An Euler path exists if and only if the graph has exactly two vertices of odd degree. View the full playlist: https://www.youtube.com/playlist?list=PLi2pCZz5m6GEftzPIxVH1ylwytux9WOGN