Network and Tree in Graph Theory
Download as PPTX, PDF1 like449 views
The document provides an overview of network flow and spanning trees within graph theory, discussing definitions, basic terms, and applications related to transportation networks. It also explains the concept of spanning trees as subgraphs that connect all vertices without forming cycles, and outlines algorithms like Prim's and Kruskal's for constructing minimum spanning trees. Additionally, it includes references for further reading on the topic.
Network and Tree in Graph Theory
- 1. NETWORK ANDTREE IN GRAPHTHEORY 1 Presented by: The Camouflage BSc.CSIT 2nd Semester Gagan Puri Rabin BK Bikram Bhurtel Prism Karki
- 2. Network flow Spanning Tree Tree traversal References 2
- 3. Also known as a transportation network It is a directed graph where each edge has a capacity and each edge receives a flow Often in operations research, a directed graph is called a network, the vertices are called nodes and the edges are called arcs Network Flow 3
- 4. A network is a graph G = (V, E), where V are a set of vertices and E are a set of edges a non-negative function c: V × V → ℝ∞ is called the capacity function If two nodes in G are distinguished, a source s and a sink t, then (G, c, s, t) is called a flow network Definition 4
- 5. Source : no incoming edges Sink : no outgoing edges Slack : ( capacity – flow ) Augmented path : a path (u1, u2, ..., uk), where u1 = s, uk = t, and cf (ui, ui + 1) > 0 Saturated : if flow = capacity Unsaturated : if flow < capacity Basic terms 5
- 6. Example: To find a maximum flow : A network is at maximum flow if and only if there is no augmenting path in the residual network Gf. s → v → t s → v → u → t Fig: optimal solution 6
- 7. Used to model traffic in road system, circulation with diamonds, fluids in pipes and anything similar in which sth travels through a network of nodes Electrical distribution systems Ecosystem network analysis Maximum flow Multi – commodity flow problem Minimum cost flow problem Circulation problem Network with gains / generalised network Source localization problem Applications 7
- 8. For a simple graph G, Spanning tree is a subgraph of G. It contains every vertex of G. All the vertices are connected The connected edges must not form a cycle 8 Spanning Tree Simple graph Spanning tree Not a tree since there are small circuits and closed cycles It is a tree because there is no closed cycle and there is a single path to every edge.
- 9. It is also a spanning tree It has the smallest possible sum of weights of its edges Algorithms used for constructing the minimum spanning tree Prim's Algorithm Kruskal's Algorithm 9 Minimum spanning tree
- 10. Step 1: Select the vertex at random Step 2: Find all the edges that connect the tree to new vertices, find the minimum and add it to the tree Step 3: Keep repeating step 2 until we get a minimum spanning tree 10 Prim's AlgorithmPrim's Algorithm
- 11. 11
- 12. Step 1:Sort all the edges from low weight to high Step 2: Take the edge with the lowest weight and add it to the spanning tree. If adding the edge created a cycle, then reject this edge Step 3: Keep adding edges until all vertices are not included 12 Kruskal's Algorithm
- 13. 13
- 14. 14
- 15. 15
- 16. 16
- 17. 17
- 18. 18
- 19. 19
- 20. 20
- 21. References Textbook reference: • Discrete Math and its application,Kenneth H. Rosen Web Reference: http://scanftree.com/Data_Structure/prim%27s-algorithm https://www.cse.ust.hk/~dekai/271/notes/L07/L07.pdf http://www.tutorvista.com/content/math/prims-algorithm/ 21
- 22. Queries 22