Unit 3 graph chapter6
- 1. CHAPTER 6 GRAPHS Dr. R. Khanchana Assistant Professor Department of Computer Science Sri Ramakrishna College of Arts and Science for Women CHAPTER 6 1 http://icodeguru.com/vc/10book/books/book1/chap06.htm
- 2. CHAPTER 6 2 Definition A graph G consists of two sets – a finite, nonempty set of vertices V(G) – a finite, possible empty set of edges E(G) – G(V,E) represents a graph
- 3. Directed Graph Vs Undirected Graph CHAPTER 6 3 An undirected graph is one in which the pair of vertices in a edge is unordered, (v0, v1) = (v1,v0) A directed graph is one in which each edge is a directed pair of vertices, <v0, v1> != <v1,v0> tail head
- 4. CHAPTER 6 4 Examples for Graph 1 2 3 4 1 2 3 4 5 6 7 G1 G2 G3 complete graph incomplete graph
- 5. CHAPTER 6 5 Complete Graph A complete graph is a graph that has the maximum number of edges – for undirected graph with n vertices, the maximum number of edges is n(n-1)/2 – for directed graph with n vertices, the maximum number of edges is n(n-1) – Example: – G1 is a complete undirected graph – G2 is a complete directed graph 1 2 3 4 G1 G2
- 6. CHAPTER 6 6 Adjacent and Incident in Undirected Graph If (v1, v2) is an edge in an undirected graph - v1 and v2 are adjacent – The edge (v1, v2) is incident on vertices v1 and v2 – Example In E(G), the vertices adjacent to vertex 2 is 4,5 and 1 – The edges incident on vetex3 in E(G) are (1,3), (3,6) and (3,7)
- 7. CHAPTER 6 7 Adjacent and Incident in Directed Graph If <v0, v1> is an edge in a directed graph – v0 is adjacent to v1, and v1 is adjacent from v0 – The edge <v0, v1> is incident on v0 and v1 – Example In E(G), <3, 2> - the vertex 3 is adjacent to vertex 2 where vertex 2 is adjacent from vertex 3 – The edge <3, 2> is incident to 3 and 2. E(G)
- 8. CHAPTER 6 8 0 2 1 (a) 3 2 1 4 (b) self edge multigraph: multiple occurrences of the same edge Figure 6.3 Multigraph is not a graph
- 9. CHAPTER 6 9 A subgraph of G is a graph G’ such that V(G’) is a subset of V(G) and E(G’) is a subset of E(G) Subgraph 1 2 3 4 G1 G3
- 10. CHAPTER 6 10 A simple path is a path in which all vertices except possibly the first and last are distinct The length of a path is the number of edges on it Path in Graph
- 11. CHAPTER 6 11 A cycle is a simple path in which the first and the last vertices are the same Cycle in Graph Cycle Path
- 12. CHAPTER 6 12 A connected component of an undirected graph is a maximal connected subgraph. In an undirected graph G, two vertices, v0 and v1, are connected if there is a path in G from v0 to v1 Connected Components
- 13. CHAPTER 6 13 A strongly connected component is a maximal subgraph that is strongly connected. A directed graph is strongly connected if there is a directed path from vi to vj and also from vj to vi. Strongly Connected Component G3
- 14. CHAPTER 6 14 0 1 2 3 0 1 2 3 4 5 6 G1 G2 Tree (acyclic graph) A tree is a graph that is connected and acyclic Acyclic Graph Connected
- 15. CHAPTER 6 15 Degree The degree of a vertex is the number of edges incident to that vertex For directed graph, – in-degree of a vertex v is the number of edges that have v as the head – out-degree of a vertex v is the number of edges that have v as the tail
- 16. CHAPTER 6 16 Degree in Undirected graph 0 1 2 3 4 5 6 G1 G2 3 2 3 3 1 1 1 1 0 1 2 3 33 3
- 18. CHAPTER 6 18 Degree in Directed graph Directed graph in-degree out-degree G3 indegree:1 outdegree: 1 indegree: 1 outdegree: 2 indegree: 1 outdegree: 0
- 20. CHAPTER 6 20 Graph Representations Adjacency Matrix Adjacency Lists Adjacency Multilists
- 21. CHAPTER 6 21 Adjacency Matrix Let G=(V,E) be a graph with n vertices. The adjacency matrix of G is a two-dimensional n * n array, say A(i,j) If the edge (vi, vj) is in E(G), A(i,j) =1 If there is no such edge in E(G), A(i,j) =0 The adjacency matrix for an undirected graph is symmetric; the adjacency matrix for a digraph need not be symmetric
- 22. CHAPTER 6 22 Examples for Adjacency Matrix 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 G1 G2 G4 1 2 3 4 1 2 3 2 1 3 4 5 6 7 8 symmetric e is the No. of Edges undirected:e<< n2/2 directed: e<n2
- 23. The representation the n rows of the adjacency matrix are represented as n linked lists. There is one list for each vertex in G. Each node has at least two fields: VERTEX and LINK The VERTEX fields contain the indices of the vertices adjacent to vertex i. CHAPTER 6 23 Adjacency Lists Each row in adjacency matrix is represented as an adjacency list.
- 24. CHAPTER 6 24 Adjacency Lists- Undirected Graph The Adjacency lists for G1 is shown
- 25. CHAPTER 6 25 Adjacency Lists –Directed Graph The Adjacency lists for G3 is shown
- 26. CHAPTER 6 26 Adjacency Lists- Connected Components The adjacency lists for G4 is shown
- 27. CHAPTER 6 27 Inverse Adjacency Lists Inverse adjacency lists for G3 Each node would now have four fields and would represent one edge. The node structure would be tail head column link for head row link for tail
- 28. CHAPTER 6 28 Adjacency Multilists An edge in an undirected graph is represented by two nodes in adjacency list representation. Adjacency Multilists –lists in which nodes may be shared among several lists. (an edge is shared by two different paths)
- 30. Adjacency Multilists CHAPTER 6 30 Adjacency List for G1
- 32. CHAPTER 6 32 TRAVERSALS Traversal Given G=(V,E) and vertex v, find all wV, such that w connects v. – Depth First Search (DFS) preorder tree traversal – Breadth First Search (BFS) level order tree traversal Connected Components Spanning Trees
- 33. CHAPTER 6 33 Depth First Search (DFS) If a depth first search is initiated from vertex v1, then the vertices of G are visited in the order: v1, v2, v4, v8, v5, v6, v3, v7.
- 34. CHAPTER 6 34 Breadth First Search (BFS) A breadth first search beginning at vertex v1 of the graph. First visit v1 and then v2 and v3. Next vertices v4, v5, v6 and v7 will be visited and finally v8.
- 35. CHAPTER 6 35 Graph and its Adjacency List
- 36. CHAPTER 6 36 Connected Components
- 37. CHAPTER 6 37 Spanning Trees When graph G is connected, a depth first or breadth first search starting at any vertex will visit all vertices in G A spanning tree is any tree that consists solely of edges in G and that includes all the vertices E(G): T (tree edges) + N (nontree edges) where T: set of edges used during search N: set of remaining edges
- 38. CHAPTER 6 38 Examples of Spanning Tree G1 Possible spanning trees
- 39. CHAPTER 6 39 Spanning Trees Either DFS or BFS can be used to create a spanning tree – When DFS is used, the resulting spanning tree is known as a depth first spanning tree – When BFS is used, the resulting spanning tree is known as a breadth first spanning tree While adding a nontree edge into any spanning tree, this will create a cycle
- 40. CHAPTER 6 40 DFS VS BFS Spanning Tree DFS Spanning BFS Spanning
- 41. CHAPTER 6 41 Minimum Cost Spanning Tree The cost of a spanning tree of a weighted undirected graph is the sum of the costs of the edges in the spanning tree A minimum cost spanning tree is a spanning tree of least cost Three different algorithms can be used – Kruskal – Prim – Sollin Select n-1 edges from a weighted graph of n vertices with minimum cost.
- 42. Minimum Cost Spanning Tree CHAPTER 6 42
- 45. SHORTEST PATHS AND TRANSITIVE CLOSURE Single Source All Destinations CHAPTER 6 45
- 46. Analysis of Algorithm SHORTEST PATH CHAPTER 6 46
- 47. Analysis of Algorithm SHORTEST PATH CHAPTER 6 47
- 48. CHAPTER 6 48
- 49. All Pairs Shortest Paths Ak(i,j) = min {Ak-1(i,j), Ak-1(i,k) + Ak -1(K,j)} K>=1 and Ao(i,j) = COST(i,j) CHAPTER 6 49
- 50. All Pairs Shortest Paths CHAPTER 6 50
- 51. Cost Matrix CHAPTER 6 51
- 52. Transitive Closure CHAPTER 6 52 Figure 6.25 Graph G and Its Adjacency Matrix A, A+ and A* A+(i,j) = 1 if path length >0 A*(i,j) = 1 if path length >=0