chapter22.ppt
Download as PPT, PDF1 like4,090 views
The document discusses various graph algorithms and representations including: - Adjacency lists and matrices for representing graphs - Breadth-first search (BFS) which explores edges from a source vertex s level-by-level - Depth-first search (DFS) which explores "deeper" first, producing a depth-first forest - Classifying edges as tree, back, forward, or cross based on vertex colors in DFS - Topological sorting of directed acyclic graphs (DAGs) - Strongly connected components (SCCs) in directed graphs and using the transpose
![Adjacency List Representation of a Graph For each u є V , the adjacency list Adj [ u ] contains all the vertices v such that there is an edge ( u , v ) є E .](https://image.slidesharecdn.com/chapter22ppt3693/85/chapter22-ppt-3-320.jpg)
![Breath-first Search The color of each vertex u is stored in the variable color [ u ]. The predecessor of u is stored in the variable π[ u ]. The distance from the source s to vertex u computed by the algorithm is stored in d [ u ]](https://image.slidesharecdn.com/chapter22ppt3693/85/chapter22-ppt-10-320.jpg)
![Depth-first Search Each vertex v has two timestamps: the first timestamp d [ v ] records when v is first discovered (and grayed), and the second timestamp f [ v ] records when the search finishes examining v 's adjacency list (and blackens v ).](https://image.slidesharecdn.com/chapter22ppt3693/85/chapter22-ppt-17-320.jpg)
Ad
More Related Content
What's hot (20)
Viewers also liked (20)
Ad
Similar to chapter22.ppt (11)
![Depth first search [dfs]](https://cdn.slidesharecdn.com/ss_thumbnails/depthfirstsearchdfs-190926145304-thumbnail.jpg?width=560&fit=bounds)
Ad
More from Tareq Hasan (20)
chapter22.ppt
- 1. Introduction to Algorithms Second Edition by Cormen, Leiserson, Rivest & Stein Chapter 22
- 2. Graph Two standard ways to represent a graph Adjacency List Adjacency Matrix
- 3. Adjacency List Representation of a Graph For each u є V , the adjacency list Adj [ u ] contains all the vertices v such that there is an edge ( u , v ) є E .
- 4. Adjacency Matrix Representation of a Graph The adjacency-matrix representation of a graph G consists of a | V | × | V | matrix A = ( aij ) such that a ij =1 if (i,j) є E =o otherwise
- 5. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
- 6. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
- 7. Breath-first Search Breadth-first search is one of the simplest algorithms for searching a graph Given a graph G = ( V , E ) and a distinguished source vertex s , breadth-first search systematically explores the edges of G to "discover" every vertex that is reachable from s .
- 8. Breath-first Search T he algorithm discovers all vertices at distance k from s before discovering any vertices at distance k + 1.
- 9. Breath-first Search To keep track of progress, breadth-first search colors each vertex white, gray, or black. All vertices start out white and may later become gray and then black. A vertex is discovered the first time it is encountered during the search, at which time it becomes nonwhite.
- 10. Breath-first Search The color of each vertex u is stored in the variable color [ u ]. The predecessor of u is stored in the variable π[ u ]. The distance from the source s to vertex u computed by the algorithm is stored in d [ u ]
- 11. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
- 12. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
- 13. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
- 14. Depth-first Search The strategy followed by depth-first search is, as its name implies, to search "deeper" in the graph whenever possible.
- 15. Depth-first Search The predecessor sub-graph of a depth-first search forms a depth-first forest composed of several depth-first trees . The edges in Eπ are called tree edges .
- 16. Depth-first Search Each vertex is initially white It is grayed when it is discovered in the search and is blackened when it is finished , that is, when its adjacency list has been examined completely. This technique guarantees that each vertex ends up in exactly one depth-first tree, so that these trees are disjoint.
- 17. Depth-first Search Each vertex v has two timestamps: the first timestamp d [ v ] records when v is first discovered (and grayed), and the second timestamp f [ v ] records when the search finishes examining v 's adjacency list (and blackens v ).
- 18. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
- 19. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
- 20. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
- 21. Property of Depth-First search An important property of depth-first search is that discovery and finishing times have parenthesis structure .
- 22. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
- 23. Classification of Edges We can define four edge types in terms of the depth-first forest Gπ produced by a depth-first search on G . 1. Tree edges are edges in the depth-first forest Gπ . Edge ( u , v ) is a tree edge if v was first discovered by exploring edge ( u , v ). 2. Back edges are those edges ( u , v ) connecting a vertex u to an ancestor v in a depth first tree. Self-loops, which may occur in directed graphs, are considered to be back edges.
- 24. Classification of Edges 3. Forward edges are those non-tree edges ( u , v ) connecting a vertex u to a descendant v in a depth-first tree. 4. Cross edges are all other edges.
- 25. Classification of Edges The DFS algorithm can be modified to classify edges as it encounters them. The key idea is that each edge ( u , v ) can be classified by the color of the vertex v that is reached when the edge is first explored . 1. WHITE indicates a tree edge, 2. GRAY indicates a back edge, and 3. BLACK indicates a forward or cross edge.
- 26. Topological Sort A topological sort of a graph can be viewed as an ordering of its vertices along a horizontal line so that all directed edges go from left to right.
- 27. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
- 28. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
- 29. Strongly Connected Components A strongly connected component of a directed graph G=(V,E) is a maximal set of vertices such that for every pair of vertices u and v in C , we have both u ~v and v ~u
- 30. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
- 31. Strongly Connected Components The transpose of a graph G, G T =(V,E T ) is used for finding the strongly connected components of a graph G=(V,E) It is interesting to observe that G and G T have exactly the same strongly connected components
- 32. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.