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chapter22.ppt

  1. 1. Introduction to Algorithms Second Edition by Cormen, Leiserson, Rivest & Stein Chapter 22
  2. 2. Graph <ul><li>Two standard ways to represent a graph </li></ul><ul><li>Adjacency List </li></ul><ul><li>Adjacency Matrix </li></ul>
  3. 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. 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. 5. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
  6. 6. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
  7. 7. Breath-first Search <ul><li>Breadth-first search is one of the simplest algorithms for searching a graph </li></ul><ul><li>Given a graph G = ( V , E ) and a distinguished source vertex s , breadth-first search systematically explores the edges of G to &quot;discover&quot; every vertex that is reachable from s . </li></ul>
  8. 8. Breath-first Search <ul><li>T he algorithm discovers all vertices at distance k from s before discovering any vertices at distance k + 1. </li></ul>
  9. 9. Breath-first Search <ul><li>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. </li></ul><ul><li>A vertex is discovered the first time it is encountered during the search, at which time it becomes nonwhite. </li></ul>
  10. 10. Breath-first Search <ul><li>The color of each vertex u is stored in the variable color [ u ]. </li></ul><ul><li>The predecessor of u is stored in the variable π[ u ]. </li></ul><ul><li>The distance from the source s to vertex u computed by the algorithm is stored in d [ u ] </li></ul>
  11. 11. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
  12. 12. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
  13. 13. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
  14. 14. Depth-first Search The strategy followed by depth-first search is, as its name implies, to search &quot;deeper&quot; in the graph whenever possible.
  15. 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. 16. Depth-first Search <ul><li>Each vertex is initially white </li></ul><ul><li>It is grayed when it is discovered in the search </li></ul><ul><li>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 </li></ul><ul><li>disjoint. </li></ul>
  17. 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. 18. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
  19. 19. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
  20. 20. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
  21. 21. Property of Depth-First search An important property of depth-first search is that discovery and finishing times have parenthesis structure .
  22. 22. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
  23. 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. 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. 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. 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. 27. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
  28. 28. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
  29. 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. 30. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
  31. 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. 32. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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