1 / 13

Chapter 9: Graphs

Mark Allen Weiss: Data Structures and Algorithm Analysis in Java. Chapter 9: Graphs. Breadth-First and Depth-First Search. Lydia Sinapova, Simpson College. Breadth-First and Depth-First Search. BFS Basic Algorithm BFS Complexity DFS Algorithm DFS Implementation

talasi
Télécharger la présentation

Chapter 9: Graphs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Mark Allen Weiss: Data Structures and Algorithm Analysis in Java Chapter 9: Graphs Breadth-First and Depth-First Search Lydia Sinapova, Simpson College

  2. Breadth-First and Depth-First Search • BFS Basic Algorithm • BFS Complexity • DFS Algorithm • DFS Implementation • Relation between BFS and DFS

  3. BFS – Basic Idea Given a graph with N vertices and a selected vertex A:  for(i = 1; there are unvisited vertices ; i++) Visit all unvisited vertices at distance i (i is the length of the shortest path between A and currently processed vertices) Queue-based implementation

  4. BFS – Algorithm BFS algorithm 1. Store source vertex S in a queue and mark as processed 2. while queue is not empty Read vertex vfrom the queue for all neighbors w: If w is not processed Mark as processed Append in the queue Record the parent of w to be v(necessary only if we need the shortest path tree)

  5. Breadth-first traversal: 1, 2, 3, 4, 6, 5 1: starting node 2, 3, 4 : adjacent to 1 (at distance 1 from node 1) 6 : unvisited adjacent to node 2. 5 : unvisited, adjacent to node 3 Example 1 Adjacency lists 1: 2, 3, 4 2: 1, 3, 6 3: 1, 2, 4, 5, 6 4: 1, 3, 5 5: 3, 4 6: 2, 3 4 2 3 5 The order depends on the order of the nodes in the adjacency lists 6

  6. Shortest Path Tree Consider the distance table: The table defines the shortest path tree, rooted at A.

  7. BFS – Complexity Step 1 : read a node from the queue O(V) times. Step 2 : examine all neighbors, i.e. we examine all edges of the currently read node. Not oriented graph: 2*E edges to examine Hence the complexity of BFS isO(V + 2*E)

  8. Depth-First Search Proceduredfs(s) mark all vertices in the graph as not reached invoke scan(s) Procedurescan(s) mark and visit s for each neighbor w of s if the neighbor is not reached invoke scan(w)

  9. Depth-First Search with Stack Initialization: mark all vertices as unvisited, visit(s) while the stack is not empty: pop (v,w) if w is not visited add (v,w) to tree T visit(w) Procedure visit(v) mark v as visited for each edge (v,w) push (v,w) in the stack

  10. Recursive DFS DepthFirst(Vertex v) visit(v); for each neighborw of v if (w is not visited) add edge (v,w) to tree T DepthFirst(w) Visit(v) mark v as visited

  11. Example Depth first traversal: 1, 2, 6, 3, 5, 4 the particular order is dependent on the order of nodes in the adjacency lists 1 4 Adjacency lists 1: 2, 3, 4 2: 6, 3, 1 3: 1, 2, 6, 5, 4 4: 1, 3, 5 5: 3, 4 6: 2, 3 2 3 5 6

  12. bfs(G) list L = empty tree T = empty choose a starting vertex x visit(x) while(L nonempty) remove edge (v,w) from beginning of L if w not visited add (v,w) to T visit(w) BFS and DFS dfs(G) list L = empty tree T = empty choose a starting vertex x visit(x) while(L nonempty) remove edge (v,w) from end of L if w not visited add (v,w) to T visit(w) Visit ( vertex v) mark v as visited for each edge (v,w) add edge (v,w) to end of L

  13. Applications of DFS • Trees: preorder traversal • Graphs: • Connectivity • Biconnectivity – articulation points • Euler graphs

More Related