1 / 10

Chapter 15 Graphs

Modified. Chapter 15 Graphs. Part D – Graph Algorithms 2. Basic Graph Algorithms. Un-weighted graphs/digraphs Spanning tree Topological sort. Constructing a spanning tree. Assume an undirected graph

bryson
Télécharger la présentation

Chapter 15 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. Modified Chapter 15 Graphs Part D – Graph Algorithms 2

  2. Basic Graph Algorithms • Un-weighted graphs/digraphs • Spanning tree • Topological sort Java Software Structures, 4th Edition, Lewis/Chase

  3. Java Software Structures, 4th Edition, Lewis/Chase

  4. Constructing a spanning tree • Assume an undirected graph • A spanning tree is a subgraphof the original graph which includes all of the vertices but only some of the edges. • If the graph has N vertices, the spanning tree will have N-1 edges. • It will have no cycles; it is a tree • It uses the minimum number of edges to connect all of the vertices together • Lots of practical applications • If a network contains any cycles, an edge can be removed and the graph is still connected. Java Software Structures, 4th Edition, Lewis/Chase

  5. Finding a spanning tree • Do a depth first traversal and record all edges traversed to get to a “new” edge. Java Software Structures, 4th Edition, Lewis/Chase

  6. Find Spanning tree – DFT starting from vertex v create a stack S Initialize spanning tree edge list to empty mark v as visited and push v onto S while S is non-empty peek at the top u of S if u has an (unvisited) neighbor w, add edge from u to w to spanning tree, mark w and push it onto S else pop S Java Software Structures, 4th Edition, Lewis/Chase

  7. Java Software Structures, 4th Edition, Lewis/Chase

  8. Topological sort A diagraph is acyclic if it has no cycles. Such a graph is often referred to as a directed acyclic graph, or DAG, for short. DAGs are used in many applications to indicate precedence among events. A directed graph G is acyclic if and only if a depth-first search of G yields no back edges. Topological sort of a DAG: a linear ordering (sequence) of vertices such that if (u, v) is an edge of the DAG, then u appears somewhere before v in that ordering. Java Software Structures, 4th Edition, Lewis/Chase

  9. Topological sort • Do a DFT of the DAG and put vertices onto the front of a linked list as they are finished. • https://www.cs.usfca.edu/~galles/visualization/TopoSortDFS.html Java Software Structures, 4th Edition, Lewis/Chase

  10. Alternate algorithm for top. sort of DAG • Compute the indegrees of all vertices • Find a vertex U with indegree 0 and append it to the ordering • If there is no such vertex then there is a cycleand the vertices cannot be ordered. Stop. • Remove U and all its edges (U,V) from the graph. • Update the indegrees of the remaining vertices. • Repeat steps 2 through 4 while there are vertices to be processed. Java Software Structures, 4th Edition, Lewis/Chase

More Related