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CSC 213

A. L 0. A. B. C. D. L 1. B. C. D. E. F. L 2. E. F. CSC 213. Lecture 21: Graph Traversals. Subgraphs. A subgraph S of a graph G is a graph such that All edges & vertices in S also exist within G Spanning subgraph of G contains all of G ’s vertices. Subgraph.

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CSC 213

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  1. A L0 A B C D L1 B C D E F L2 E F CSC 213 Lecture 21: Graph Traversals

  2. Subgraphs • A subgraph S of a graph G is a graph such that • All edges & vertices in S also exist within G • Spanning subgraph of G contains all of G’svertices Subgraph Spanning subgraph

  3. Graph is connected path exists between every pair of vertices Does not require an edge between all vertices, however Connected component is a maximal connected subgraph So you could not add another connected node to the subgraph Connectivity Connected graph Graph with two connected components

  4. Tree • Tree is a graph • Must be undirected • Must be connected • Cannot contain a cycle • Forest is graph containing a number of trees Tree Forest

  5. Spanning Tree • Spanning tree of a connected graph is a subgraph containing every vertex and no cycles Graph Spanning tree

  6. Depth- & Breadth-First Search • Common techniques to traverse a graph • DFS and BFS traversal of a graph G • Visits all vertices and edges in G • Computes if G is connected and, if it is not, finds connected components • Computes spanning tree/spanning forest • They help solve many graph problems

  7. DFS and Maze Traversal • Classic strategy for exploring maze • Mark the intersections, corners and dead ends visited • We mark corridors (edges) followed • Can follow lines to get back to where we started the maze

  8. A A B D E B D E C C Example unexplored vertex A visited vertex A unexplored edge discovery edge back edge A B D E C

  9. A A A B D E B D E B D E C C C A B D E C Example (cont.)

  10. Property 1 Visits all vertices and edges within a connected component Property 2 Edges followed during the depth-first search form a spanning tree for the connected component A B D E C Properties of DFS

  11. Visitor Pattern • Another commonly used coding method • Often uses DFS as a Template Method • Defines at least 5 methods: • initResult() – called at start of traversal • startVisit() – called at start of vertex (node) processing • 1 or more methods to analyze data during the traversal • finishVisit() – called at end of vertex processing • result() – called to get result of traversal

  12. L0 A L1 B C D E F Example unexplored vertex A visited vertex A unexplored edge discovery edge cross edge L0 L0 A A L1 L1 B C D B C D E F E F

  13. L0 L0 A A L1 L1 B C D B C D L2 E F E F L0 L0 A A L1 L1 B C D B C D L2 L2 E F E F Example (cont.)

  14. L0 L0 A A L1 L1 B C D B C D L2 L2 E F E F Example (cont.) L0 A L1 B C D L2 E F

  15. Property 1 Visits all the vertices and edges within connected component Property 2 Discovery edges form a spanning tree of the component Property 3 For each vertex v in Li Path from s to v in spanning treehas exactlyi edges Paths from s to v in G has at leasti edges Properties of BFS A B C D E F L0 A L1 B C D L2 E F

  16. L0 A L1 B C D L2 E F DFS vs. BFS A B C D E F DFS BFS

  17. Back edge(v,w) w is an ancestor of v in the tree of discovery edges Cross edge(v,w) w is in the same level as v or in the next level in the tree of discovery edges A L0 A B C D L1 B C D E F L2 E F DFS vs. BFS (cont.) DFS BFS

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