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Graph Searching

This lecture covers graph searching methodologies including Breadth-First Search (BFS) and Depth-First Search (DFS), their applications in finding properties of graphs like spanning trees and connected components, and practical examples of graph traversal. The DFS algorithm is detailed, emphasizing the recursive marking process for identifying connected components. Additionally, the implementation of adjacency lists and the use of DFS for labeling connected components are discussed. The lecture also compares BFS and DFS approaches, highlighting their performance metrics and the ability to find minimum spanning trees in weighted graphs.

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Graph Searching

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  1. Graph Searching CSE 373 Data Structures Lecture 16

  2. Readings • Reading • Sections 9.5 and 9.6 Graph Searching - Lecture 16

  3. Graph Searching • Find Properties of Graphs • Spanning trees • Connected components • Bipartite structure • Biconnected components • Applications • Finding the web graph – used by Google and others • Garbage collection – used in Java run time system • Alternating paths for matching Graph Searching - Lecture 16

  4. Graph Searching Methodology Breadth-First Search (BFS) • Breadth-First Search (BFS) • Use a queue to explore neighbors of source vertex, then neighbors of neighbors etc. • All nodes at a given distance (in number of edges) are explored before we go further Graph Searching - Lecture 16

  5. Graph Searching Methodology Depth-First Search (DFS) • Depth-First Search (DFS) • Searches down one path as deep as possible • When no nodes available, it backtracks • When backtracking, it explores side-paths that were not taken • Uses a stack (instead of a queue in BFS) • Allows an easy recursive implementation Graph Searching - Lecture 16

  6. Depth First Search Algorithm • Recursive marking algorithm • Initially every vertex is unmarked DFS(i: vertex) mark i; for each j adjacent to i do if j is unmarked then DFS(j) end{DFS} Marks all vertices reachable from i Graph Searching - Lecture 16

  7. DFS Application: Spanning Tree • Given a (undirected) graph G(V,E) a spanning tree of G is a graph G’(V’,E’) • V’ = V, the tree touches all vertices (spans) the graph • E’ is a subset of E such G’ is connected and there is no cycle in G’ • A graph is connected if given any two vertices u and v, there is a path from u to v Graph Searching - Lecture 16

  8. Example of DFS: Graph connectivity and spanning tree 2 DFS(1) 1 7 3 5 6 4 Graph Searching - Lecture 16

  9. Example Step 2 2 DFS(1) DFS(2) 1 7 3 5 6 4 Red links will define the spanning tree if the graph is connected Graph Searching - Lecture 16

  10. Example Step 5 2 DFS(1) DFS(2) DFS(3) DFS(4) DFS(5) 1 7 3 5 6 4 Graph Searching - Lecture 16

  11. Example Steps 6 and 7 2 DFS(1) DFS(2) DFS(3) DFS(4) DFS(5) DFS(3) DFS(7) 1 7 3 5 6 4 Graph Searching - Lecture 16

  12. Example Steps 8 and 9 2 DFS(1) DFS(2) DFS(3) DFS(4) DFS(5) DFS(7) 1 7 3 5 6 Now back up. 4 Graph Searching - Lecture 16

  13. Example Step 10 (backtrack) 2 DFS(1) DFS(2) DFS(3) DFS(4) DFS(5) 1 7 3 5 Back to 5, but it has no more neighbors. 6 4 Graph Searching - Lecture 16

  14. Example Step 12 2 DFS(1) DFS(2) DFS(3) DFS(4) DFS(6) 1 7 3 5 6 Back up to 4. From 4 we can get to 6. 4 Graph Searching - Lecture 16

  15. Example Step 13 2 DFS(1) DFS(2) DFS(3) DFS(4) DFS(6) 1 7 3 5 From 6 there is nowhere new to go. Back up. 6 4 Graph Searching - Lecture 16

  16. Example Step 14 2 DFS(1) DFS(2) DFS(3) DFS(4) 1 7 3 5 Back to 4. Keep backing up. 6 4 Graph Searching - Lecture 16

  17. Example Step 17 2 DFS(1) 1 7 All the way back to 1. Done. 3 5 6 4 All nodes are marked so graph is connected; red links define a spanning tree Graph Searching - Lecture 16

  18. Adjacency List Implementation • Adjacency lists 2 G 1 M 7 1 2 3 4 5 6 7 0 2 4 6 0 3 1 7 3 0 4 5 0 5 6 1 3 5 0 3 7 4 0 6 1 4 0 4 5 2 Index next Graph Searching - Lecture 16

  19. Another Use for Depth First Search: Connected Components 2 1 3 7 10 4 9 5 6 8 11 3 connected components Graph Searching - Lecture 16

  20. Connected Components 2 1 3 7 10 4 9 5 6 8 11 3 connected components are labeled Graph Searching - Lecture 16

  21. Depth-first Search for Labeling Connected components Main { i : integer for i = 1 to n do M[i] := 0; label := 1; for i = 1 to n do if M[i] = 0 then DFS(G,M,i,label); label := label + 1; } DFS(G[]: node ptr array, M[]: int array, i,label: int) { v : node pointer; M[i] := label; v := G[i]; // first neighbor // while v  null do if M[v.index] = 0 then DFS(G,M,v.index,label); v := v.next; // next neighbor // } Graph Searching - Lecture 16

  22. Connected Components for Image Analysis 4 1 2 3 Graph Searching - Lecture 16

  23. Performance DFS • n vertices and m edges • Storage complexity O(n + m) • Time complexity O(n + m) • Linear Time! Graph Searching - Lecture 16

  24. Breadth-First Search BFS Initialize Q to be empty; Enqueue(Q,1) and mark 1; while Q is not empty do i := Dequeue(Q); for each j adjacent to i do if j is not marked then Enqueue(Q,j) and mark j; end{BFS} Graph Searching - Lecture 16

  25. Can do Connectivity using BFS • Uses a queue to order search 2 1 7 3 5 6 4 Queue = 1 Graph Searching - Lecture 16

  26. Beginning of example 2 1 7 3 5 6 4 Mark while on queueto avoid putting inqueue more than once Queue = 2,4,6 Graph Searching - Lecture 16

  27. Depth-First vs Breadth-First • Depth-First • Stack or recursion • Many applications • Breadth-First • Queue (recursion no help) • Can be used to find shortest paths from the start vertex • Can be used to find short alternating paths for matching Graph Searching - Lecture 16

  28. Minimum Spanning Tree • Edges are weighted: find minimum cost spanning tree • Applications • Find cheapest way to wire your house • Find minimum cost to wire a message on the Internet Graph Searching - Lecture 16

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