Graph Searching CSE 373 Data Structures Lecture 20 Readings - - PowerPoint PPT Presentation

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Graph Searching CSE 373 Data Structures Lecture 20 Readings - - PowerPoint PPT Presentation

Apr 09, 2023 301 likes •593 views

Graph Searching CSE 373 Data Structures Lecture 20 Readings Reading Sections 9.5 and 9.6 3/7/03 Graph Searching - Lecture 20 2 Graph Searching Find Properties of Graphs Spanning trees Connected components

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Graph Searching

CSE 373 Data Structures Lecture 20

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3/7/03 Graph Searching - Lecture 20 2

Readings

  • Reading

› Sections 9.5 and 9.6

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Graph Searching

  • Find Properties of Graphs

› Spanning trees › Connected components › Bipartite structure › Biconnected components

  • Applications

› Finding the web graph – used by Google and

  • thers

› Garbage collection – used in Java run time system › Alternating paths for matching

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3/7/03 Graph Searching - Lecture 20 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

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3/7/03 Graph Searching - Lecture 20 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

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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

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3/7/03 Graph Searching - Lecture 20 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

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3/7/03 Graph Searching - Lecture 20 8

Example of DFS: Graph connectivity and spanning tree

1 2 7 5 4 6 3 DFS(1)

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Example Step 2

1 2 7 5 4 6 3 DFS(1) DFS(2)

Red links will define the spanning tree if the graph is connected

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Example Step 5

1 2 7 5 4 6 3 DFS(1) DFS(2) DFS(3) DFS(4) DFS(5)

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Example Steps 6 and 7

1 2 7 5 4 6 3 DFS(1) DFS(2) DFS(3) DFS(4) DFS(5) DFS(3) DFS(7)

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Example Steps 8 and 9

1 2 7 5 4 6 3 DFS(1) DFS(2) DFS(3) DFS(4) DFS(5) DFS(7)

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Example Step 10 (backtrack)

1 2 7 5 4 6 3 DFS(1) DFS(2) DFS(3) DFS(4) DFS(5)

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Example Step 12

1 2 7 5 4 6 3 DFS(1) DFS(2) DFS(3) DFS(4) DFS(6)

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Example Step 13

1 2 7 5 4 6 3 DFS(1) DFS(2) DFS(3) DFS(4) DFS(6)

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Example Step 14

1 2 7 5 4 6 3 DFS(1) DFS(2) DFS(3) DFS(4)

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Example Step 17

1 2 7 5 4 6 3 DFS(1)

All nodes are marked so graph is connected; red links define a spanning tree

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Adjacency List Implementation

  • Adjacency lists

1 2 3 4 5 6 7 2 4 6 3 1 7 4 5 5 6 1 3 7 4 3 1 4 5 2

1 2 7 5 4 6 3

G M Index next

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Connected Components

1 2 3 9 8 6 10 4 5 7 11

3 connected components

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Connected Components

1 2 3 9 8 6 10 4 5 7 11

3 connected components are labeled

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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]; while v ≠ null do if M[v.index] = 0 then DFS(G,M,v.index,label); v := v.next; }

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Performance DFS

  • n vertices and m edges
  • Storage complexity O(n + m)
  • Time complexity O(n + m)
  • Linear Time!
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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}

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Can do Connectivity using BFS

  • Uses a queue to order search

Queue = 1 1 2 7 5 4 6 3

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Beginning of example

1 2 7 5 4 6 3 Queue = 2,4,6

Mark while on queue to avoid putting in queue more than once

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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

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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