Graph Search Methods Spring 2007 CSE, POSTECH. Graph Search Methods A vertex u is reachable from...

Graph Search Methods

Spring 2007CSE, POSTECH


A vertex u is reachable from vertex v iff there is a path from v to u.

A search method starts at a given vertex v and visits/labels/marks every vertex that is reachable from v.

23

5

4

1

8

9

67

11

10


Many graph problems solved by a search method– Finding a path from one vertex to another.– Determining whether a graph is connected– Find a spanning tree– Finding a minimum-cost path/spanning tree

Commonly used search methods– Breadth-first search (BFS)– Depth-first search (DFS)

Breadth-First Search (BFS) Algorithm

BFS algorithm is the method of starting at a vertex and identifying all vertices reachable from it

Read Section 16.8.1 and the pseudo-code of BFS in Figure 16.7

Similar to the level-order traversal of a binary tree

Breadth-First Search (BFS) Algorithm

Visit the starting vertex and put into a FIFO queue Repeatedly remove a vertex from the queue,

visit its unvisited adjacent vertices,put newly visited vertices into the queue

All vertices reachable from the start vertex (including the start vertex) are visited

When the queue becomes empty, the search is terminated

Breadth-First Search Example

Start search at vertex 1.

23

5

4

1

8

9

67

11

10


Mark/label start vertex and put in a FIFO queue. Remove 1 from Queue;

Visit adjacent unvisited vertices;Put the visited vertices in Queue.

23

5

4

1

8

9

67

11

10

FIFO queue1


Remove 2 from Queue;Visit adjacent unvisited vertices;Put the visited vertices in Queue.

Are there any newly visited vertices?

23

5

4

1

8

9

67

11

10

FIFO queue2 4




23

5

4

1

8

9

67

11

10

FIFO queue4 5 3 6




23

5

4

1

8

9

67

11

10

FIFO queue5 3 6




23

5

4

1

8

9

67

11

10

FIFO queue3 6 9 7




23

5

4

1

8

9

67

11

10

FIFO queue6 9 7




23

5

4

1

8

9

67

11

10

FIFO queue9 7


Remove 7 from Queue;Visit adjacent unvisited vertices;Put in Queue.

23

5

4

1

8

9

67

11

10

FIFO queue7 8


23

5

4

1

8

9

67

11

10

FIFO queue8

Remove 8 from Queue;Visit adjacent unvisited vertices;Put in Queue.


Queue is empty, thus the search terminates.

23

5

4

1

8

9

67

11

10

FIFO queue


What was the order of visited vertices?

1, 2, 4, 5, 3, 6, 9, 7, 8 What is the subgraph formed by the edges used to reach n

ew vertices during the BFS? 2

3

5

4

1

8

9

67

Time Complexity

Each visited vertex is put on the queue exactly once. When a vertex is removed from the queue,

we examine its adjacent vertices.– O(n) if adjacency matrix used.– O(vertex degree) if adjacency lists used.

Total time– O(n2) if adjacency matrix used– O(n+e) if adjacency lists used

where e is the sum of vertex degrees andtherefore e is also the number of edges.

Path Finding Problem from vertex u to vertex v

Start a breadth-first search at vertex u. Terminate if either of the following occurs

– successfully when vertex v is visited or– unsuccessfully when queue becomes empty but v is not visited

Time– O(n2) when adjacency matrix used– O(n+e) when adjacency lists used

(e is number of edges)

Is a Graph Connected?

Start a breadth-first search at any vertex of the graph. A graph is connected iff all n vertices are visited. Time

– O(n2) when adjacency matrix used– O(n+e) when adjacency lists used


Connected Components

Start a breadth-first search at any unvisited vertex of the graph.

Newly visited vertices (plus edges between them)define a component.

Repeat until all vertices are visited. Time

– O(n2) when adjacency matrix used– O(n+e) when adjacency lists used


Connected Components

23

5

4

1

8

9

67

11

10

Breath-First Spanning Tree

Start a breadth-first search at any vertex of the graph. If a graph is connected, the n-1 edges used to get to

unvisited vertices define a spanning tree breadth-first spanning tree

Time– O(n2) when adjacency matrix used– O(n+e) when adjacency lists used


Breadth-First Spanning Tree

Digraph Example BF Spanning Tree of Digraph (a)

Depth-First Search (DFS) Algorithm

DFS algorithm is an alternative to BFS Similar to the pre-order traversal of a binary tree See Figure 16.18 for the pseudo-code of DFS

Visit the starting vertex v and mark it as reached Select an unreached vertex u adjacent from v If such a vertex does not exist, the search terminates, other

wise a depth-first search from u is initiated When this search is completed, we select another unreach

ed vertex adjacent to from v If such a vertex does not exist, then the search terminates,

otherwise a depth-first search starts from this vertex and so on…..

Depth-First Search (DFS) Algorithm

Start search at vertex 1. Label vertex 1 as reached and do a depth-first search

from either 2 or 4.

23

5

4

1

8

9

67

11

10

Depth-First Search Example

Label vertex 2 and do a depth-first searchfrom either 3, 5 or 6.

23

5

4

1

8

9

67

11

10


Label vertex 5 and do a depth-first searchfrom either 3, 7 or 9.

23

5

4

1

8

9

67

11

10


Label vertex 9 and do a depth-first searchfrom either 6 or 8.

23

5

4

1

8

9

67

11

10


Label vertex 8 and return to vertex 9. From vertex 9 do a DFS(6).

23

5

4

1

8

9

67

11

10


Label vertex 6 and do a depth-first searchfrom either 4 or 7.

23

5

4

1

8

9

67

11

10


Label vertex 4 and return to 6. From vertex 6 do a DFS(7).

23

5

4

1

8

9

67

11

10


Label vertex 7 and return to 6. Return to 9.

23

5

4

1

8

9

67

11

10


Return to 5.

23

5

4

1

8

9

67

11

10


Do a DFS(3).

23

5

4

1

8

9

67

11

10


Label 3 and return to 5. Return to 2. Return to 1.

23

5

4

1

8

9

67

11

10


DFS is done and return to invoking method.

23

5

4

1

8

9

67

11

10



What was the order of visited vertices?

1, 2, 5, 9, 8, 6, 4, 7, 3 What is the subgraph formed by the edges used to reach n

ew vertices during the BFS? 2

3

5

4

1

8

9

67

Same complexity as BFS. Same properties with respect to path finding,

connected components, and spanning trees. Edges used to reach unlabeled vertices

define a depth-first spanning treewhen the graph is connected.

Depth-First Search Properties

1

2

6

4

5

37

10

28

1416

25 24

22

1812

Exercise 16.41 – for graph of Figure 16.4(a)

Draw a linked adjacency-list representation

Starting from vertex 4, the order of visited vertices using BFS are?

Show the subgraph formed in part (b)

Redo parts (b) & (c) using DFS

(a) Draw a linked adjacency-list representation

Exercise 16.41 – solution (a) & (b)

(b) Starting from vertex 4, the order of visited vertices are

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

(c) Show the subgraph formed in part (b)

Exercise 16.41 – solution (c) & (d)

(d) Redo parts (b) & (c) using DFS

DFS order: 4,3,2,1,6,5,7

Do Exercise 16.41 using Figure 16.16(a) starting from vertex 1

Read Chapter 16

READING

Graph Search Methods Spring 2007 CSE, POSTECH. Graph Search Methods A vertex u is reachable from...

Documents

Transcript of Graph Search Methods Spring 2007 CSE, POSTECH. Graph Search Methods A vertex u is reachable from...