BreadthFirst Search

1
Breadth-First Search
Seminar Networking Algorithms CS and EE
Dept. Lulea University of Technology 27 Jan.
2005 Mohammad Reza Akhavan
2
Outline (BFS)

• Rooted Tree
• Spanning Tree
• Breadth-First Search
• BFS Algorithm
• Example

3
Rooted Tree

• A rooted tree is a tree in which a special node
  is singled out. This node is called the "root. A
  tree which is not rooted is sometimes called a
  free tree.
• Rooted Tree
  Free Tree

4
Spanning Tree

• A spanning tree in a graph G with n nodes and m
  edges
• A sub-graph that connects all the nodes.
• A sub-graph with no cycles.
• A sub-graph with mn-1 edges.

5
Breadth-First Search

• Visits all the nodes and edges of G
• Determines whether G is connected
• Computes the connected components of G
• Find and report a path with the minimum number of
  edges between two given nodes
• Find a simple cycle, if there is one
• Provide the shortest path from a given root to
  all other nodes of the network

6
BFS for Shortest Path (i0)
s
0

• The algorithm uses a mechanism for setting and
  getting labels of nodes and edges.
• Nodes whose distance from s is 0 are labeled.

7
BFS for Shortest Path (i1)
s
1
0
1
1
Nodes whose distance from s is 1 are labeled.
8
BFS for Shortest Path (i2)
1
s
0
2
2
1
2
1
Nodes whose distance from s is 2 are labeled.
9
BFS for Shortest Path (i3)
Nodes whose distance from s is 3 are labeled. The
next iteration finds out that the whole graph is
labeled and therefore the procedure stops.
10
The BFS Tree
11
BFS Algorithm

• The algorithm uses a mechanism for setting and
  getting labels of nodes and edges
• Search a graph by increasing distance from the
  starting vertex (or from the starting vertices in
  case of several connected components).
• Can think of creating one level after the other
  (by increasing depth).

12
BFS Asynchronies Mode

0
1
1
1
3
2
3
2
2
2
4
2
3
3
4
3
4
2
3
4
4
5
4
7
3
5
7
4
7
6
6
13
References

• Introduction to Distributed Algorithms
• by Gerard Tel.
• Google !