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 m=n-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 (i=0) 0 s 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 (i=1) 0 1 1 1 s Nodes whose distance from s is 1 are labeled.

8. BFS for Shortest Path (i=2) 0 1 1 2 2 1 2 s Nodes whose distance from s is 2 are labeled.

9. BFS for Shortest Path (i=3) 0 1 1 2 2 1 2 s 3 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 0 1 1 2 2 1 2 s 3 0 1 1 2 2 1 2 s 3

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 3 32 0 4 1 1 1 4 22 2 3 4 4 3 2 7 2 5 3 7 6 4 3 4 5 4 3 7 6

13. References: Introduction to Distributed Algorithms by Gerard Tel. Google !