Breadth First Search. cse2011 section 13.3 of textbook

Transcription

1 Breadth irst Search cse section. of textbook

2 Graph raversal (.) Application example Given a graph representation and a vertex s in the graph, find all paths from s to the other vertices. wo common graph traversal algorithms: Breadth-irst Search (BS) Idea is similar to level-order traversal for trees. Implementation uses a queue. Gives shortest path from a vertex to another. Depth-irst Search (DS) Idea is similar to preorder traversal for trees (visit a node then visit its children recursively). Implementation uses a stack (implicitly via recursion).

3 BS and Shortest Path Problem Given any source vertex s, BS visits the other vertices at increasing distances away from s. In doing so, BS discovers shortest paths from s to the other vertices. What do we mean by distance? he number of edges on a path from s (unweighted graph). Example Consider s=vertex s Nodes at distance?,,, Nodes at distance?,,, Nodes at distance?

4 How Does BS Work? Similarly to level-order traversal for trees. he BS code we will discuss works for both directed and undirected graphs.

5 Skeleton of BS Algorithm output v;

6 BS Algorithm flag[ ]: visited or not output v;

7 BS Example Adjacency List Visited able (/) source Initialize visited table (all alse) Q = { } Initialize Q to be empty

8 Adjacency List source Visited able (/) Q = { } lag that has been visited Place source on the queue

9 Adjacency List Visited able (/) Neighbors source Mark neighbors as visited,, Q = {} {,, } Dequeue. Place all unvisited neighbors of on the queue

10 Adjacency List Visited able (/) source Neighbors Mark newly visited neighbors, Q = {,, } {,,, } Dequeue. -- Place all unvisited neighbors of on the queue. -- Notice that is not placed on the queue again, it has been visited!

11 Adjacency List Visited able (/) Neighbors source Mark newly visited neighbors, Q = {,,, } {,,,, } Dequeue. -- Place all unvisited neighbors of on the queue. -- Only nodes and haven t been visited yet.

12 Adjacency List Visited able (/) source Neighbors Q = {,,,, } {,,, } Dequeue. -- has no unvisited neighbors!

13 Adjacency List Visited able (/) Neighbors source Q = {,,, } {,, } Dequeue. -- has no unvisited neighbors!

14 Adjacency List Visited able (/) source Neighbors Q = {,, } {, } Dequeue. -- has no unvisited neighbors!

15 Adjacency List Visited able (/) Neighbors source Mark new visited Vertex Q = {, } {, } Dequeue. -- place neighbor on the queue.

16 Adjacency List Visited able (/) source Neighbors Mark new visited Vertex Q = {, } {, } Dequeue. -- place neighbor on the queue

17 Adjacency List Visited able (/) source Neighbors Q = {, } { } Dequeue. -- no unvisited neighbors of

18 Adjacency List Visited able (/) source Neighbors Q = { } { } Dequeue. -- no unvisited neighbors of

19 Adjacency List Visited able (/) source What did we discover? Q = { } SOP!!! Q is empty!!! Look at visited tables. here exists a path from source vertex to all vertices in the graph

20 Assume adjacency list Running ime of BS V = number of vertices; E = number of edges Each vertex will enter Q at most once. dequeue is O(). he for loop takes time proportional to deg(v).

21 Running ime of BS () Recall: Given a graph with E edges Σ vertex v deg(v) = E he total running time of the while loop is: O( Σ vertex v ( + deg(v)) ) = O(V+E) his is the sum over all the iterations of the while loop! Homework: What is the running time of BS if we use an adjacency matrix?

22 BS and Unconnected Graphs L N P O Q R A graph may not be connected (strongly connected) enhance the above BS code to accommodate this case. M D E s A C G B K A graph with components H

23 Recall the BS Algorithm output ( v );

24 Enhanced BS Algorithm A graph with components N L M A B C H K We can re-use the previous BS(s) method to compute the connected components of a graph G. BSearch( G ) { i = ; // component number for every vertex v flag[v] = false; for every vertex v if ( flag[v] == false ) { print ( Component + i++ ); BS( v ); } }

25 Applications of BS What can we do with the BS code we just discussed? Is there a path from source s to a vertex v? Check flag[v]. Is an undirected graph connected? Scan array flag[ ]. If there exists flag[u] = false then Is a directed graph strongly connected? Scan array flag[ ]. If there exists flag[u] = false then

26 Next lecture Depth irst Search (DS) Review inal exam