Breadth First Search. Graph Traversal. CSE 2011 Winter Application examples. Two common graph traversal algorithms

Transcription

1 Breadth irst Search CSE Winter Graph raversal Application examples Given a graph representation and a vertex s in the graph ind all paths from s to the other vertices wo common graph traversal algorithms Breadth-irst Search (BS) Depth-irst Search (DS)

2 BS and Shortest Path Problem Given any 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? How Does BS Work? Similarly to level-order traversal for trees Examples: Code: similar to code for topological sort (see the next slide) flag[v] = false: we have not visited v flag[v] = true: we already visited v Why does BS need a flag for each vertex (topological sort does not)? he code works for both directed and undirected graphs

3 opological Sort ind all starting points Decrement indegree(w) Place new start vertices on the Q BS Algorithm flag[ ]: visited or not

4 BS Example Visited able (/) Q = { } Initialize visited table (all alse) Initialize Q to be empty Visited able (/) Q = { } lag that has been visited Place on the queue

5 Visited able (/) Mark neighbors as visited,, Q = {} {,, } Dequeue. Place all unvisited neighbors of on the queue Visited able (/) 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!

6 Visited able (/) Mark newly visited neighbors, Q = {,,, } {,,,, } Dequeue. -- Place all unvisited neighbors of on the queue. -- Only nodes and haven t been visited yet. Visited able (/) Q = {,,,, } {,,, } Dequeue. -- has no unvisited neighbors!

7 Visited able (/) Q = {,,, } {,, } Dequeue. -- has no unvisited neighbors! Visited able (/) Q = {,, } {, } Dequeue. -- has no unvisited neighbors!

8 Visited able (/) Mark new visited Vertex Q = {, } {, } Dequeue. -- place neighbor on the queue. Visited able (/) Mark new visited Vertex Q = {, } {, } Dequeue. -- place neighbor on the queue

9 Visited able (/) Q = {, } { } Dequeue. -- no unvisited neighbors of Visited able (/) Q = { } { } Dequeue. -- no unvisited neighbors of

10 Visited able (/) What did we discover? Q = { } SOP!!! Q is empty!!! Look at visited tables. here exists a path from vertex to all vertices in the graph Applications of BS What can we do with the BS code we just discussed? Is there a path from 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 o output the contents (e.g., the vertices) of a connected (strongly connected) graph What if the graph is not connected (weakly connected)? Add just a little bit of code and invoke method BS( ) discussed later.

11 Other Applications of BS o find the shortest path from a vertex s to a vertex v in an unweighted graph o find the length of such a path o find out if a graph contains cycles o find the connected components of a graph that is not connected o construct a BS tree/forest from a graph Running ime of BS Assume adjacency list 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).

12 Running ime of BS (cont d) 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? Applications of BS Is there a path from s to a vertex v? Is an undirected graph connected? Is a directed graph strongly connected? o output the contents (e.g., the vertices) of a connected (strongly connected) graph Discuss non-connected (weakly connected) graphs later o find the shortest path from a vertex s to a vertex v in an unweighted graph o find the length of such a path o find out if a graph contains cycles o find the connected components of a graph that is not connected o construct a BS tree/forest from a graph