CS 312: BFS and DFS implementation

Dan Sheldon

February 10, 2015

Plan

I Generic Graph Traversal

I Graph Data Structure: Adjacency Lists

I Directed Graphs and Toplogical Sort

Generic Graph Traversal

Maintain set of explored nodes and discovered

I Explored = have seen this node and explored all of its outgoingedges

I Discovered = the “frontier”. Have seen the node, but notexplored its outgoing edges.

Picture on board

Generic Graph Traversal: Node Version

Let A = data structure of discovered nodes

Traverse(s)

Put s in Awhile A is not empty do

Take a node v from Aif v is not marked ”explored” then

Mark v as ”explored”for each edge (v, w) incident to v do

Put w in Aend for

end if

end while

BFS: A is a queue (LIFO)DFS: A is a stack (LIFO)

Generic Graph Traversal

Let A = data structure of discovered nodes

Traverse(s)

Put s in Awhile A is not empty do

Take a node v from Aif v is not marked ”explored” then

Mark v as ”explored”for each edge (v, w) incident to v do

Put w in Aend for

end if

end while

BFS: A is a queue (FIFO)DFS: A is a stack (LIFO)

Examples

“Proof by example”

I BFS with queue

I DFS with stack

Generic Graph Traversal: Edge Version

To return a tree, put edges instead of nodes on TODO list

Traverse(s)

T = {} . BFS/DFS treePut (�, s) in Awhile A is not empty do

Take an edge (u, v) from Aif v is not marked ”explored” then

T = T [ (u, v)Mark v as ”explored”for each edge (v, w) incident to v do

Put edge (v, w) in Aend for

end if

end while

Return T

Running Time?

Traverse(s)

Put s in Awhile A is not empty do

Take a node v from Aif v is not marked ”explored” then

Mark v as ”explored”for each edge (v, w) incident to v do

Put w in Aend for

end if

end while

Discuss on board. Running time is O(m+ n), pending correct datastructure

n2m

2m2m

2m

2m

Representing Graphs: Adjacency List

Adjacency list. Each node keeps a (linked) list of neighbors.!

How long to find a specific edge?!How long to find all edges incident on a node?

1

3

5 4

21

2

3

4

5

2 4 5

1 3 4

2 5

1 2

1 3

Graph Traversal: Summary

BFS/DFS: O(m+ n)

I Is G connected?I Find connected components of GI Find distance of every vertex from sourceI Get BFS/DFS trees (useful in other applications)

BFS: explore by distance; layers; queueDFS: explore deeply; recursive; stack