Breadth-first search (BFS)
Description:
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BFS starts visiting vertices of a graph at an arbitrary vertex by marking it as visited.
It visits graph’s vertices by across to all the neighbors of the last visited vertex
Instead of a stack, BFS uses a queue
Similar to level-by-level tree traversal
“Redraws” graph in tree-like fashion (with tree edges and cross edges for undirected
graph)
Algorithm:
ALGORITHM BFS (G)
//implements BFS traversal of a given graph
//i/p: Graph G = { V, E}
//o/p: BFS tree/forest
Mark each vertex in V with 0 as a mark of being “unvisited”
count  0
for each vertex v in V do
if v is marked with 0
bfs(v)
bfs(v)
count  count + 1
mark v with count and initialize a queue with v
while the queue is NOT empty do
for each vertex w in V adjacent to front’s vertex v do
if w is marked with 0
count  count + 1
mark w with count
add w to the queue
remove vertex v from the front of the queue
Example:
Starting at vertex A traverse the following graph using BFS traversal method:
A
B
C
D
E
F
G
H
Solution:
Step
Graph
Remarks
Insert A into queue
1
A
A(1)
2
A
Insert B, E into queue
B
A(1), B (2), E(3)
B (2), E(3)
E
3
4
A
B
E
F
Insert F, G into queue
G
NO unvisited adjacent vertex for E, backtrack
B(2), E(3), F(3), G(4)
E(3), F(3), G(4)
Delete E from queue
F(3), G(4)
5
NO unvisited adjacent vertex for F, backtrack
Delete F from queue
G(4)
6
A
B
C
E
F
G
A
B
C
Insert C, H into queue
H
G(4), C(5), H(6)
C(5), H(6)
7
8
G
F
E
D
Insert D into queue
H
C(5), H(6), D(7)
H(6), D(7)
NO unvisited adjacent vertex for H, backtrack
Delete H from queue
D(7)
9
NO unvisited adjacent vertex for D, backtrack
Delete D from queue
Queue becomes empty. Algorithm stops as all the
nodes in the given graph are visited
The BFS tree is as follows: (dotted lines are cross edges)
A
B
E
G
F
C
H
D
Applications of BFS:
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To check connectivity of a graph (number of times queue becomes empty tells the
number of components in the graph)
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To check if a graph is acyclic. (no cross edges indicates no cycle)
To find minimum edge path in a graph
Efficiency:
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Depends on the graph representation:
o Array : Θ(n2)
o List: Θ(n + e)
Difference between DFS & BFS:
DFS
Data structure Stack
No. of vertex orderings 2 orderings
Edge types Tree edge
Back edge
Applications Connectivity
Acyclicity
Articulation points
Efficiency for Θ(n2)
adjacency matrix
Efficiency for Θ(n + e)
adjacency lists
BFS
Queue
1 ordering
Tree edge
Cross edge
Connectivity
Acyclicity
Minimum edge paths
Θ(n2)
Θ(n + e)