CS473-Algorithms I
Lecture 14-A
Graph Searching: Breadth-First Search
CS 473
Lecture 14
1
Graph Searching: Breadth-First Search
Graph G (V, E), directed or undirected with adjacency list repres.
GOAL: Systematically explores edges of G to
• discover every vertex reachable from the source vertex s
• compute the shortest path distance of every vertex
from the source vertex s
• produce a breadth-first tree (BFT) G with root s
 BFT contains all vertices reachable from s
 the unique path from any vertex v to s in G
constitutes a shortest path from s to v in G
IDEA: Expanding frontier across the breadth -greedy• propagate a wave 1 edge-distance at a time
• using a FIFO queue: O(1) time to update pointers to both ends
CS 473
Lecture 14
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Breadth-First Search Algorithm
Maintains the following fields for each u  V
• color[u]: color of u
 WHITE : not discovered yet
 GRAY : discovered and to be or being processed
 BLACK: discovered and processed
• [u]: parent of u (NIL of u  s or u is not discovered yet)
• d[u]: distance of u from s
Processing a vertex  scanning its adjacency list
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Lecture 14
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Breadth-First Search Algorithm
BFS(G, s)
for each u  V {s} do
color[u]  WHITE
[u]  NIL; d [u]  
color[s]  GRAY
[s]  NIL; d [s]  0
Q  {s}
while Q   do
u  head[Q]
for each v in Adj[u] do
if color[v]  WHITE then
color[v]  GRAY
[v]  u
d [v]  d [u]  1
ENQUEUE(Q, v)
DEQUEUE(Q)
color[u]  BLACK
CS 473
Lecture 14
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Breadth-First Search
Sample Graph:
s
FIFO
queue Q
0
a
d
c
b
a
CS 473
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e
f
g
just after
processing vertex
i
h
Lecture 14
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Breadth-First Search
s
d
FIFO
queue Q
e
a
a,b,c
0
a
1
c
1
b
f
g
CS 473
just after
processing vertex
a
i
h
Lecture 14
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Breadth-First Search
s
FIFO
queue Q
0
a
d
1
c
a
a,b,c
a,b,c,f
1
b
e
2
f
g
CS 473
just after
processing vertex
a
b
i
h
Lecture 14
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Breadth-First Search
s
FIFO
queue Q
0
a
d
1
c
1
2
b
e
2
f
g
CS 473
i
a
a,b,c
a,b,c,f
a,b,c,f,e
just after
processing vertex
a
b
c
h
Lecture 14
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Breadth-First Search
s
FIFO
queue Q
0
a
d
1
c
1
2
b
e
2
f
g
a
a,b,c
a,b,c,f
a,b,c,f,e
a,b,c,f,e,g,h
a
b
c
f
h
3
CS 473
i
just after
processing vertex
3
Lecture 14
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Breadth-First Search
s
0
a
FIFO
queue Q
3
d
1
c
1
2
b
e
2
f
i
3
g
h
3
just after
processing vertex
a
a,b,c
a,b,c,f
a,b,c,f,e
a,b,c,f,e,g,h
a,b,c,f,e,g,h,d,i
a
b
c
f
e
3
all distances are filled in after processing e
CS 473
Lecture 14
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Breadth-First Search
s
0
a
FIFO
queue Q
3
d
1
c
1
2
b
e
2
f
i
3
g
h
3
CS 473
just after
processing vertex
a
a,b,c
a,b,c,f
a,b,c,f,e
a,b,c,f,e,g,h
a,b,c,f,e,g,h,d,i
a
b
c
f
g
3
Lecture 14
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Breadth-First Search
s
0
a
FIFO
queue Q
3
d
1
c
1
2
b
e
2
f
i
3
g
h
3
CS 473
just after
processing vertex
a
a,b,c
a,b,c,f
a,b,c,f,e
a,b,c,f,e,g,h
a,b,c,f,e,g,h,d,i
a
b
c
f
h
3
Lecture 14
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Breadth-First Search
s
0
a
FIFO
queue Q
3
d
1
c
1
2
b
e
2
f
i
3
g
h
3
CS 473
just after
processing vertex
a
a,b,c
a,b,c,f
a,b,c,f,e
a,b,c,f,e,g,h
a,b,c,f,e,g,h,d,i
a
b
c
f
d
3
Lecture 14
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Breadth-First Search
s
0
a
FIFO
queue Q
3
d
1
c
1
2
b
e
2
f
i
3
g
h
3
just after
processing vertex
a
a,b,c
a,b,c,f
a,b,c,f,e
a,b,c,f,e,g,h
a,b,c,f,e,g,h,d,i
a
b
c
f
i
3
algorithm terminates: all vertices are processed
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Lecture 14
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Breadth-First Search Algorithm
Running time: O(VE)  considered linear time in graphs
• initialization: (V)
• queue operations: O(V)
 each vertex enqueued and dequeued at most once
 both enqueue and dequeue operations take O(1) time
• processing gray vertices: O(E)
 each vertex is processed at most once and
 | Adj[u ] | ( E )
uV
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Lecture 14
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Breadth-First Tree Generated by BFS
LEMMA 4: predecessor subgraph G(V, E) generated by
BFS(G, s) , where V {v  V: [v]  NIL}{s} and
E {([v],v)  E: v  V {s}}
is a breadth-first tree such that
 V consists of all vertices in V that are reachable from s
 v  V , unique path p(v, s) in G constitutes a sp(s, v) in G
PRINT-PATH(G, s, v)
if v  s then print s
else if [v]  NIL then
print no “sv path”
else
PRINT-PATH(G, s, [v] )
print v
CS 473
Lecture 14
Prints out vertices on a
sv shortest path
16
Breadth-First Tree Generated by BFS
BFS(G,a) terminated
BFT generated by BFS(G,a)
s
0
a
3
d
1
0
s
a
3
d
1
c
c
1
2
b
1
e
2
b
e
2
2
f
i
f
i
3
g
h
3
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g
3
h
3
Lecture 14
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