lec 09-graphs-bfs-dfs.ppt

Graphs
David Kauchak
cs302
Spring 2013

Graphs
What is a graph?
A
B
C
E
D
F
G

Graphs
A graph is a set of vertices V and a set of
edges (u,v)  E where u,v  V
A
B
C
E
D
F
G

Graphs
How do graphs differ? What are graph
characteristics we might care about?
A
B
C
E
D
F
G

Different types of graphs
Undirected – edges do not have a direction
A
B
C
E
D
F
G

Directed – edges do have a direction
A
B
C
E
D
F
G

7
Graph Variations
 More variations:
 A multigraph allows multiple edges between the
same vertices
 E.g., the call graph in a program (a function can get
called from multiple other functions)

Weighted – edges have an associated weight
A
B
C
E
D
F
G
8
2
7
20
1
7
2

Terminology
Path – A path is a list of vertices p1,p2,…pk
where there exists an edge (pi,pi+1)  E
A
B
C
E
D
F
G

A
B
C
E
D
F
G
{A, B, D, E, F}
Terminology

A
B
C
E
D
F
G
{C, D}
Terminology

A
B
C
E
D
F
G
A simple path contains
no repeated vertices
(often this is implied)
Terminology

Cycle – A subset of the edges that form a path
such that the first and last node are the same
A
B
C
E
D
F
G
Terminology

A
B
C
E
D
F
G
{A, B, D}
Terminology

A
B
C
E
D
F
G
not a cycle
Terminology

Cycle – A path p1,p2,…pk where p1 = pk
A
B
C
E
D
F
G
cycle
Terminology

Connected – every pair of vertices is connected
by a path
A
B
C
E
D
F
G
connected
Terminology

Connected (undirected graphs) – every pair of
vertices is connected by a path
A
B
C
E
D
F
G
not connected
Terminology

Strongly connected (directed graphs) –
Every two vertices are reachable by a path
A
B
C
E
D
F
G
not strongly
connected
Terminology

A
B
E
D
F
G
not strongly
connected
Terminology

A
B
E
D
F
G
strongly
connected
Terminology

What is a tree (in our terminology)?
A
B
C
E
D
F
G
H

Tree – connected, undirected graph without
any cycles
A
B
C
E
D
F
G
H

Tree – connected, undirected graph without
any cycles
A
B
C
E
D
F
G
H
need to specify root

DAG – directed, acyclic graph
A
B
C
E
D
F
G
H

Complete graph – an edge exists between
every node
A
B
C
D
F

Bipartite graph – a graph where every vertex can be partitioned into
two sets X and Y such that all edges connect a vertex u  X and a
vertex v  Y
A
B
C
E
D
F
G

When do we see graphs in
real life problems?
 Transportation networks (flights, roads, etc.)
 Communication networks
 Web
 Social networks
 Circuit design
 Bayesian networks

33
Graphs
 We will typically express running times in
terms of |E| and |V| (often dropping the |’s)
 If |E|  |V|2 the graph is dense
 If |E|  |V| the graph is sparse
 If you know you are dealing with dense or
sparse graphs, different data structures may
make sense

Representing graphs
Adjacency list – Each vertex u  V contains an
adjacency list of the set of vertices v such that
there exists an edge (u,v)  E
A
B
C
E
D
A: B D
B: A D
C: D
D: A B C E
E: D

Representing graphs
Adjacency list – Each vertex u  V contains an
adjacency list of the set of vertices v such that
there exists an edge (u,v)  E
A
B
C
E
D
A: B
B:
C: D
D: A B
E: D

Representing graphs
Adjacency matrix – A |V|x|V| matrix A such that:
A
B
C
E
D
A B C D E
A 0 1 0 1 0
B 1 0 0 1 0
C 0 0 0 1 0
D 1 1 1 0 1
E 0 0 0 1 0


 

otherwise
0
)
,
(
if
1 E
j
i
aij

Representing graphs
A
B
C
E
D
A B C D E
A 0 1 0 1 0
B 1 0 0 1 0
C 0 0 0 1 0
D 1 1 1 0 1
E 0 0 0 1 0


 

otherwise
0
)
,
(
if
1 E
j
i
aij

Representing graphs
A
B
C
E
D
A B C D E
A 0 1 0 1 0
B 1 0 0 1 0
C 0 0 0 1 0
D 1 1 1 0 1
E 0 0 0 1 0
Is it always
symmetric?


 

otherwise
0
)
,
(
if
1 E
j
i
aij

Representing graphs
A B C D E
A 0 1 0 0 0
B 0 0 0 0 0
C 0 0 0 1 0
D 1 1 0 0 0
E 0 0 0 1 0
A
B
C
E
D


 

otherwise
0
)
,
(
if
1 E
j
i
aij

43
Graphs: Adjacency List
 How much storage is required?
 The degree of a vertex v = # incident edges
 Directed graphs have in-degree, out-degree
 For directed graphs, # of items in adjacency lists is
 out-degree(v) = |E|
takes (V + E) storage
 For undirected graphs, # items in adj lists is
 degree(v) = 2 |E|
also (V + E) storage
 So: Adjacency lists take O(V+E) storage

Adjacency list vs.
adjacency matrix
Adjacency list Adjacency matrix
Sparse graphs (e.g. web)
Space efficient
Must traverse the adjacency
list to discover is an edge
exists
Dense graphs
Constant time lookup to
discover if an edge exists
Simple to implement
For non-weighted graphs,
only requires boolean matrix
Can we get the best of both worlds?

Sparse adjacency matrix
Rather than using an adjacency list, use an adjacency
hashtable
A
B
C
E
D
A:
B:
C:
D:
E:
hashtable [B,D]
hashtable [A,D]
hashtable [D]
hashtable [A,B,C,E]
hashtable [D]

Sparse adjacency matrix
Constant time lookup
Space efficient
Not good for dense graphs, why?
A
B
C
E
D
A:
B:
C:
D:
E:
hashtable [B,D]
hashtable [A,D]
hashtable [D]
hashtable [A,B,C,E]
hashtable [D]

Weighted graphs
Adjacency list
 store the weight as an additional field in the list
A
B
C
E
D
8
2
3
13
10
A: B:8 D:3

Weighted graphs
Adjacency matrix
A
B
C
E
D
8
2
3
13
10


 

otherwise
0
)
,
(
if E
j
i
weight
aij
A B C D E
A 0 8 0 3 0
B 8 0 0 2 0
C 0 0 0 10 0
D 3 2 10 0 13
E 0 0 0 13 0

Graph algorithms/questions
 Graph traversal (BFS, DFS)
 Shortest path from a to b
 unweighted
 weighted positive weights
 negative/positive weights
 Minimum spanning trees
 Are all nodes in the graph connected?
 Is the graph bipartite?

50
Graph Searching
 Given: a graph G = (V, E), directed or undirected
 Goal: methodically explore every vertex and
every edge
 Ultimately: build a tree on the graph
 Pick a vertex as the root
 Choose certain edges to produce a tree
 Note: might also build a forest if graph is not
connected

51
Breadth-First Search
 “Explore” a graph, turning it into a tree
 One vertex at a time
 Expand frontier of explored vertices across the
breadth of the frontier
 Builds a tree over the graph
 Pick a source vertex to be the root
 Find (“discover”) its children, then their children,
etc.

52
Breadth-First Search
 Will associate vertex “colors” to guide the
algorithm
 White vertices have not been discovered
 All vertices start out white
 Grey vertices are discovered but not fully explored
 They may be adjacent to white vertices
 Black vertices are discovered and fully explored
 They are adjacent only to black and gray vertices
 Explore vertices by scanning adjacency list of
grey vertices

55
Breadth-First Search: Example








r s t u
v w x y

56


0





r s t u
v w x y
s
Q:

57
1

0
1




r s t u
v w x y
w
Q: r

58
1

0
1
2
2


r s t u
v w x y
r
Q: t x

59
1
2
0
1
2
2


r s t u
v w x y
Q: t x v

60
1
2
0
1
2
2
3

r s t u
v w x y
Q: x v u

61
1
2
0
1
2
2
3
3
r s t u
v w x y
Q: v u y

62
1
2
0
1
2
2
3
3
r s t u
v w x y
Q: u y

63
1
2
0
1
2
2
3
3
r s t u
v w x y
Q: y

64
1
2
0
1
2
2
3
3
r s t u
v w x y
Q: Ø

65
BFS: The Code Again
Total running time: O(V+E)
Overhead for initialization: O(V)
Each vertex is queued atmost once
and de-queued atmost once O(V)
Adjacency lists of the vertex is only
scanned when it is dequeued. The
sum of the length of all adjacency
list is O(E)
Total space used:
O(max(degree(v))) = O(E)

66
Breadth-First Search:
Properties
 BFS calculates the shortest-path distance to
the source node
 Shortest-path distance (s,v) = minimum number
of edges from s to v, or  if v not reachable from s
 Proof given in the book (p. 472-5)
 BFS builds breadth-first tree, in which paths
to root represent shortest paths in G
 Thus can use BFS to calculate shortest path from
one vertex to another in O(V+E) time

67
Depth-First Search
Explore “deeper” in the
graph whenever possible
Edges are explored out
of the most recently
discovered vertex v that
still has unexplored
edges

68
Depth-First Search
 Vertices initially colored white
 Then colored gray when discovered
 Then black when finished

74
DFS Example
1 | | |
|
|
3 |
2 | |
source
vertex
d f

75
DFS Example
1 | | |
|
|
3 | 4
2 | |
source
vertex
d f

76
DFS Example
1 | | |
|
5 |
3 | 4
2 | |
source
vertex
d f

77
DFS Example
1 | | |
|
5 | 6
3 | 4
2 | |
source
vertex
d f

78
DFS Example
1 | 8 | |
|
5 | 6
3 | 4
2 | 7 |
source
vertex
d f

79
DFS Example
1 | 8 | |
|
5 | 6
3 | 4
2 | 7 9 |
source
vertex
d f
What is the structure of the grey vertices?
What do they represent?

80
DFS Example
1 | 8 | |
|
5 | 6
3 | 4
2 | 7 9 |10
source
vertex
d f

81
DFS Example
1 | 8 |11 |
|
5 | 6
3 | 4
2 | 7 9 |10
source
vertex
d f

82
DFS Example
1 |12 8 |11 |
|
5 | 6
3 | 4
2 | 7 9 |10
source
vertex
d f

83
DFS Example
1 |12 8 |11 13|
|
5 | 6
3 | 4
2 | 7 9 |10
source
vertex
d f

84
DFS Example
1 |12 8 |11 13|
14|
5 | 6
3 | 4
2 | 7 9 |10
source
vertex
d f

85
DFS Example
1 |12 8 |11 13|
14|15
5 | 6
3 | 4
2 | 7 9 |10
source
vertex
d f

86
DFS Example
1 |12 8 |11 13|16
14|15
5 | 6
3 | 4
2 | 7 9 |10
source
vertex
d f

87
DFS: Kinds of edges
 DFS introduces an important distinction
among edges in the original graph:
 Tree edge: encounter new (white) vertex
 The tree edges form a spanning forest

88
DFS Example
1 |12 8 |11 13|16
14|15
5 | 6
3 | 4
2 | 7 9 |10
source
vertex
d f
Tree edges

89
DFS: Kinds of edges
 Back edge: from descendent to ancestor
 Encounter a grey vertex (grey to grey)

90
DFS Example
1 |12 8 |11 13|16
14|15
5 | 6
3 | 4
2 | 7 9 |10
source
vertex
d f
Tree edges Back edges

91
DFS: Kinds of edges
 Forward edge: from ancestor to descendent
 Not a tree edge, though
 From grey node to black node

92
DFS Example
1 |12 8 |11 13|16
14|15
5 | 6
3 | 4
2 | 7 9 |10
source
vertex
d f
Tree edges Back edges Forward edges

93
DFS: Kinds of edges
 Cross edge: between a tree or subtrees
 From a grey node to a black node

94
DFS Example
1 |12 8 |11 13|16
14|15
5 | 6
3 | 4
2 | 7 9 |10
source
vertex
d f
Tree edges Back edges Forward edges Cross edges

95
DFS: Kinds of edges
 Cross edge: between a tree or subtrees
 Note: tree & back edges are important; most
algorithms don’t distinguish forward & cross

96
DFS And Graph Cycles
 Thm: An undirected graph is acyclic iff a DFS
yields no back edges
 Thus, can run DFS to find whether a graph
has a cycle
 What will be the running time?
 A: O(V+E)

97
Directed Acyclic Graphs
 A directed acyclic graph or DAG is a directed
graph with no directed cycles:
directed graph G is acyclic iff a DFS of G yields no back edges:

98
Topological Sort
 Topological sort of a DAG:
 Linear ordering of all vertices in graph G such that
vertex u comes before vertex v if edge (u, v)  G
 Real-world example: getting dressed

101
Topological Sort Algorithm
Topological-Sort()
{
Run DFS
When a vertex is finished, output it
Vertices are output in reverse topological
order
}
 Time: O(V+E)

102
 G
 GT
Strongly Connected Components

104
Assignment: Strongly Connected
Components
1. Call DFS(G) to compute finishing times f[u]
for each vertex u;
2. Compute GT
3. Call DFS(GT), but in the main loop of DFS,
consider the vertices in order of decreasing
f[u].
4. Output the vertices for each tree in the
depth-first forest of step 3 as a separate
strongly connected component.

105
Graphs: Adjacency Matrix
 Example:
1
2 4
3
a
d
b c
A 1 2 3 4
1
2
3
??
4

106
Graphs: Adjacency Matrix
 Example:
1
2 4
3
a
d
b c
A 1 2 3 4
1 0 1 1 0
2 0 0 1 0
3 0 0 0 0
4 0 0 1 0

lec 09-graphs-bfs-dfs.ppt

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