Download 01_GraphIntro

Intro to Graphs
CSIT 402
Data Structures II
Graphs
• Graphs are composed of
node
› Nodes (vertices)
• Can be labeled
› Edges (arcs)
• Directed (arrowhead)
• Undirected
• Can be labeled
CSIT 402 Graph Introduction
edge
2
Graphs: Data Structure Progression
Consider previous data structures
•Linked List: Nodes with 1 incoming
edge and 1 outgoing edge
node
Value Next
node
Value Next
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•Binary Trees/Heaps: Nodes with 1
incoming edge and 2 outgoing edges
•General Trees: Nodes with 1 incoming
edge and any number of outgoing edges
Idea: All of the above are graphs
More examples on next seven slides…
CSIT 402 Graph Introduction
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10
96
99
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Course Prerequisites
461
373
413
321
326
142
415
410
322
143
370
341
417
378
Nodes = courses
Directed edge = prerequisite
CSIT 402 Graph Introduction
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401
4
Representing a Maze
Directed or Undirected?
S
S
B
E
E
Nodes = locations
Edge = paths between locations
CSIT 402 Graph Introduction
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Representing Electrical
Circuits
Battery
Switch
Directed or Undirected?
Nodes = battery, switch, resistor, etc.
Edges = connections
CSIT 402 Graph Introduction
Resistor
6
Program statements
Directed or Undirected?
x1=q+y*z
x2=y*z-q
Naive:
x1
x2
+
*
q
y
y*z calculated twice
common
subexpression
eliminated:
*
Nodes = symbols/operators
Edges = relationships
CSIT 402 Graph Introduction
z
x1
x2
+
-
q
*
y
q
q
z
7
Precedence
S1
S2
S3
S4
S5
S6
a=0;
b=1;
c=a+1
d=b+a;
e=d+1;
e=c+d;
Directed or Undirected?
6
5
Which statements must execute before S6?
S1, S2, S3, S4
3
4
Values
Nodes = statements
Edges = precedence requirements
1
CSIT 402 Graph Introduction
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8
Information Transmission in a Computer Network
Directed or Undirected?
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Tokyo
Seattle
Seoul
128
16
30
Sydney
New York
181
140
L.A. Nodes = computers
Edges = transmission rates
Values
CSIT 402 Graph Introduction
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Traffic Flow on Highways
Directed or Undirected?
UW
Nodes = cities
Edges = # vehicles on
connecting highway
CSIT 402 Graph Introduction
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Formal Definition
• A graph G is a tuple (V, E) where
›
›
›
V is a set of vertices or nodes
E is a set of edges
• An Edge e is a pair (v1,v2), where vi ε V
Example:
• V = {A, B, C, D, E, F}
• E = {(A,B), (A,D), (B,C), (C,D), (C,E), (D,E)}
B
C
A
F
D
CSIT 402 Graph Introduction
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Directed vs Undirected Graphs
• If the order of edge pairs (vi, vj) is important (e.g
prerequisites), the graph is directed (also called a
digraph): (vi, vj)  (vj, vi) , i  j
vj
vi
• If the order of edge pairs (vi, vj) is unimportant (e.g.
maze), the graph is called an undirected graph (never
called an undigraph): In this case, (vi, vj) = (vj, vi) , i  j
vi
vj
CSIT 402 Graph Introduction
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Undirected Graph Terminology
• Two vertices u and v are adjacent in an undirected graph G if
{u,v} is an edge in G
› edge e = {u,v} is incident (in contact) with vertex u and vertex v
• The degree of a vertex in an undirected graph is the number of
edges incident with it
› a self-loop counts twice (both ends count)
B is adjacent to C and C is adjacent to B
› denoted with deg(v)
B
(A,B) is incident
to A and to B
C
Self-loop
A
F
Deg(D) = 3
D
CSIT 402 Graph Introduction
E
Deg(F) = 0
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Directed Graph Terminology
• Vertex u is adjacent to vertex v in a directed
graph G if (u,v) is an edge in G
› vertex u is the initial vertex of (u,v)
• Vertex v is adjacent from vertex u
› vertex v is the terminal (or end) vertex of (u,v)
• Degree
› in-degree is the number of edges with the vertex
as the terminal vertex
› out-degree is the number of edges with the vertex
as the initial vertex
CSIT 402 Graph Introduction
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Directed Graph Terminology
B adjacent to C and C adjacent from B
B
C
A
F
D
E
In-degree = 2
Out-degree = 1
CSIT 402 Graph Introduction
In-degree = 0
Out-degree = 0
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Handshaking Theorem
• Let G=(V,E) be an undirected graph with
|E|=e edges. Then
2e   deg(v)
Add up the degrees of all vertices.
vV
• Every edge contributes +1 to the degree of
each of the two vertices it is incident with
› number of edges is exactly half the sum of deg(v)
› the sum of the deg(v) values must be even
CSIT 402 Graph Introduction
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Graph Representations
• Space and time are analyzed in terms of:
• Number of vertices = |V| and
• Number of edges = |E|
• There are at least two ways of representing
graphs:
• The adjacency matrix representation
• The adjacency list representation
CSIT 402 Graph Introduction
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Adjacency Matrix
B
C
A
F
D
E
1 if (v, w) is in E
M(v, w) =
0 otherwise
A
B
C
D
E
F
A
0
1
0
1
0
0
B
1
0
1
0
0
0
C
0
1
0
1
1
0
D
1
0
1
0
1
0
E
0
0
1
1
0
0
F
0
0
0
0
0
0
Space = |V|2
CSIT 402 Graph Introduction
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Adjacency Matrix for a Digraph
B
C
A
F
D
E
1 if (v, w) is in E
M(v, w) =
0 otherwise
A
B
C
D
E
F
A
0
1
0
1
0
0
B
0
0
1
0
0
0
C
0
0
0
1
1
0
D
0
0
0
0
1
0
E
0
0
0
0
0
0
F
0
0
0
0
0
0
Space = |V|2
CSIT 402 Graph Introduction
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Adjacency List
For each v in V, L(v) = list of w such that (v, w) is in E
a
b
B
C
A
F
D
E
list of
neighbors
A
B
D
B
A
C
C
B
D
E
D
A
C
E
E
C
D
F
CSIT 402 Graph Introduction
Space = |V| + 2 |E|
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Adjacency List for a Digraph
For each v in V, L(v) = list of w such that (v, w) is in E
a
b
B
C
A
F
D
E
A
B
B
C
C
D
D
E
D
E
E
F
CSIT 402 Graph Introduction
Space = |V| + |E|
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Paths and Cycles
• Given a directed graph G = (V,E), a path is a
sequence of vertices v1,v2, …,vk such that:
› (vi,vi+1) in E for 1 < i < k
• path length = number of edges in the path
• path cost = sum of costs of each edge (if edges have costs)
• A graph contains a cycle if, there exists a path v1,v2,
…,vk, k > 1 such that vk = v1
• G is acyclic if it has no cycles.
CSIT 402 Directed
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Graphs