1. Graphs

2. What is a Graph? A graph G = (V,E) is composed of: V: set of vertices
E: set of edges connecting the vertices in V
An edge e = (u1,u2) is a pair of vertices
Examples:
Bus routes
Transitions in a finite state machine
Course prerequisites

3. What is a Graph? If the elements of E(G) are ordered pairs
then G is called a directed graph or digraph; else G is called an undirected graph
In an undirected graph, the pairs (u1,u2) and (u2,u1) represent the same edge

4. What is a Graph? 2 3 1 4 5 2 3 1 4 5
Undirected Graph
Directed Graph

5. Applications Electronic circuits Transportation networks
Printed circuit board
Integrated circuit
Transportation networks
Highway network
Flight network
Computer networks
Local area network
Internet
Web
JFK
LAX
LAX
STL
HNL
DFW

6. Terminology Adjacent and Incident
If (v0, v1) is an edge in an undirected graph,
v0 and v1 are adjacent
The edge (v0, v1) is incident on vertices v0 and v1
If (v0, v1) is an edge in a directed graph
v0 is adjacent to v1, and v1 is adjacent from v0
The edge (v0, v1) is incident on v0 and v1 v0 v1 v0 v1

7. Terminology (cont.) Degree of a Vertex
The degree of a vertex is the number of edges incident to that vertex
For directed graph,
- the in-degree of a vertex v is the number of edges that have v as the head
- the out-degree of a vertex v is the number of edges that have v as the tail
if di is the degree of a vertex i in a graph G with n vertices and e edges, the number of edges is
Why? Since adjacent vertices each
count the adjoining edge, it will be
counted twice

8. Examples 1 2 1 3 2 G1 G2 Undirected Graph Directed Graph 3
in:1, out: 1 3 1 2 1
in: 1, out: 2 3 3 3 2
in: 1, out: 0 G1 G2

9. Terminology (cont.) Path Simple path Examples P1 X U V W Z Y a c b e d
sequence of vertices v1,v2,. . .vk such that consecutive vertices vi and vi+1 are adjacent.
Simple path
no repeated vertices
Examples
P1=(V, X, Z) is a simple path
P2=(U, W, X, Y, W, V) is a path that is not simple P1 X U V W Z Y a c b e d f g h P2

10. Terminology (cont.) Cycle Simple cycle Examples C1 X U V W Z Y a c b e
circular sequence of vertices, where that the last vertex is the same as the first vertex .
Simple cycle
simple path, except that the last vertex is the same as the first vertex
Examples
C1=(V, X, Y, W, U, V) is a simple cycle
C2=(U, W, X, Y, W, V, U) is a cycle that is not simple C1 X U V W Z Y a c b e d f g h C2

11. Terminology (cont.) Subgraph Spanning subgraph
A subgraph S of a graph G is a graph such that
The vertices of S are a subset of the vertices of G
The edges of S are a subset of the edges of G
Spanning subgraph
A spanning subgraph of G is a subgraph that contains all the vertices of G
Subgraph S
Spanning subgraph

12. Subgraphs Examples 1 2 3 G1 1 2 G2 (a) Some of the subgraph of G11 2 3 1 2 3 3 1 2 1 2
(i) (ii) (iii) (iv)
(a) Some of the subgraph of G1 G1 1 2 1 1 2 1 2
(i) (ii) (iii) (iv)
(b) Some of the subgraph of G2 G2

13. Non connected graph with two connected components
Terminology (cont.)
Connected graph
Non connected graph with two connected components
Connected graph
there is a path between every pair of vertices
Connected component
it is the maximal connected subgraph of G

14. Terminology (cont.) Tree Forest it is a connected graph without cycles
// This definition is different from the rooted tree (in chapter 9)
Forest
collection of trees (i.e. it is an undirected graph without cycles)
tree
forest

15. Terminology (cont.)
A spanning tree of a connected graph is a spanning subgraph that is a tree
A spanning tree is not unique unless the graph is a tree
Spanning trees have applications to the design of communication networks
A spanning forest of a graph is a spanning subgraph that is a forest
Graph
Spanning tree

16. Terminology (cont.)
A complete graph is a graph in which all pairs of vertices are adjacent
How many total edges in a complete graph?
Each of the n vertices is incident to n-1 edges, however, we would have counted each edge twice! Therefore, intuitively,
m = n(n -1)/2 edges.
Therefore, if a graph is not complete, m < n(n -1)/2
n = 5
m = (5*4)/2

17. Graph Representation Two standard ways Adjacency-matrix representation
Space required O(n2).
Adjacency-list representation
Space required O(|E|)
Depending on problems, both representations are useful.

18. Adjacency-matrix representation
Assume that the nodes are numbered 1, 2, …, n.
The adjacency-matrix consists of a |V ||V | matrix A=(aij) such that
if (i,j)  E
aij= 1
else aij= 0. 1 2 3 4 5 2 1 5 4 3

19. Adjacency-matrix representation
Space required O(n2)
The adjacency matrix for an undirected graph is symmetric. Hence, we may store only lower or upper triangle 1 2 3 4 5

20. Examples for Adjacency Matrix1 2 1 2 3 4 5 6 2 1 3 G2 7 G1
symmetric
undirected: n2/2
directed: n2 G3

21. Adjacency Matrix Analysis
Operation
Time
The degree of a vertex v is
O(n)
For a digraph (= directed graph), the row sum is the out degree, while the column sum is the in degree
Find the edges incident to a given vertex v
Check that two vertices v and w are adjacent
O(1)
Insert (or remove) a Vertex v
O(n2)
Insert (or remove) and Edge e

22. Adjacency-list representation
Let G=(V, E) be a graph
V– set of nodes (vertices)
E– set of edges.
For each wV, the adjacency list A[w] contains all nodes in V that are adjacent to u. 2 5 / 1 2 1 5 4 3 2 1 5 3 4 / 3 2 4 / 4 2 5 3 / 5 4 1 2 /
(a)
(b)

23. Examples for Adjacency List1 2 3 4 5 6 7 1 2 3 1 2 3 1 2 3 2 3 1 3 1 2 3 4 5 6 7 1 2 1 2 3 3 G1 1 2 1 2 5 1 2 1 4 6 2 5 7 6 G2 G3
An undirected graph with n vertices and e edges ==> n head nodes and 2e list nodes

24. Adjacency List Analysis
Operation
Time
Find the edges incident to a given vertex v
O(degree(v))
Check that two vertices v and w are adjacent
O(min
(degree(v), degree(w)))
Insert a Vertex v
O(1)
Remove a Vertex v
Insert an Edge e

25. Example Graph in JAVA public class Graph<E> {
// 1. The vertex numbers range from 0 to labels.length-1.
// 2. For each vertex number i, labels[i] contains the label for vertex i.
// 3. For any two vertices i and j, edges[i][j] is true if there is a
// vertex from i to j; otherwise edges[i][j] is false.
private boolean[ ][ ] edges;
private E[ ] labels; :
// Graph methods appear here }

26. Example Graph in JAVA
public Graph(int n) { edges = new boolean[n][n]; // All values initially false labels = (E[ ]) new Object[n]; // All values initially null } public void addEdge(int source, int target) edges[source][target] = true;

27. Example Graph in JAVA
public Object getLabel(int vertex) { return labels[vertex]; } public boolean isEdge(int source, int target) return edges[source][target];

28. Example Graph in JAVA
public int[ ] neighbors(int vertex) { int i; int count; int[ ] answer; // First count how many edges have the vertex as their source count = 0; for (i = 0; i < labels.length; i++) if (edges[vertex][i]) count++; } :

29. Example Graph in JAVA
public int[ ] neighbors(int vertex) { : : // Allocate the array for the answer answer = new int[count]; // Fill the array for the answer count = 0; for (i = 0; i < labels.length; i++) { if (edges[vertex][i]) answer[count++] = i; } return answer;

30. Example Graph in JAVA
public void removeEdge(int source, int target) { edges[source][target] = false; } public void setLabel(int vertex, E newLabel) labels[vertex] = newLabel; public int size( ) return labels.length;

31. Graph Traversals No obvious order of traversal (like trees)
No obvious starting point (no root)
Traversals may not reach every vertex by following edges (connectedness)
Traversals may return to a vertex (cycles)
References:
Data Structures and Algorithms in JAVA
By M. Goodrich and R. Tamassia
Chapters 9 & 10

32. Graph Traversals Traversal Algorithms: References:
- Depth-first search DFS.
- Breadth-first search BFS.
References:
Data Structures and Algorithms in JAVA
By M. Goodrich and R. Tamassia
Chapters 9 & 10

33. Depth-first search DFS
Visits a vertex, then
visit a neighbor of the vertex,
visit a neighbor of the neighbor,
Etc.
Advance as possible from the original vertex
Then back up by one vertex
Considers the next neighbor

34. Example unexplored vertex visited vertex unexplored edgeB A C E A
visited vertex
unexplored edge
discovery edge
back edge D B A C E D B A C E

35. Example (cont.) D B A C E D B A C E D B A C E D B A C E

36. DFS Algorithm Algorithm DFS( v) Input A vertex v of a graph G
Output labeling of the edges of G in the connected component of v as discovery edges and back edges
label v as visited
for each edge e incident on v do
if edge e is unexplored then
let w be the other endpoint of e
if vertex w is unvisited then
Label e as a discovery edge
recursively call DFS(w)
else
label e as a back edge

37. Data Structures and Algorithms in JAVA, M. Goodrich and R. Tamassia
Example 2
(a)
(b)
(c)
(d)
(e)
(f)
Data Structures and Algorithms in JAVA, M. Goodrich and R. Tamassia

38. Properties of DFS Property 1 Property 2
DFS(G, v) visits all the vertices in the connected component of v
Property 2
The discovery edges labeled by DFS(G, v) form a spanning tree of the connected component of v v D B A C E

39. Properties of DFS Property 3
Let G be a graph with n vertices and m edges. A DFS traversal of G can be performed in time O(n+m). Also, there exist O(n+m) time algorithms based on DFS for the following problems:
Computing a spanning tree of G, if G is connected
Computing the connected components of G
Computing a path from one given vertex to another in G
Computing a cycle in G, or reporting that G has no cycles

40. Breadth-first search BFS
Visits a vertex, then
visit each of the vertex's neighbors before advancing

41. Note: BFS subdivides the vertices into levels (L0, L1,….etc.)
Example
unexplored vertex C B A E D L0 L1 F A
visited vertex A
unexplored edge
discovery edge
cross edge L0 L0 A A L1 L1 B C D B C D E F E F
Note: BFS subdivides the vertices into levels (L0, L1,….etc.)

42. Note: BFS subdivides the vertices into levels (L0, L1,….etc.)
Example C B A E D L0 L1 F C B A E D L0 L1 F L2 C B A E D L0 L1 F L2 C B A E D L0 L1 F L2
Note: BFS subdivides the vertices into levels (L0, L1,….etc.)

43. Note: BFS subdivides the vertices into levels (L0, L1,….etc.)
Example C B A E D L0 L1 F L2 L0 A L1 B C D L2 E F C B A E D L0 L1 F L2
Note: BFS subdivides the vertices into levels (L0, L1,….etc.)

44. Breadth-first search BFS
Algorithm BFS(v)
Input A vertex v of a graph G
Output labeling of the edges of G in the connected component of v as discovery edges and cross edges
initialize container L0 to contain vertex v
i  0
while Li is not empty do
create container Li+1 to initially be empty
for each vertex v in Li do
for each edge e incident on v do
if edge e is unexplored then
let w be the other endpoint of e
if vertex w is unexplored then
label e as a discovery edge
insert w into Li+1
else label e as a cross edge
i  i + 1

45. Data Structures and Algorithms in JAVA, M. Goodrich and R. Tamassia
Example
Data Structures and Algorithms in JAVA, M. Goodrich and R. Tamassia

46. Properties L0 L1 L2 A Property 1
BFS(s) visits all the vertices in the connected component of s
Property 2
The discovery edges form a spanning tree Ts of connected component of s
Property 3
For each vertex v in Li
The path of Ts from s to v has i edges
Any other path from s to v in Gs has at least i edges B C D E F L0 A L1 B C D L2 E F

47. Properties
Property 4
Let G be a graph with n vertices and m edges. A BFS traversals of G takes time O(n+m). Also, there exist O(n+m) time algorithms based on BFS for the following problems:
Testing whether G is connected
Computing a spanning tree of G, if G is connected
Computing the connected components of G
Computing, for every vertex v of G, the minimum number of edges of any path between s and v

48. Another Implementation for BFS
Algorithm BFS(v) Input A vertex v of a graph G Output processing the vertices of the connected component of v vertexQueue = a new queue to hold vertices as they are visited Mark v as visited vertexQueue.enqueue (v) while (!vertexQueue.isEmpty ()) do frontVertex = vertexQueue.dequeue () //process frontVertex while (frontVertex has a neighbor) do nextNeighbor = next neighbor of frontVertex if (nextNeighbor is not visited) then Mark nextNeighbor as visited vertexQueue.enqueue (nextNeighbor)

49. Applications: Cycle Finding
We can specialize the DFS algorithm to find a simple cycle
We use a stack S to keep track of the path between the start vertex and the current vertex
As soon as a back edge (v, w) is encountered, we return the cycle as the portion of the stack from the top to vertex w

50. Applications: Path Finding
Find path from source vertex s to destination vertex d
Use graph search starting at s and terminating as soon as we reach d
Need to remember edges traversed
Use depth first search DFS?
Use breath first search BFS?

51. DFS vs. BFS D C B B B A A A A G D B A F B A start DFS Process E G D C
destination D
Call DFS on D C
DFS on C B
DFS on B B B
Return to call on B A
DFS on A A A A
found destination - done!
Path is implicitly stored in DFS recursion
Path is: A, B, D, G G
Call DFS on G D B A

52. DFS vs. BFS A B D C D G F B A start E BFS Process G D C destination
rear
front
rear
front
rear
front
rear
front A B D C D
Initial call to BFS on A
Add A to queue
Dequeue A
Add B
Dequeue B
Add C, D
Dequeue C
Nothing to add
rear
front G
found destination - done!
Path must be stored separately
Dequeue D
Add G

53. Digraphs E D C B A A digraph is a graph whose edges are all directed
Short for “directed graph”
Applications
one-way streets
flights
task scheduling
A graph G=(V,E) such that
Each edge goes in one direction: Edge (a,b) goes from a to b, but not b to a. A C E B D

54. Digraph Application
Scheduling: edge (a,b) means task a must be completed before b can be started
cs21
cs22
cs23
cs51
cs53
cs52
cs181
cs131
cs141
cs121
cs171
The good life
cs151

55. Reachability
Directed Graph rooted at v : vertices reachable from v via directed paths v E D v E D C A C F v A B E D C F A B

56. Strong Connectivity Each vertex can reach all other vertices a g c d eb e f g

57. Strong Connectivity Algorithm
Pick a vertex v in G.
Perform a DFS from v in G.
If there’s a w not visited, print “no”.
Let G’ be G with edges reversed.
Perform a DFS from v in G’.
Else, print “yes”.
Running time: O(n+m). c d e b f a
G’: g c d e b f

58. Transitive Closure
Given a digraph G, the transitive closure of G is the digraph G* such that
G* has the same vertices as G
if G has a directed path from u to v (u  v), G* has a directed edge from u to v
The transitive closure provides reachability information about a digraph B A D C E G B A D C E G*

59. Computing the Transitive Closure
We can perform DFS starting at each vertex
Based on the idea that, if there's a way to get from A to B and from B to C, then there's a way to get from A to C.
O(n(n+m))
Alternatively ... Use The Floyd-Warshall Algorithm

60. Floyd-Warshall Transitive Closure
Floyd-Warshall’s algorithm numbers the vertices of G as v1 , …, vn and computes a the transitive closure in a series of rounds
Initialize G0=G
Begin a computation of the rounds, beginning with round 1
In generic round k , we construct Gk as:
Gk = Gk-1
Add a directed edge (vi, vj) to Gk , if Gk-1 contains both the edges (vi, vk) and (vk, vj)
After n rounds, we have that Gn = G*
Running time: O(n3), assuming adjacency matrix representation (i.e. checking whether 2 vertices are adjacent is O(1) )

61. Floyd-Warshall - Example
JFK
BOS
MIA
ORD
LAX
DFW
SFO v 2 1 3 4 5 6
G0=G

62. Floyd-Warshall - Iteration 1G1
BOS v
ORD 4
JFK v 2 v 6
SFO
DFW
LAX v 3 v 1
MIA v 5

63. Floyd-Warshall - Iteration 2G2
JFK
BOS
MIA
ORD
LAX
DFW
SFO v 2 1 3 4 5 6

64. Floyd-Warshall - Iteration 3G3
BOS v
ORD 4
JFK v 2 v 6
SFO
DFW
LAX v v 3 1
MIA v 5

65. Floyd-Warshall - Iteration 4G4
BOS v
ORD 4
JFK v 2 v 6
SFO
DFW
LAX v v 3 1
MIA v 5

66. Floyd-Warshall - Iteration 5G5
BOS v
ORD 4
JFK v 2 v 6
SFO
DFW
LAX v 3 v 1
MIA v 5

67. Floyd-Warshall - Iteration 6
JFK
MIA
ORD
LAX
DFW
SFO v 2 1 3 4 5 6
BOS G6

68. Floyd-Warshall - Conclusion
JFK
MIA
ORD
LAX
DFW
SFO v 2 1 3 4 5 6
BOS G7

69. Floyd-Warshall - Algorithm
Algorithm FloydWarshall(G)
Input digraph G with n vertices
Output transitive closure G* of G
i := 1
for all v  G
denote v as vi
i := i + 1
G0 := G
for k := 1 to n do
Gk := Gk - 1
for i := 1 to n (i  k) do
for j := 1 to n (j  i, k) do
if both edges (vi, vk) and (vk, vj)  Gk - 1 then
add edge (vi, vj) to Gk. (if it is not already present)
return Gn

70. DAGs and Topological Ordering
A directed acyclic graph (DAG) is a digraph that has no directed cycles
A topological ordering of a digraph is a numbering
v1 , …, vn
of the vertices such that for every edge (vi , vj), we have i < j
Example: in a task scheduling digraph, a topological ordering a task sequence that satisfies the precedence constraints
Theorem
A digraph might has a topological ordering if and only if it is a DAG B A D C E
DAG G
Topological ordering of G v1 v2 v3 v4 v5

71. Topological Sorting
Number vertices, so that (u,v) in G implies u < v 1
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eat
study computer sci.
A typical student day 4 5
nap
more c.s. 7
play 8
write c.s. program 6 9
work out
make cookies for professors 10
sleep 11
dream about graphs

72. Algorithm for Topological Sorting
Algorithm TopologicalSort(G)
H := G // Temporary copy of G
n := the number of vertices in G
while H is not empty do
Let v be a vertex with no outgoing edges
Label v := n
n := n - 1
Remove v from H
Running time: O(n + m)

73. Topological Sorting Example

74. Topological Sorting Example9

75. Topological Sorting Example9

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79. Topological Sorting Example7 8 9

80. Topological Sorting Example6 7 8 9

81. Topological Sorting Example6 7 8 9

82. Topological Sorting Example6 5 7 8 9

83. Topological Sorting Example6 5 7 8 9

84. Topological Sorting Example4 6 5 7 8 9

85. Topological Sorting Example4 6 5 7 8 9

86. Topological Sorting Example3 4 6 5 7 8 9

87. Topological Sorting Example3 4 6 5 7 8 9

88. Topological Sorting Example2 1 3 4 6 5 7 8 9

89. Topological Sorting Example2 1 3 4 6 5 7 8 9