Graphs I Kruse and Ryba Chapter 12. Undirected Graphs An undirected graph is a set of nodes and a set of edges between nodes; e.g.: The graph is undirected.

Graphs I

Kruse and Ryba

Chapter 12

Undirected Graphs

An undirected graph is a set of nodes and a set of edges between nodes; e.g.:

The graph is undirected because its edges do not have a particular direction.

Edges

We sometimes need to refer to the edges ei of a graph:

e0

e1

e3

e4

e5

e2

Vertices

We also need to refer to the vertices vj of a graph:

v1

e0

e1

e3

e4

e5

e2v2

v3

v4

v0

Undirected Graph Definition

• An undirected graph G = (V, E) is a finite set of vertices V, together with a finite set of edges E.

• Both sets might be empty (no vertices and no edges), which is called the empty graph.

• Each edge has associated with it two vertices.• If two vertices are connected by an edge, they are

said to be adjacent.

Undirected Graph Example: Subway Map

Another Undirected Graph: vBNS (fast internet backbone)

Directed Graphs

In a directed graph each vertex has an associated direction; e.g.:

Each edge is associated with two vertices: its source vertex and target vertex.

Directed Graph Examples

• Flow graph: vertices are pumping stations and edges are pipelines connecting them.

• Task network: each project has a number of component activities called tasks. Each task has a duration (amount of time needed to complete task).

Loops

A loop is an edge that connects a vertex with itself; for example:

Loops can occur in directed graphs or undirected graphs.

Multiple Edges

In principle, a graph may have two or more edges connecting the same two vertices in the same direction:

These are called multiple edges. In a graph diagram, each edge is drawn separately.

Simple Graphs

A graph is said to be a simple graph, if:1. the graph has no loops, and

2. no multiple edges

Path

• A path in a graph is a sequence of vertices, v0, v1, … vm, such that each adjacent pair of vertices in the list are connected by an edge.

v1

v2

v3

v4

v0

Path


v1

v2

v3

v4

v0Example path:

v0, v3, v4

Path


v1

v2

v3

v4

v0Example path:

v0, v3, v4

v0, v3, v2, v1, v4

Path

• In a directed graph, the connection must go from source vi to target vi+1.

v1

v2

v3

v4

v0Is there a path from

v0 to v4?v3 to v1? v1 to v2? v2 to v1?

Example Task Network

Start

Find a realtor

Look at homes

Choose a home to buy

Apply for financing and

establish price limit

Investigateneighborhood

schools

Prepare offer

Investigatetermite and soil

reports

Presentoffer

0

20

5 9

2

15

10

3

2

How long will it take to buy a house? (The duration of each task isshown below each task).

Critical Path

Start

Find a realtor

Look at homes

Choose a home to buy

Apply for financing and

establish price limit

Investigateneighborhood

schools

Prepare offer

Investigatetermite and soil

reports

Presentoffer

0

20

5 9

2

15

10

3

2

The task will take as long as the maximum duration path from start to finish.

Cycles

• A cycle in a graph is a path that leads back to the first node in the path.

v1

v2

v3

v4

v0Example cycles:

v1, v2, v3, v2, v1

Cycles

• A cycle in a graph is a path that leads back to the first node in the path.

v1

v2

v3

v4

v0Example cycles:

v1, v2, v3, v2, v1

v0, v3, v4, v1, v0

Does this graph have any cycles?

Cycles in Directed Graphs

A cycle in a directed graph must follow edge directions; e.g.:

DAG: Directed Acyclic GraphA directed acyclic graph (DAG) must have no cycles;

e.g.:

In other words, there is no path that starts and ends at the same node.

Complete Graph

An undirected graph which has an edge between every pair of vertices.

Connected Graph

An undirected graph which has a path between every pair of vertices.

1

0

2 3

Strongly Connected Graph

A directed graph which has a path between every pair of vertices.

Weakly Connected Graph

A directed graph which is connected, but not not strongly connected.

A DAG that is not Connected

Connected Components

Review of Terms

• Undirected vs. directed graphs• Multiple edges, loops• Simple graph• Path• Cycle• Directed, acyclic graph (DAG)• Connected, strongly connected, and weakly

connected graphs• Connected component

Graph Implementations

Adjacency Matrix

01

3

2

0 1 2 3 0123

Adjacency Matrix

01

3

2 10 1 2 3

0123

Adjacency Matrix

01

3

2 1 0 0 1 2 3

0123

Adjacency Matrix

01

3

2 1 0 1 0 1 2 3

0123

Adjacency Matrix

01

3

2 1 0 1 10 1 2 3

0123

Adjacency Matrix

01

3

2 1 0 1 1

0 0 0 1

0 1 2 3 0123

Adjacency Matrix

01

3

2 1 0 1 1

0 0 0 1

1 0 0 0

0 1 2 3 0123

Adjacency Matrix

01

3

2 1 0 1 1

0 0 0 1

1 0 0 0

0 1 0 0

0 1 2 3 0123

C++ Adjacency Matrix

01

3

2 1 0 1 1

0 0 0 1

1 0 0 0

0 1 0 0

0 1 2 3 0123

bool graph_adjacency[4][4];

Adjacency Matrix: Weighted Edges

01

3

2 5.8 0 6.0 1.5

0 0 0 2.5

2.5 0 0 0

0 4.2 0 0

0 1 2 3

0123

double graph_adjacency[4][4];

6.0

2.5

4.2

2.5

5.8

1.5

Weighted Edges: An Example

$320$245

San FransiscoBoston

Chicago

$452

$60 $180Dallas

Flights. What’s the cheapest way to fly San Fransisco Boston?


18001200

San FransiscoBoston

Chicago

2800

2300 1600Dallas

Mileage for various hops from San Fransisco Boston*

*Mileage is not real, but for sake of example.


Fran Pat

ADSL provider100Kbs

Available internet bandwidth on various paths between an ADSL provider and customers*.

*This is a simplication for sake of example, and bit rates are fictional.

1Gbs 100Kbs

500Kbs

Adjacency Matrix: Undirected Graph

01

3

2

0 1 2 3 0123

bool graph_adjacency[4][4];


01

3

2 0

0

0

0

0 1 2 3 0123

No loops zero diagonal.


01

3

2 0 1 1 1

0

0

0

0 1 2 3 0123


01

3

2 0 1 1 1

1 0 0 1

0

0

0 1 2 3 0123


01

3

2 0 1 1 1

1 0 0 1

1 0 0 0

0

0 1 2 3 0123


01

3

2 0 1 1 1

1 0 0 1

1 0 0 0

1 1 0 0

0 1 2 3 0123

Symmetry of Adjacency Matrix of Undirected Graph

01

3

2 0 1 1 1

1 0 0 1

1 0 0 0

1 1 0 0

0 1 2 3 0123

a[i][j]== a[j][i]

Non-symmetric Adjacency Matrix for Directed Graph

01

3

2 1 0 1 1

0 0 0 1

1 0 0 0

0 1 0 0

0 1 2 3 0123

a[i][j] a[j][i]

Symmetric Adjacency Matrix

• An undirected graph will always have a symmetric adjacency matrix.

• A directed graph may not have a symmetric adjacency matrix.

• If a directed graph has a symmetric matrix, what does that mean?

Non-symmetric Adjacency Matrix Unweighted Directed Graph

01

3

2 1 0 1 1

0 0 0 1

1 0 0 0

0 1 0 0

0 1 2 3 0123

a[i][j] a[j][i]

Add an Edge

01

3

2 1 0 1 1

0 0 0 1

1 0 0 0

1 1 0 0

0 1 2 3 0123

a[i][j]=a[j][i]

Adjacency Matrix for Weighted Directed Graph

01

3

2 5 0 3 6

0 0 0 4

3 0 0 0

6 4 0 0

0 1 2 3 0123

a[i][j]=a[j][i]

3

32

6

6

4

45

Time Complexity Analysis

Given N we have vertices in a graph, and we use the adjacency matrix representation.

1. What is the worst case and average complexity of inserting an edge in the graph?

2. What is the worst case and average complexity of removing an edge in the graph?

3. What is the complexity of retrieving a list of all nodes that share and edge with a particular vertex?

Representing Graphs with Edge Lists

01

3

2

2

3Null

Edge list for vertex 0

Null


0


1Null


Null


Given N we have vertices in a graph, and we use the edge list representation.




Representing Graphs with Edge Sets

01

3

2

Assume a Set template class is available:

Declare array of sets, one per node in graph:

Set<int> connections[N];

Each node’s set contains the indices of all vertices that it is connected to.


Given N we have vertices in a graph, and we use the edge set representation.




Graph Implementations Summary

• Adjacency matrix

• Edge lists

• Edge sets

Graph Searching

Graph Search

• Choice of container– If a stack is used as the container for adjacent

vertices, we get depth first search.– If a list is used as the container adjacent

vertices, we get breadth first search.

Breadth First AlgorithmGiven graph G=(V,E) and source vertex s VCreate a queue QFor each vertex u V – {s}

color[u] whitecolor[s] grayQ {s}While Q {

u head[Q];for each v Adjacent[u]

if color[v] = white { color[v] gray Enqueue(Q,v)}

Dequeue(Q)color[u] black;

}

0

3

2

1

5

4






}

0

3

2

1

5

4s

Q =






}

0

3

2

1

5

4s

Q =






}

0

3

2

1

5

4s

Q = 0






}

0

3

2

1

5

4

u

Q = 0






}

0

3

2

1

5

4

u

Q = 0 1






}

0

3

2

1

5

4

u

Q = 0 1 2






}

0

3

2

1

5

4

u

Q = 0 1 2 3






}

0

3

2

1

5

4

u

Q = 01 2 3






}

0

3

2

1

5

4

u

Q = 01 2 3






}

0

3

2

1

5

4

u

Q = 01 2 3 4






}

0

3

2

1

5

4

u

Q = 012 3 4






}

0

3

2

1

5

4u

Q = 012 3 4






}

0

3

2

1

5

4u

Q = 012 3 4






}

0

3

2

1

5

4u

Q = 0123 4






}

0

3

2

1

5

4

u

Q = 0123 4






}

0

3

2

1

5

4

u

Q = 0123 4 5






}

0

3

2

1

5

4

u

Q = 01234 5






}

0

3

2

1

5

4

u

Q = 01234 5






}

0

3

2

1

5

4

u

Q = 012345






}

0

3

2

1

5

4

uQ = 012345






}

0

3

2

1

5

4

uQ =

Graph Search




Depth First AlgorithmGiven graph G=(V,E) and source vertex s VCreate a stack SFor each vertex u V – {s}

color[u] whitecolor[s] grayS {s}While S {

u = Pop(S)for each v Adjacent[u]

if color[v] = white { color[v] gray Push(S,v)}

color[u] black;}

0

3

2

1

5

4s

S =





color[u] black;}

0

3

2

1

5

4s

S =





color[u] black;}

0

3

2

1

5

4s

S = 0





color[u] black;}

0

3

2

1

5

4

u

S =





color[u] black;}

0

3

2

1

5

4

u

S = 1





color[u] black;}

0

3

2

1

5

4

u

S = 32 1





color[u] black;}

0

3

2

1

5

4

u

S = 3 2 1





color[u] black;}

0

3

2

1

5

4

u

S = 3 2 1





color[u] black;}

0

3

2

1

5

4

u

S = 32 1





color[u] black;}

0

3

2

1

5

4

u

S = 32 15





color[u] black;}

0

3

2

1

5

4

u

S = 32 15

3





color[u] black;}

0

3

2

1

5

4

u

S = 52 1

3





color[u] black;}

0

3

2

1

5

4

u

S = 52 14

3





color[u] black;}

0

3

2

1

5

4

u

S = 52 14





color[u] black;}

0

3

2

1

5

4

u

S = 4352 1





color[u] black;}

0

3

2

1

5

4

u

S = 4352 1





color[u] black;}

0

3

2

1

5

4

u

S = 43521





color[u] black;}

0

3

2

1

5

4

u

S = 43521





color[u] black;}

0

3

2

1

5

4

u

S =





color[u] black;}

0

3

2

1

5

4

u

S =

Graph Search








color[u] black;}

0

3

2

1

5

4

u

S =

Graphs I Kruse and Ryba Chapter 12. Undirected Graphs An undirected graph is a set of nodes and a set of edges between nodes; e.g.: The graph is undirected.

Documents

vertices v j

finite set of vertices

source v i

undirected graph example

graph diagram

multiple edges slide

undirected graph definition

undirected graph g