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Data Structures and Algorithms

(CS210/ESO207/ESO211)

Lecture 19

Graphs Definitions, notations, and terminologies

Data structures for graphs

A few algorithmic problems in graphs

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Why Graphs ??

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Finding shortest route between cities

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Given a network of roadsconnecting various cities, design an algorithm to

compute the shortest route between any two cities.

Just imagine how you would solve/approach this problem.

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Embedding an integrated circuit on

mother board

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How would you embed portsof various ICs on a plane and make connectionsamong

them so that

No two connections intersect each other

The total length of all the connections is minimal

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A social network or world wide web (WWW)

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Can we make some useful observations about the structure (diameter, degree

distribution, etc.) of such networks ?

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The three applications can be modeled as Graph

VerticesEdges

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Graph

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Definitions, notations, and terminologies

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Graph

A graph Gis defined by two sets

V : set of vertices

E : set of edges

Note that E (VV)

Notation: A graph consisting of vertices V and edges E is denoted by G= (V,E)

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Types of graphs

Undirected Graph Directed Graph

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V= {1,2,3,4,5,6}

E= {(1,2), (1,5),

(2,5), (2,3),

(3,4),

(4,5), (4,6)}

V= {0,1,2,3,4,5}

E= { (0,1), (0,4),(1,2),

(2,0), (2,1),

(3,2),

(4,5),

(5,4) }

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Notations

Notations:

= |V|

m= |E|

Note: For directed graphs, m ??.

For undirected graphs, m

??

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(-1)

(-1)/2

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Terminologies

is a walkfrom1 to4.

is nota walk.

is a walkfrom 1to 6. is a pathfrom 1to 6.

is a cycle.

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Terminologies

Connected component:

A maximalsubset of connected vertices

You can not add any more vertex to the subsetand still keeping it connected.

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Data Structures for Graphs

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Link based data structure for graph

Undirected Graph Adjacency Lists

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V= {1,2,3,4,5,6}

E= {(1,2), (1,5),

(2,5), (2,3),

(3,4),

(4,5), (4,6)}

1 3 5

2 5

2 4

4

3 5 6

1 2 4

1

2

3

4

5

6

Size = O(n+m)

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Link based data structure for graph

Advantage of Adjacency Lists :

Space efficient

All the neighbors of a vertex can be computed in optimal time.

Disadvantage of Adjacency Lists :

To determine if there is an edge fromxto y, it may require scanning the

entire adjacency list of vertexx(Hence O(n) time in the worst case).

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Array based data structure for graph

Undirected Graph Adjacency Matrix

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V= {1,2,3,4,5,6}

E= {(1,2), (1,5),

(2,5), (2,3),

(3,4),

(4,5), (4,6)}

1

2

3

4

5

6

0 1 0 0 1 0

1 2 3 4 5 6

1 0 1 0 1 0

0 1 0 1 0 0

0 0 1 0 1 1

1 1 0 1 0 0

0 0 0 1 0 0

Size = O(2)

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Array based data structure for graph

Advantage of Adjacency Matrix :

It takes O(1) time to determine whether there is an edge fromxto y for

any two verticesxand y.

Disadvantage of Adjacency Matrix :

It takes O() time to compute/enumerate all neighbors of a given vertexx.

It takes O() space. So we can not store Adjacency Matrix in these days

RAM even for graphs having 1 million vertices.

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Which data structure is commonly used for

storing graphs ?

Majority of applications of graphs store them in adjacency lists mainly due to

the following compelling reasons.

Graphs in real life are sparse ( ). So Adjacency lists representation

offers a much more compact storage compared to Adjacency matrixrepresentation.

Most of the algorithms on graphs require enumerating and processing all

neighbors of each vertex. As a result adjacency matrix will enforce O()

bound on the time complexity for the algorithm irrespective of the othercomputation.

**There are some exceptions where the algorithm does not require enumeration of all

neighbors of each vertex. One such exception is discussed at the end of this lecture.

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A sample of Graph algorithmic Problems

Almost every second algorithm is a graph algorithms due to wide applications

of graphs for modeling various computational problems. The student is

encouraged to ponder over the following problems. All of them and many

more will be discussed during this course and the course CS345.

Determine if there is a path between any two given vertices in a graph.

Compute all connected components in a graph.

Determine if there is a cycle in a graph.

Compute the largest possible set of edges from the graph whose removal

does not destroy the connectivity of the graph Compute a path of shortest length between two vertices.

Determine if there is a cycle passing through all vertices.

Compute the cycle of shortest length in a graph.

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An interesting problem(Finding a sink)

A vertexxin a given directed graph is said to be a sinkif

There is no edge emanating from (or leaving)x

Every other vertex has an edge intox.

Given a directed graph G=(V,E) given in an adjacency matrix representation,

design an O(n) time algorithm to determine if there is any sinkin G.

You are encouraged to solve the problem mentioned above. It does not require

anything more than what has been discussed in this lecture. We shall discuss its

solution in the next class.

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