data structure lecture 8 graph[1]

7/28/2019 Data Structure Lecture 8 Graph[1]

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Lecture-8(graph)


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CS 103 2

Definition of Graphs

A graph is a finite set of nodes with edges

between nodes

Formally, a graph G is a structure (V,E)

consisting of

a finite set V called the set of nodes, and

a set E that is a subset of VxV. That is, E is a set of

pairs of the form (x,y) where x and y are nodes in

V


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What is a Graph?

Graphs are Generalization of Trees.

A simple graph G = (V, E) consists of a non-empty set V, whose members

are called the vertices of G, and a set E of pairs of distinct vertices from V,

called the edges of G.

Undirected Directed (Digraph) Weighted


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CS 103 4

Examples of Graphs

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

E={(0,1), (1,2), (0,3), (3,0), (2,2), (4,3)}

01

4

2

3

When (x,y) is an edge,

we say that x is adjacent to y, and y is

adjacent from x.


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Some Example Applications of Graph

Finding the least congested route between two phones, given connectionsbetween switching stations.

Determining if there is a way to get from one page to another, just byfollowing links.

Finding the shortest path from one city to another.

As a traveling sales-person, finding the cheapest path that passes throughall the cities that the sales person must visit.

Determining an ordering of courses so that prerequisite courses arealways taken first.


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Graphs Terminologies

Adjacent Vertices: there is a connecting edge.

A Path: A sequence of adjacent vertices.

A Cycle: A path in which the last and first vertices are adjacent.

Connected graph: There is a path from any vertex to every other vertex.

Path Cycle Connected Disconnected


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More Graph Terminologies

Path and cycles in a digraph: must move in the direction specified by the arrow.

Connectedness in a digraph: strong and weak.

Strongly Connected: If connected as a digraph - following the arrows.

Weakly connected: If the underlying undirected graph is connected (i.e.ignoring the arrows).

Directed Cycle Strongly Connected Weakly Connected


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Further Graph Terminologies

Emanate: an edge e = (v, w) is said to emanate from v.

A(v) denotes the set of all edges emanating from v.

Incident: an edge e = (v, w) is said to be incident to w.

I(w) denote the set of all edges incident to w.

Out-degree: number of edges emanating from v-- |A(v)|

In-degree: number of edges incident to w-- |I(w)|.

Directed Graph Undirected Graph


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Graph Representations

For vertices:

an array or a linked list can be used

For edges:

Adjacency Matrix (Two-dimensional array)

Adjacency List (One-dimensional array of linked lists)

Linked List (one list only)


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Adjacency Matrix Representation

Adjacency Matrix uses a 2-D array of dimension |V|x|V| foredges. (For vertices, a 1-D array is used)

The presence or absence of an edge, (v, w) is indicated by the

entry in row v, column w of the matrix.

For an unweighted graph, boolean values could be used.

For a weighted graph, the actual weights are used.


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Notes on Adjacency Matrix

For undirected graph, the adjacency matrix is always symmetric.

In a Simple Graph, all diagonal elements are zero (i.e. no edge from avertex to itself).

However, entries in the matrix can be accessed directly.


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Adjacency List Representation

This involves representing the set of vertices adjacent to each vertex as a

list. Thus, generating a set of lists.

This can be implemented in different ways.

Our representation:

Vertices as a one dimensional array

Edges as an array of linked list (the emanating edges of vertex 1 will be in thelist of the first element, and so on,

1

2

3

4

vertices

edges

2 3 Null

3 4 Null

1 3 Null2

Empty


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Simple List Representation

Vertices are represented as a 1-D array or a linked list

Edges are represented as one linked list

Each edge contains the information about its two vertices

1

2

3

4

vertices edges

edge(1,2)

edge(1,3)

edge(2,3)

edge(2,4)edge(4,1)

edge(4,2)

edge(4,3)

.

.

.

.

.

.


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Breadth First Search Algorithm

Pseudocode: Uses FIFO Queue Q


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DFS Algorithm

Pseudocode

DFS(s)

for each vertex u Vdo color[u] White ; not visited

time

1 ; time stampfor each vertex u Vdo if color[u]=White

then DFS-Visit(u,time)

DFS-Visit(u,time)

color[u] Gray ; in progress nodesd[u] time ; d=discover timetime time+1for each v Adj[u] do

if color[u]=White

then DFS-Visit(v,time)

color[u] Blackf[u] time time+1 ; f=finish time


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The Floyd-Warshall Algorithm

Represent the directed, edge-weighted

graph in adjacency-matrix form.

W= matrix of weights =

w w w

w w w

w w w

11 12 13

21 22 23

31 32 33

wijis the weight of edge (i, j), or if there is no suchedge.

Return a matrix D, where each entry dij is d(i,j).

Could also return a predecessor matrix, P, where eachentry pij is the predecessor of j on the shortest pathfrom i.


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Floyd-Warshall:

Consider intermediate vertices of a path:

i j

Say we know the length of the shortest path from i

to j whose intermediate vertices are only those with

numbers 1, 2, ..., k-1.


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Floyd-Warshall

Two possibilities:

1. Going through the vertex kdoesnthelp the path through vertices 1...k-1 is

still the shortest. 2. There is a shorter path consisting of two

subpaths, one from ito kand one from ktoj. Each subpath passes only through

vertices numbered 1 to k-1.

j

k

i


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8a-ShortestPathsMore

Floyd-Warshall

Thus,

(since there are no intermediate vertices.)

When k= |V|, were done.

ijk

ijk

ikk

kjk

ij ij

d d d d

d w

( ) ( ) ( ) ( )

( )

min( , )

1 1 1

0Also,


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Floyd-Warshall

Floyd-Warshall is a dynamic programming

algorithm:

Compute and store solutions to sub-

problems. Combine those solutions to

solve larger sub-problems.

Here, the sub-problems involve finding the

shortest paths through a subset of the

vertices.


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Code for Floyd-Warshall

Floyd-Warshall(W)

)(

)1()1()1()(

)0(

return7

),min(6

to1fordo5

to1ifordo4

to1for3

2

verticesofnumber//][1

n

k

kj

k

ik

k

ij

k

ij

D

nj

n

nk

WD

Wrowsn

dddd


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Example of Floyd-Warshall

data structure lecture 8 graph[1]

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