directed vs undirected graph

7/30/2019 Directed vs Undirected Graph

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Directed and Undirected Graphs

A graph is a mathematical structure consisting of a set of vertices and a setof edges connecting the vertices.

Formally: G = (V,E), where V is a set and E V V.

G = (V,E) undirected if for all v,w V:

(v,w) E (w, v) E.

Otherwise directed.


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A directed graph

G = (V,E) with vertex set

V=

0, 1, 2, 3, 4, 5, 6

and edge set

E=

(0, 2), (0, 4), (0, 5), (1, 0), (2, 1), (2, 5), (3, 1), (3, 6), (4, 0), (4, 5), (6, 3), (6, 5).

4 5 6

32

0 1


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An undirected graph

4

0

1

2

3

6

5

7

8

9

1011

12


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Examples of graphs in real lifeRoad Maps.Edges represent streets and vertices represent crossings.


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Examples (contd)

Airline route maps.

Vertices represent airports, and there is an edge from vertexA

to vertexB

ifthere is a direct flight from the airport represented by A to the airport representedby B.


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Examples (contd)Electrical Circuits.Vertices represent diodes, transistors, capacitors, switches, etc., and edges

represent wires connecting them.


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Examples (contd)

Computer Networks.Vertices represent computers and edges represent network connections (cables)

between them.

The World Wide Web.Vertices represent webpages, and edges represent hyperlinks.


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Examples (contd)

Flowcharts.Vertices represent boxes and edges represent arrows.

Conflict graphs in scheduling problems.Example: Assignment of time slots to exams. No overlapping time slots for examstaken by the same students.

Modeled by graph whose vertices represent the exams, with an edge between twovertices if there is a student who has to take both exams.


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Examples (contd)

Molecules.Vertices are atoms, edges are bonds between them.

H C O H

H

C

H

H

H


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Examples (contd)

Binary Relations in Mathematics.

For example, the is a proper divisor relation on the integers.

15

10 2

4

8

6

7

3 9


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The adjacency matrix data structure

Let G = (V,E) be a graph with n vertices. Vertices ofG numbered 0, . . . , n 1.

The adjacency matrix ofG is the n n matrix A = (aij)0i,jn1 with

aij =

1 if there is an edge from vertex i to vertex j

0 otherwise.


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Example

4 5 6

32

0 1

0 0 1 0 1 1 01 0 0 0 0 0 00 1 0 0 0 1 0

0 1 0 0 0 0 11 0 0 0 0 1 00 0 0 0 0 0 00 0 0 1 0 1 0


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The adjacency list data structure

Array with one entry for each vertex v, which is a list of all vertices adjacent to v.

Example

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0 10

1

2

4

3

5

6

2 5 4

0

1 5

1 6

0 5

5 3


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Adjacency matrices vs adjacency lists

n = number of vertices, m = number of edges

adjacency matrix adjacency list

Space (n2) (n + m)

Time to check if

w adjacent to v

(1) (out-degree(v))

Time to visit allw adjacent to v

(n) (out-degree(v))

Time to visit alledges (n

2

) (n + m)


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Sparse and dense graphs

G = (V,E) graph with n vertices and m edges

Observation: m n2

G dense ifm close to n2

G sparse ifm much smaller than n2


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Graph traversalsA traversal is a strategy for visiting all vertices of a graph.

BFS = breadth-first search

DFS = depth-first search

General strategy:

1. Let v be an arbitrary vertex

2. Visit all vertices reachable from v

3. If there are vertices that have not been visited, let v be such a vertex and goback to (2)


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BFS

Visit all vertices reachable from v in the following order:

v

all neighbours ofv

all neighbours of neighbours ofv that have not been visited yet

all neighbours of neighbours of neighbours ofv that have not been visited yet

etc.


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BFS (contd)

Algorithm bfs(G)

1. Initialise Boolean array visited by setting all entries to false

2. Initialise QueueQ

3. for all v V do

4. if visited[v] = false then

5. bfsFromVertex(G, v

)


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BFS (contd)Algorithm bfsFromVertex(G, v)

1. visited[v] = true

2. Q.enqueue(v)

3. while not Q.isEmpty() do

4. v Q.dequeue()

5. for all w adjacent to v do

6. if visited[w] = false then

7. visited[w] = true

8. Q.enqueue(w)


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DFS

Visit all vertices reachable from v in the following order:

v

some neighbor w ofv that has not been visited yet

some neighbor x ofw that has not been visited yet

etc., until the current vertex has no neighbour that has not been visited yet

Backtrack to the first vertex that has a yet unvisited neighbour v.

Continue with v

, a neighbour, a neighbour of the neighbour, etc., backtrack,etc.


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DFS (contd)

Algorithm dfs(G)

1. Initialise Boolean array visited by setting all entries to false

2. Initialise Stack S

3. for all v V do


5. dfsFromVertex(G, v)


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DFS (contd)

Algorithm dfsFromVertex(G, v)

1. S.push(v)

2. while not S.isEmpty() do

3. v S.pop()


5. visited[v] = true

6. for all w adjacent to v do

7. S.push(w)

directed vs undirected graph

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