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Dr. Ahmad R. Hadaegh

A.R. Hadaegh Cali for nia State University San Marcos (CSUSM) Page 1

Graphs

TopologicalSort

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Dr. Ahmad R. Hadaegh

A.R. Hadaegh Cali for nia State University San Marcos (CSUSM) Page 2

Topological Sorting

When a set of tasks has to be performed, you may need to order thetasks based on priority or dependency

For example, a student cannot take CSII unless he/she takes CSI

First.

Or for example in a program, procedure 2 cannot be executed

unless procedure 1 is executed; however, order of procedure 3 and

procedure 4 may not be important

A topological sort creates a linear structure of a diagram; that is it

labels all its vertices with number 1, 2, , |V| so that i

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Dr. Ahmad R. Hadaegh

A.R. Hadaegh Cali for nia State University San Marcos (CSUSM) Page 3

The diagram must not include cycle; otherwise, the topologicalsort is impossible

The algorithm for topological sort is To find a vertex vwith no outgoing edge called a sinkor aminimal vertexand then disregard all edges leading from anyvertex to v

To summarize the above:TopologicalSort (diagraph)for i=1 to |V|{

find a minimal vertex v;num(v) = i;

}remove from diagraph vertex v and all edges incident with v

The next example shows a graph that goes through a sequence ofdeletion to find the topological sort of the vertices

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Dr. Ahmad R. Hadaegh

A.R. Hadaegh Cali for nia State University San Marcos (CSUSM) Page 4

c

a

f

d

b

g

e c

a

f

d

b

e

First minimalvertex is g

c

a

f

d

b

c

a

f

dc

a

d

c

a

aNULL

nextoneis e

next one is bnext one is f

nextone

is d

next one is c next one is a

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Dr. Ahmad R. Hadaegh

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TS(v)

num(v) = i++;

for all vertices u adjacent to v

if num(u) = 0TS(u)

else if TSNum(u) = 0 // cycle is detected

error;

TSNum(v) =j++

topologicalSorting(digraph)

for all vertices v

num(v) = TSNum(v) = 0

i = j = 1;while there is a vertex v such that num(v) = 0

TS(v)

Output vertices according to their TSNums

This is a differentalgorithm thattravels through the

vertices and marksthe vertices basedon their dependencyto find a topologicalsort of vertices

This algorithm doesnot remove anyvertex or edge from

the graph

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Dr. Ahmad R. Hadaegh

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c

a

f

d

b

g

e c(2)

a(1)

d(3)

d

f

f(7)b(4) e

e(5)

g(6)

The vertices are numbered as they are ordered

The topological sort of the vertices can be found if youprint this tree in post order traversal

g e b f d c a

TSNum(g) = 1

TSNum(b) = 3

TSNum(e) = 2

TSNum(f) = 4

TSNum(d) = 5

TSNum(c) = 6

TSNum(a) = 7