Clear Explanation about the topological sort

1. TOPOLOGICAL SORTING
GROUP MEMBERS
A.SHEREEN FATHIMA(140071601068)
D.STELLA(140071601073)
CSE “B”
ALGORITHM DESIGN AND ANAYLSIS LAB(CSB3105)
B.S ABDUR RAHMAN UNIVERSITY
1

2. PREFACE
Problem
Description
Student
Activities
Topological
sort
Algorithm of
topology
sorting
Complexity
Of an algorithm
2

3. PROBLEM DESCRIPTION
3
A Student has various activities to be performed on
time to maintain effectiveness in both educational
and professional development. So in order to keep
the activities organized in a effective manner, the
order of the activities to be done is sorted
systematically using the topological sorting
algorithms.

4. WAKE
UP STUDY
ALGORITHM
ANAYSIS
BREAKFAST
NAP
STUDY
PROGRAMS
WORK
OUT
AT
COLLEG
E
ATTENDING
CLASSES
PLAY
SLEE
P
DREAM
ABOUT
FUTURE
ACTIVITIES OF STUDENT
4

5. AT
COLLEGE
STUDY
PROGRAM
WORK
OUT
DREAM
ABOUT
FUTURE
BREAK
FAST
PLAY
SLEEP
STUDY
ADA
NAP
ATTEND
CLASSES
WAKE
UP
5
Student activity graph

6. DEFINITION :
6
A Topology sort or topological ordering of a directed graph is
a linear ordering of its vertices such that for every directed
edge UV from vertex U to vertex V,Ucomes before V in the
ordering

7. ALGORITHM
1. Compute the indegrees of all vertices
2. Find a vertex U with indegree 0 and print it (store it in
the ordering)If there is no such vertex then there is a
cycle and the vertices cannot be ordered. Stop.
3. Remove U and all its edges (U,V) from the graph.
4. Update the indegrees of the remaining vertices.
5. Repeat steps 2 through 4 while there are vertices to be
processed
7

8. IMPROVED ALGORITHM
After the initial scanning to find a
vertex of degree 0, we need to scan only those vertices whose
updated indegrees have become equal to zero.
1. Store all vertices with indegree 0 in a queue
2. Get a vertex U and place it in the sorted sequence
(array or another queue).
3. For all edges (U,V) update the indegree of V, and put
V in the queue if the updated indegree is 0.
4. Perform steps 2 and 3 while the queue is not empty.
8

9. 7
AT
COLLEGE
5
STUDY
PROGRAM
6
WORK
OUT
11
DREAM
ABOUT
FUTURE
4
BREAK
FAST
9
PLAY
10
SLEEP
2
STUDY
ADA
3
NAP
8
ATTEND
CLASSES
1
WAKE
UP
9
Student activity graph
Indegree of 1 = 0
Indegree of 3 = 2
Indegree of 8 = 1
Indegree of 5 = 2
Indegree of 10 = 1
Indegree of 2 = 1
Indegree of 7 = 2
Indegree of 9 = 2
Indegree of 4 = 2
Indegree of 6 = 1
Indegree of 11 = 1

10. 7
AT
COLLEGE
5
STUDY
PROGRAM
6
WORK
OUT
11
DREAM
ABOUT
FUTURE
4
BREAK
FAST
9
PLAY
10
SLEEP
2
STUDY
ADA
3
NAP
8
ATTEND
CLASSES
10
Student activity graph
Indegree of 3 = 1
Indegree of 8 = 1
Indegree of 5 = 2
Indegree of 10 = 1
Indegree of 2 = 0
Indegree of 7 = 2
Indegree of 9 = 2
Indegree of 4 = 1
Indegree of 6 = 1
Indegree of 11 = 1
1

11. 7
AT
COLLEGE
5
STUDY
PROGRAM
6
WORK
OUT
11
DREAM
ABOUT
FUTURE
4
BREAK
FAST
9
PLAY
10
SLEEP
3
NAP
8
ATTEND
CLASSES
11
Student activity graph
Indegree of 3 = 0
Indegree of 8 = 1
Indegree of 5 = 2
Indegree of 10 = 1
Indegree of 7 = 2
Indegree of 9 = 2
Indegree of 4 = 1
Indegree of 6 = 1
Indegree of 11 = 1
1 2

12. 12
1
WAKE
UP
2
STUDY
ADA
3
NAP
4
BREAK
FAST
5
STUDY
PROGRAM
6
WORK
OUT
7
AT
COLLEGE
8
ATTENDING
CLASS
9
PLAY
10
SLEEP
11
DREAM
ABOUT
FUTURE
SORTED ACTIVITIES

13. COMPLEXITY OF AN ALGORITHM
For input graph G = (V,E), Run Time = ?
Break down into total time required to:
§ Initialize In-Degree array: O(|E|)
§ Find vertex with in-degree 0:|V| vertices,
each takes O(|V|) to search In-Degree array.
§ Total time = O(|V|²)
§ Reduce In-Degree of all vertices adjacent to
a vertex: O(|E|)
§ Output and mark vertex: O(|V|)
§ Total time= O(|V|² + |E|) Quadratic time! 13

14. 14