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1. Topological Sort
Strongly Connected Components

2.  For a directed acyclic graph G=(V,E)
 A topological sort is an ordering of all of G’svertices
 v1, v2, v3,….,vn such that …
 Vertex u comes before vertex v if edge (u,v) belongs to G
 So,formally for every egde (vi,vk) in E ,i<k
WHAT IS TOPOLOGICAL SORT ?

3. TOPOLOGICAL SORT
• Definition:
A topological sort or topological ordering of a directed ayclic
graph is a linear ordering of its vertices such that for every
directed edge (u,v) from vertex u to v, u comes before v in the
ordering.

4. 0 1
5
2
4
3
• There can be more then one
topological sorting for a graph.
• A topological sorting of the given
graph can be
• “5 4 2 3 1 0”
• “4 5 2 3 1 0”
• “5 2 3 4 1 0”
• “5 2 3 4 0 1”
• The first vertex in topological
sorting is always a vertex with
in-degree as “0”
• ( a vertex with no incoming edges).

5. Topological sorting Application
• The vertices of the graph may represent tasks to be performed, and the
edges may represent constraints that one task must be performed before
another;
• In this application, a topological ordering is just a valid sequence for the
tasks.
• A topological ordering is possible if and only if the graph has no
directed cycles, that is, if it is a directed acyclic graph (DAG).

6. Topological sorting Application

7. The graph shown to the left has many valid
topological sorts, including:
• 5, 7, 3, 11, 8, 2, 9, 10 (visual left-to-right,top-to-bottom)
• 3, 5, 7, 8, 11, 2, 9, 10 (smallest-numbered available vertexfirst)
• 5, 7, 3, 8, 11, 10, 9, 2 (fewest edges first)
• 7, 5, 11, 3, 10, 8, 9, 2 (largest-numbered available vertexfirst)
• 5, 7, 11, 2, 3, 8, 9, 10 (attempting top-to-bottom,left-to-right)
• 3, 7, 8, 5, 11, 10, 2, 9 (arbitrary)

8. Algorithms
Kahn's Algorithm
One of these algorithms, first described by Kahn (1962) works by choosing vertices
in the same order as the eventual topological sort.
L ← Empty list that will contain the sorted elements
S ← Set of all nodes with no incoming edge
while S is non-empty do
remove a node n from S
add n to tail of L
for each node m with an edge e from n to m do
remove edge e from the graph
if m has no other incoming edges then
insert m into S
if graph has edges then
return error (graph has at least one cycle)
else
return L (a topologically sorted order)
If the graph is a DAG (Directed Acyclic Graph) a solution will be contained in the list L

9. Example
1 2
4
3 5
6 7
Identify nodes having in degree ‘0’
Select a node and delete it with its
edges then add node to output
Select Node : 1
Output :
1

10. Contd…
2
4
3 5
6 7
Identify nodes having in degree ‘0’
Select a node and delete it with its
edges then add node to output
Select Node : 2
Output :
1 2

11. Contd…
4
3 5
6 7
Identify nodes having in degree ‘0’
Select a node and delete it with its
edges then add node to output
Select Node : 5
Output :
1 2 5

12. Contd…
4
3
6 7
Identify nodes having in degree ‘0’
Select a node and delete it with its
edges then add node to output
Select Node : 4
Output :
1 2 5 4

13. Contd…
3
6 7
Identify nodes having in degree ‘0’
Select a node and delete it with its
edges then add node to output
Select Node : 3
Output :
1 2 5 4 3

14. Contd…
6 7
Identify nodes having in degree ‘0’
Select a node and delete it with its
edges then add node to output
Select Node : 6, 7
Output :
1 2 5 4 3 6 7
1 2
4
3 5
6 7

15. Topological sort using DFS
Topological_Sort(G)
1. Call DFS(G) to compute finishing time v.f for each vertex v.
2. As each vertex is finished, insert it into the front of a linked list.
3. Return the linked list of vertices.
Time complexity : 𝛳(V+E) as DFS takes 𝛳(V+E) time.
It takes O(1) time to insert each of the |V| vertices onto the front of the
linked list.

16. Example
e a b d f g c h i
a e
d
b
c g
h
i
f
11/16
12/15
6/7
1/8
2/5
3/4
17/18
13/14
9/10
DFS Traversal
3/4
2/5
6/7
1/8 9/10
13/14
12/15
11/16
17/18

17. Strongly Connected
Components
A strongly connected component of a directed graph
G=(V,E) is a maximal set of vertices such that
for every pair of vertices u and v in C , we have both
u~v and v~u
C V

18. STRONGLY-CONNECTED-COMPONENTS(G)
1. call DFS(G) to compute finishing times f[u] for each vertex u
2. compute GT
GT = (V, ET), where ET = {(u,v): (v,u)ϵ E}. That is, ET consists of
the edges of G with their directions reversed.
3. call DFS(GT), but in the main loop of DFS, consider the vertices
in order of decreasing f[u] (as computed in line 1)
4. output the vertices of each tree in the depth-first forest formed in
line 3 as a separate strongly connected component

19. (a) A directed graph G. Each shaded
region is a strongly connected
component of G. Each vertex is
labeled with its discovery and
finishing times in a depth-first search,
and tree edges are shaded.
(b) The graph GT, the transpose of G,
with the depth-first forest computed
in line 3 of STRONGLY-
CONNECTED-COMPONENTS
shown and tree edges shaded. Each
strongly connected component
corresponds to one depth-first tree.
Vertices b, c, g, and h, which are
heavily shaded, are the roots of the
depth-first trees produced by the
depth-first search of GT.
(c) The acyclic component graph GSCC
obtained by contracting all edges
within each strongly connected
component of G so that only a single
vertex remains in each component.

20. Strongly Connected Components
The transpose of a graph G, GT=(V,ET) is used for finding the strongly
connected components of a graph G=(V,E)
It is interesting to observe that G and GT have exactly the same strongly
connected components

21. • The complexity of algorithm is 𝛳(V+E).
Strongly Connected Components