Find Transitive closure of a Graph Using Warshall's Algorithm

Oct 16, 2019Download as pptx, pdf0 likes4,515 views

Here I actually describe how we can find transitive closure of a graph using warshall' algorithm. It will be easy to learn about transitive closure, their time complexity, count space complexity.

Find Transitive Closure Using Warshall’s Algorithm
Md. Safayet Hossain
M.Sc student of CSE department , KUET.

1. It is transitive
2. It contains R
3. Minimal relation satisfies (1) & (2)
RT = R ∪ { (a, c) | (a, b) ∈ R ,(b, c) ∈ R }
Example:
A = { 1, 2, 3}
R ={ (1, 2), (2, 3) }
RT = { (1, 2), (2, 3) , (1, 3)}
Transitive closure
1 2
3
1 2
3

Input: Input the given graph as adjacency matrix
Output: Transitive Closure matrix.
Begin
copy the adjacency matrix into another matrix named T
for any vertex k in the graph, do
for each vertex i in the graph, do
for each vertex j in the graph, do
T [ i, j] = T [i, j] OR (T [ i, k]) AND T [ k, j])
done
done
done
Display the T
End
Algorithm to find transitive closure using Warshall’s algorithm

0 1
23
0 1 1 0
0 0 1 0
0 0 0 1
0 0 0 0
T =
MAdj =
0 1 1 0
0 0 1 0
0 0 0 1
0 0 0 0
0 1 2 3
0
1
2
3
copy

0 1 1 0
0 0 1 0
0 0 0 1
0 0 0 0
T =
for (k = 0; k < V; k++) {
for (i = 0; i < V; i++) {
for (j = 0; j < V; j++) {
//T[i][j] = T[i][j] || ( T[i][k] && T[k][j] );
T[0][0] = T[0][0] || ( T[0][0] && T[0][0] );
}
}
}
0
k =0
i =0
j = 0
T =
Transitive matrix when k = 0

0 1 1 0
0 0 1 0
0 0 0 1
0 0 0 0
T =
for (k = 0; k < V; k++) {
for (i = 0; i < V; i++) {
for (j = 0; j < V; j++) {
//T[i][j] = T[i][j] || (T[i][k] && T[k][j]);
T[0][1] = T[0][1] || (T[0][0] && T[0][1]);
}
}
}
0 1
k =0
i =0
j = 1
T =
Transitive matrix when k = 0

0 1 1 0
0 0 1 0
0 0 0 1
0 0 0 0
T =
for (k = 0; k < V; k++) {
for (i = 0; i < V; i++) {
for (j = 0; j < V; j++) {
//T[i][j] = T[i][j] || (T[i][k] && T[k][j]);
T[0][2] = T[0][2] || (T[0][0] && T[0][2]);
}
}
}
0 1 1
k =0
i =0
j = 2
T =
Transitive matrix when k = 0

0 1 1 0
0 0 1 0
0 0 0 1
0 0 0 0
T =
for (k = 0; k < V; k++) {
for (i = 0; i < V; i++) {
for (j = 0; j < V; j++) {
//T[i][j] = T[i][j] || (T[i][k] && T[k][j]);
T[0][3] = T[0][3] || (T[0][0] && T[0][3]);
}
}
}
0 1 1 0
k =0
i =0
j = 3
T =
Transitive matrix when k = 0

0 1 1 0
0 0 1 0
0 0 0 1
0 0 0 0
T =
for (k = 0; k < V; k++) {
for (i = 0; i < V; i++) {
for (j = 0; j < V; j++) {
//T[i][j] = T[i][j] || (T[i][k] && T[k][j]);
T[3][3] = T[3][3] || (T[3][0] && T[0][3]);
}
}
}
0 1 1 0
0 0 1 0
0 0 0 1
0 0 0 0
k =0
i =3
j = 3
T =
Transitive matrix when k = 0

0 1 1 0
0 0 1 0
0 0 0 1
0 0 0 0
T =
for (k = 0; k < V; k++) {
for (i = 0; i < V; i++) {
for (j = 0; j < V; j++) {
//T[i][j] = T[i][j] || (T[i][k] && T[k][j]);
T[3][3] = T[3][3] || (T[3][0] && T[0][3]);
}
}
}
0 1 1 0
0 0 1 0
0 0 0 1
0 0 0 0
k =1
i =3
j = 3
T =
Transitive matrix when k = 1

0 1 1 0
0 0 1 0
0 0 0 1
0 0 0 0
T =
for (k = 0; k < V; k++) {
for (i = 0; i < V; i++) {
for (j = 0; j < V; j++) {
//T[i][j] = T[i][j] || (T[i][k] && T[k][j]);
T[0][3] = T[0][3] || (T[0][2] && T[2][3]);
}
}
}
0 1 1 1
0 0 1 0
0 0 0 1
0 0 0 0
k =2
i =0
j = 3
T =
Transitive matrix when k = 2

0 1 1 0
0 0 1 0
0 0 0 1
0 0 0 0
T =
for (k = 0; k < V; k++) {
for (i = 0; i < V; i++) {
for (j = 0; j < V; j++) {
//T[i][j] = T[i][j] || (T[i][k] && T[k][j]);
T[1][3] = T[1][3] || (T[1][2] && T[2][3]);
}
}
}
0 1 1 1
0 0 1 1
0 0 0 1
0 0 0 0
k =2
i =1
j = 3
T =
Transitive matrix when k = 2

0 1 1 0
0 0 1 0
0 0 0 1
0 0 0 0
T =
for (k = 0; k < V; k++) {
for (i = 0; i < V; i++) {
for (j = 0; j < V; j++) {
//T[i][j] = T[i][j] || (T[i][k] && T[k][j]);
T[3][3] = T[3][3] || (T[3][2] && T[2][3]);
}
}
}
0 1 1 1
0 0 1 1
0 0 0 1
0 0 0 0
k =2
i =3
j = 3
T =
Transitive matrix when k = 2

0 1 1 0
0 0 1 0
0 0 0 1
0 0 0 0
T =
for (k = 0; k < V; k++) {
for (i = 0; i < V; i++) {
for (j = 0; j < V; j++) {
//T[i][j] = T[i][j] || (T[i][k] && T[k][j]);
T[3][3] = T[3][3] || (T[3][3] && T[3][3]);
}
}
}
0 1 1 1
0 0 1 1
0 0 0 1
0 0 0 0
k =3
i =3
j = 3
T =
Transitive matrix when k = 3

0 1 1 1
0 0 1 1
0 0 0 1
0 0 0 0
T(3) =
0 1
23
Final Transitive closure of the given graph
0 1 1 1
0 0 1 1
0 0 0 1
0 0 0 0
Transitive closure matrix for k = 3
0 1 1 0
0 0 1 0
0 0 0 1
0 0 0 0
MAdj =

0 1 1 0
0 0 1 0
0 0 0 1
0 0 0 0
T =
0 1 1 0
0 0 1 0
0 0 0 1
0 0 0 0
0 1 1 0
0 0 1 0
0 0 0 1
0 0 0 0
Transitive closure matrix for k=0
Transitive closure matrix for k=1
0 1 1 1
0 0 1 1
0 0 0 1
0 0 0 0
Transitive closure matrix for k=2
0 1 1 1
0 0 1 1
0 0 0 1
0 0 0 0
Transitive closure matrix for k=3
O(n2)
O(n2)
O(n2)
O(n2)
Time complexity = n * O(n2)
= O(n3)
Space complexity = n * O(n2)
= O(n3)
T =
T =
T =
T =
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Find Transitive closure of a Graph Using Warshall's Algorithm

1. Welcome to My presentation

2. Find Transitive Closure Using Warshall’s Algorithm Md. Safayet Hossain M.Sc student of CSE department , KUET.

3. 1. It is transitive 2. It contains R 3. Minimal relation satisfies (1) & (2) RT = R ∪ { (a, c) | (a, b) ∈ R ,(b, c) ∈ R } Example: A = { 1, 2, 3} R ={ (1, 2), (2, 3) } RT = { (1, 2), (2, 3) , (1, 3)} Transitive closure 1 2 3 1 2 3

4. Input: Input the given graph as adjacency matrix Output: Transitive Closure matrix. Begin copy the adjacency matrix into another matrix named T for any vertex k in the graph, do for each vertex i in the graph, do for each vertex j in the graph, do T [ i, j] = T [i, j] OR (T [ i, k]) AND T [ k, j]) done done done Display the T End Algorithm to find transitive closure using Warshall’s algorithm

5. 0 1 23 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 T = MAdj = 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 2 3 0 1 2 3 copy

6. 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 T = for (k = 0; k < V; k++) { for (i = 0; i < V; i++) { for (j = 0; j < V; j++) { //T[i][j] = T[i][j] || ( T[i][k] && T[k][j] ); T[0][0] = T[0][0] || ( T[0][0] && T[0][0] ); } } } 0 k =0 i =0 j = 0 T = Transitive matrix when k = 0

7. 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 T = for (k = 0; k < V; k++) { for (i = 0; i < V; i++) { for (j = 0; j < V; j++) { //T[i][j] = T[i][j] || (T[i][k] && T[k][j]); T[0][1] = T[0][1] || (T[0][0] && T[0][1]); } } } 0 1 k =0 i =0 j = 1 T = Transitive matrix when k = 0

8. 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 T = for (k = 0; k < V; k++) { for (i = 0; i < V; i++) { for (j = 0; j < V; j++) { //T[i][j] = T[i][j] || (T[i][k] && T[k][j]); T[0][2] = T[0][2] || (T[0][0] && T[0][2]); } } } 0 1 1 k =0 i =0 j = 2 T = Transitive matrix when k = 0

9. 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 T = for (k = 0; k < V; k++) { for (i = 0; i < V; i++) { for (j = 0; j < V; j++) { //T[i][j] = T[i][j] || (T[i][k] && T[k][j]); T[0][3] = T[0][3] || (T[0][0] && T[0][3]); } } } 0 1 1 0 k =0 i =0 j = 3 T = Transitive matrix when k = 0

10. 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 T = for (k = 0; k < V; k++) { for (i = 0; i < V; i++) { for (j = 0; j < V; j++) { //T[i][j] = T[i][j] || (T[i][k] && T[k][j]); T[3][3] = T[3][3] || (T[3][0] && T[0][3]); } } } 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 k =0 i =3 j = 3 T = Transitive matrix when k = 0

11. 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 T = for (k = 0; k < V; k++) { for (i = 0; i < V; i++) { for (j = 0; j < V; j++) { //T[i][j] = T[i][j] || (T[i][k] && T[k][j]); T[3][3] = T[3][3] || (T[3][0] && T[0][3]); } } } 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 k =1 i =3 j = 3 T = Transitive matrix when k = 1

12. 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 T = for (k = 0; k < V; k++) { for (i = 0; i < V; i++) { for (j = 0; j < V; j++) { //T[i][j] = T[i][j] || (T[i][k] && T[k][j]); T[0][3] = T[0][3] || (T[0][2] && T[2][3]); } } } 0 1 1 1 0 0 1 0 0 0 0 1 0 0 0 0 k =2 i =0 j = 3 T = Transitive matrix when k = 2

13. 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 T = for (k = 0; k < V; k++) { for (i = 0; i < V; i++) { for (j = 0; j < V; j++) { //T[i][j] = T[i][j] || (T[i][k] && T[k][j]); T[1][3] = T[1][3] || (T[1][2] && T[2][3]); } } } 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 k =2 i =1 j = 3 T = Transitive matrix when k = 2

14. 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 T = for (k = 0; k < V; k++) { for (i = 0; i < V; i++) { for (j = 0; j < V; j++) { //T[i][j] = T[i][j] || (T[i][k] && T[k][j]); T[3][3] = T[3][3] || (T[3][2] && T[2][3]); } } } 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 k =2 i =3 j = 3 T = Transitive matrix when k = 2

15. 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 T = for (k = 0; k < V; k++) { for (i = 0; i < V; i++) { for (j = 0; j < V; j++) { //T[i][j] = T[i][j] || (T[i][k] && T[k][j]); T[3][3] = T[3][3] || (T[3][3] && T[3][3]); } } } 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 k =3 i =3 j = 3 T = Transitive matrix when k = 3

16. 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 T(3) = 0 1 23 Final Transitive closure of the given graph 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 Transitive closure matrix for k = 3 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 MAdj =

17. 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 T = 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 Transitive closure matrix for k=0 Transitive closure matrix for k=1 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 Transitive closure matrix for k=2 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 Transitive closure matrix for k=3 O(n2) O(n2) O(n2) O(n2) Time complexity = n * O(n2) = O(n3) Space complexity = n * O(n2) = O(n3) T = T = T = T = Time & space Complexity

18. Thank You