Find Transitive closure of a Graph Using Warshall's Algorithm
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The presentation discusses finding the transitive closure of a graph using Warshall's algorithm, presented by Md. Safayet Hossain. It explains the concept of transitive closure, provides examples, and details the algorithm's process, including the time and space complexities. The final transitive closure matrix is displayed, highlighting the method's efficiency.
![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](https://image.slidesharecdn.com/transitiveclosure-191016130716/85/Find-Transitive-closure-of-a-Graph-Using-Warshall-s-Algorithm-4-320.jpg)
![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](https://image.slidesharecdn.com/transitiveclosure-191016130716/85/Find-Transitive-closure-of-a-Graph-Using-Warshall-s-Algorithm-6-320.jpg)
![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](https://image.slidesharecdn.com/transitiveclosure-191016130716/85/Find-Transitive-closure-of-a-Graph-Using-Warshall-s-Algorithm-7-320.jpg)
![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](https://image.slidesharecdn.com/transitiveclosure-191016130716/85/Find-Transitive-closure-of-a-Graph-Using-Warshall-s-Algorithm-8-320.jpg)
![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](https://image.slidesharecdn.com/transitiveclosure-191016130716/85/Find-Transitive-closure-of-a-Graph-Using-Warshall-s-Algorithm-9-320.jpg)
![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](https://image.slidesharecdn.com/transitiveclosure-191016130716/85/Find-Transitive-closure-of-a-Graph-Using-Warshall-s-Algorithm-10-320.jpg)
![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](https://image.slidesharecdn.com/transitiveclosure-191016130716/85/Find-Transitive-closure-of-a-Graph-Using-Warshall-s-Algorithm-11-320.jpg)
![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](https://image.slidesharecdn.com/transitiveclosure-191016130716/85/Find-Transitive-closure-of-a-Graph-Using-Warshall-s-Algorithm-12-320.jpg)
![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](https://image.slidesharecdn.com/transitiveclosure-191016130716/85/Find-Transitive-closure-of-a-Graph-Using-Warshall-s-Algorithm-13-320.jpg)
![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](https://image.slidesharecdn.com/transitiveclosure-191016130716/85/Find-Transitive-closure-of-a-Graph-Using-Warshall-s-Algorithm-14-320.jpg)
![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](https://image.slidesharecdn.com/transitiveclosure-191016130716/85/Find-Transitive-closure-of-a-Graph-Using-Warshall-s-Algorithm-15-320.jpg)
Find Transitive closure of a Graph Using Warshall's Algorithm
- 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