DATA STRUCTURE AND ALGORITHM LMS MST KRUSKAL'S ALGORITHM
- 1. SRM Institute of Science and Technology Minimum Spanning Tree - Kruskal’s Algorithm
- 2. SRM Institute of Science and Technology A Spanning tree can be defined as a subset of a graph, which consists of all the vertices covering minimum possible edges and does not have a cycle. Spanning tree cannot be disconnected. Every connected and undirected graph has at least one spanning tree. A disconnected graph does not have a spanning tree as it is not possible to include all vertices.
- 3. NN-2 number of spanning trees. Thus in the above graph N =3, therefore, it has 3(3-2) = 3 spanning trees.
- 4. SRM Institute of Science and Technology Some of the properties of the spanning tree are listed below: •A connected graph can have more than one spanning trees. •All spanning trees in a graph have the same number of nodes and edges. •If we remove one edge from the spanning tree, then it will become minimally connected and will make the graph disconnected. •On the other hand, adding one edge to the spanning tree will make it maximally acyclic thereby creating a loop. •A spanning tree does not have a loop or a cycle.
- 5. SRM Institute of Science and Technology What Is A Minimum Spanning Tree (MST) A minimum spanning tree is the one that contains the least weight among all the other spanning trees of a connected weighted graph. There can be more than one minimum spanning tree for a graph. •Kruskal’s algorithm •Prim’s algorithm
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- 7. Minimum Spanning Tree - Kruskal’s Algorithm SRM Institute of Science and Technology •Kruskal’s Algorithm is a famous greedy algorithm. •It is used for finding the Minimum Spanning Tree (MST) of a given graph. •To apply Kruskal’s algorithm, the given graph must be weighted, connected and undirected. Kruskal’s Algorithm Implementation- The implementation of Kruskal’s Algorithm is explained in the following steps- Step-01: •Sort all the edges from low weight to high weight. Step-02: •Take the edge with the lowest weight and use it to connect the vertices of graph. •If adding an edge creates a cycle, then reject that edge and go for the next least weight edge.
- 8. SRM Institute of Science and Technology Step-03: •Keep adding edges until all the vertices are connected and a Minimum Spanning Tree (MST) is obtained. Step 1:Create a forest in such a way that each graph is a separate tree Step 2: Create a priority queue Q that contains all the edges of the graph Step 3: Repeat steps 4 and 5 while Q is NOT EMPTY Step 4: Remove an edge from Q Step 5: If the edge obtained in Step 4 connects two different trees, then Add it to the forest(for combining two trees into one tree). ELSE Discard the edge Step 6: END Kruskal’s Algorithm Note: Forest is a collection of trees
- 9. SRM Institute of Science and Technology Thumb Rule to Remember The above steps may be reduced to the following thumb rule- •Simply draw all the vertices on the paper. •Connect these vertices using edges with minimum weights such that no cycle gets formed. MST - Kruskal’s Algorithm
- 10. SRM Institute of Science and Technology PRACTICE PROBLEMS BASED ON KRUSKAL’S ALGORITHM- Problem-01: Construct the minimum spanning tree (MST) for the given graph using Kruskal’s Algorithm- Now we need to find the weight of the minimum spanning tree
- 11. Solution- To construct MST using Kruskal’s Algorithm, •Simply draw all the vertices on the paper. •Connect these vertices using edges with minimum weights such that no cycle gets formed.- Step-01: Draw all the vertices on the paper as you can see in the Fig No of edges = 9
- 12. SRM Institute of Science and Technology Write all vertex pair VERTEX PAIR WEIGHT (1,6) 10 (1,2) 28 (2,3) 16 (2,7) 14 (3,4) 12 (4,5) 22 (4,7) 18 (5,6) 25 (5,7) 24
- 13. SRM Institute of Science and Technology Vertex pair in ascending order VERTEX PAIR WEIGHT (1,6) 10 (3,4) 12 (2,7) 14 (2,3) 16 (4,7) 18 (4,5) 22 (5,7) 24 (5,6) 25 (1,2) 28
- 14. SRM Institute of Science and Technology Step-02: Step-03: VERTEX PAIR WEIGHT ACTION (1,6) 10 Accepte d (3,4) 12 Accepte d (2,7) 14 (2,3) 16 (4,7) 18 (4,5) 22 (5,7) 24 (5,6) 25 (1,2) 28 Accept the edge if it does not form cycle continue till all nodes are visited Mark 1 if visited the nodes No cycle formation No cycle formation VERTEX KNOWN 1 1 2 0 3 1 4 1 5 0 6 1 7 0
- 15. SRM Institute of Science and Technology Step-04: Step-05: VERTEX PAIR WEIGHT ACTION (1,6) 10 Accepted (3,4) 12 Accepted (2,7) 14 Accepted (2,3) 16 Accepted (4,7) 18 Rejected (4,5) 22 (5,7) 24 (5,6) 25 (1,2) 28 Accept the edge if it does not form cycle continue till all nodes are visited VERTEX KNOWN 1 1 2 1 3 1 4 1 5 0 6 1 7 1 Mark 1 if visited the nodes No cycle formation
- 16. SRM Institute of Science and Technology Step-06: Step-07: Accept the edge if it does not form cycle continue till all nodes are visited Since all the vertices have been connected / included in the MST, so we stop. Weight of the MST = Sum of all edge weights = 10 + 25 + 22 + 12 + 16 + 14 VERTEX PAIR WEIGHT ACTION (1,6) 10 Accepted (3,4) 12 Accepted (2,7) 14 Accepted (2,3) 16 Accepted (4,7) 18 Rejected (4,5) 22 Accepted (5,7) 24 Rejected (5,6) 25 Accepted (1,2) 28 VERTEX KNOWN 1 1 2 1 3 1 4 1 5 1 6 1 7 1 Mark 1 if visited the nodes All nodes visited. Stop the Algorithm No cycle formation No cycle formation
- 17. SRM Institute of Science and Technology Problem 2: Construct the minimum spanning tree (MST) for the given graph using Kruskal’s Algorithm Now we need to find the weight of the minimum spanning tree
- 18. SRM Institute of Science and Technology V1 V2 V3 V4 V6 V5 Now Let us draw the minimum Spanning Tree using Kruskal’s Algorithm Draw all the vertices on the paper as you can see in the Fig Step 1:
- 19. SRM Institute of Science and Technology Step 2 Step 3 • We shall connect the vertices using the edges with the least weight • Now, the least weight edge is 3.So the vertices V1 and V2 are connected • Next the least weight edge is 4.So the vertices V1 and V3 are connected. No cycle formation. So accepted. No cycle formation. So accepted.
- 20. SRM Institute of Science and Technology Step 4 Cycle is not allowed. So rejected. Step 5 • Next least weight edge is 5 which is connecting V3 and V2.But if we include this edge, it will contain a cycle which is not allowed. • Next the least weight edge is 6.So the vertices V2 and V4 are connected. No cycle formation. So accepted.
- 21. SRM Institute of Science and Technology Step 6 Step 7 Cycle is not allowed. So rejected. • Next least weight edge is 7 which is connecting V3 and V4.But if we include this edge, it will contain a cycle which is not allowed. No cycle formation. So accepted. • Next the least weight edge is 8.So the vertices V3 and V5 are connected.
- 22. SRM Institute of Science and Technology Step 8 Step 9 • Next least weight edge is 9 which is connecting V4 and V5.But if we include this edge, it will contain a cycle which is not allowed. • Next the least weight edge is 10. So the vertices V4 and V6 are connected. Cycle is not allowed. So rejected. No cycle formation. So accepted.
- 23. SRM Institute of Science and Technology Step 10 Cycle is not allowed. So rejected. • Next least weight edge is 11 which is connecting V5 and V6.But if we include this edge, it will contain a cycle which is not allowed. Since all the vertices have been connected / included in the MST, so we stop. Weight of the MST = Sum of all edge weights = 3 + 4 + 6 + 8 + 10 = 31 units
- 24. SRM Institute of Science and Technology NETWORK FLOW PROBLEM – FORD FULKERSON ALGORITHM
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- 27. Source S A B C D T Sink flow = 0 0/4 0/9 Flow 0/2 0/8 0/6 Initially flow = 0
- 28. Source S A B C D T Sink flow = 8 0/4 0/9 Flow 0/2 0/8 0/6 Initially flow = 0 AUGMENTING PATHS BOTTLE NECK CAPCITY S A D T 8 8 8 8 STEP: 1 Minimum capacity
- 29. Source S A B C D T Sink flow = 8+2=10 0/4 0/9 Flow 0/2 8/8 0/6 Initially flow = 0 AUGMENTING PATHS BOTTLE NECK CAPCITY 2 STEP: 2 Residual capacity = 10 - 8 =2 2 2 8+2 =10 S C D T
- 30. Source S A B C D T Sink flow = 10+4=14 0/4 2/9 Flow 0/2 8/8 0/6 AUGMENTING PATHS BOTTLE NECK CAPCITY S C D A B T 4 STEP: 3 R capacity = 4 2+4=6 2+4=6 8-4=4 0+4=4 0+4=4 Each and every vertex other than source and sink should have same In-flow and Out-flow
- 31. Source S A B C D T Sink flow = 14+2=16 4/4 6/9 Flow 0/2 4/8 0/6 AUGMENTING PATHS BOTTLE NECK CAPCITY S A D B T 2 STEP: 4 Same In-flow and Out-flow has to be maintained R capacity = 10-8=2 8+2=10 4+2=6 0+2=2 4+2=6
- 32. Source S A B C D T Sink flow = 16+3=19 4/4 6/9 Flow 0/2 6/8 2/6 AUGMENTING PATHS BOTTLE NECK CAPCITY S C D B T 3 STEP: 5 R capacity = 9-6=3 6+3=9 6+3=9 2+3=5 6+3=9
- 33. Source S A B C D T Sink flow = 16+3=19 4/4 9/9 Flow 0/2 6/8 5/6 AUGMENTING PATHS BOTTLE NECK CAPCITY S C D B T 3 STEP: 6 R capacity = 9-6=3
- 34. SRM Institute of Science and Technology SORTEST PATH ALGORITHM
- 35. SRM Institute of Science and Technology •Shortest path problem is a problem of finding the shortest path(s) between vertices of a given graph. •Shortest path between two vertices is a path that has the least cost as compared to all other existing paths. Shortest Path Problem- Shortest Path Algorithms- Shortest path algorithms are a family of algorithms used for solving the shortest path problem. Applications- •Google Maps •Road Networks •Logistics Research
- 36. SRM Institute of Science and Technology Types of Shortest Path Problem- Various types of shortest path problem are- Shortest Path Problems Single Pair Shortest Path Problems Single Source Shortest Path Problems Single destination Shortest Path Problems All Pairs Shortest Path Problems 1.Single-pair shortest path problem 2.Single-source shortest path problem 3.Single-destination shortest path problem 4.All pairs shortest path problem
- 37. SRM Institute of Science and Technology Single-Pair Shortest Path Problem- •It is a shortest path problem where the shortest path between a given pair of vertices is computed. •A* Search Algorithm is a famous algorithm used for solving single-pair shortest path problem. Single-Source Shortest Path Problem- •It is a shortest path problem where the shortest path from a given source vertex to all other remaining vertices is computed. •Dijkstra’s Algorithm and Bellman Ford Algorithm are the famous algorithms used for solving single-source shortest path problem.
- 38. SRM Institute of Science and Technology Single-Destination Shortest Path Problem- •It is a shortest path problem where the shortest path from all the vertices to a single destination vertex is computed. •By reversing the direction of each edge in the graph, this problem reduces to single-source shortest path problem. •Dijkstra’s Algorithm is a famous algorithm adapted for solving single-destination shortest path problem. All Pairs Shortest Path Problem- •It is a shortest path problem where the shortest path between every pair of vertices is computed. •Floyd-Warshall Algorithm and Johnson’s Algorithm are the famous algorithms used for solving All pairs shortest path problem.