![• Step 1: Select a starting vertex
• Step 2: Repeat Steps 3 and 4 until there are fringe
vertices
• Step 3: Select an edge 'e' connecting the tree
vertex and fringe vertex that has minimum weight
• Step 4: Add the selected edge and the vertex to
the minimum spanning tree T
• [END OF LOOP]
• Step 5: EXIT
Prim's Algorithm](https://image.slidesharecdn.com/acprogramforprimsminimumspanningtreemstalgorithm-241107164431-e9104520/85/_A-C-program-for-Prim-s-Minimum-Spanning-Tree-MST-algorithm-The-program-is-for-adjacency-m-2-pptx-8-320.jpg)
_A C program for Prim's Minimum Spanning Tree (MST) algorithm. The program is for adjacency m-2.pptx
- 1. NAME- SHREYA KHANNA ROLL NO- 202210101110196 SUBJECT- DESIGN AND ANALYSIS OF ALGORITHM COURSE- B.TECH (CS)-57/58 TOPIC-MINIMUM SPANNING TREE TEACHER- Mr. Anil Kumar COURSE CODE- BCS5001
- 2. • A Spanning Tree is a subgraph that includes all vertices of the original graph, is connected, and contains no cycles. 1.Properties: • Includes all vertices of the graph. • Minimizes the number of edges (V-1 edges for V vertices). • Does not contain cycles. Spanning Tree
- 3. Introduction to Minimum Spanning Tree A Minimum Spanning Tree (MST) is a spanning tree with the smallest possible total edge weight. • Example: Imagine connecting cities with minimum road length—MST helps find this optimized connection.
- 4. MST only exists if the graph is connected and undirected. An MST will have V-1 edges for a graph with V vertices. Uniqueness: • If all edge weights are unique, the MST is unique. • Duplicate weights may lead to multiple MSTs. Note: MST doesn’t guarantee shortest paths between individual nodes—only minimum total weight. Properties of MST
- 5. Kruskal's Algorithm Kruskal's Algorithm is used to find the minimum spanning tree for a connected weighted graph. The main target of the algorithm is to find the subset of edges by using which we can traverse every vertex of the graph. • Sort all edges by weight. • Add edges to the MST in ascending order of weight. • Only add edges that don’t create a cycle (using Union-Find data structure). Time Complexity: O(Elog E)O(E log E)O(ElogE), where EEE is the number of edges.
- 6. Kruskal's Algorithm • Step 1: Create a forest F in such a way that every vertex of the graph is a separate tree. • Step 2: Create a set E that contains all the edges of the graph. • Step 3: Repeat Steps 4 and 5 while E is NOT EMPTY and F is not spanning • Step 4: Remove an edge from E with minimum weight • Step 5: IF the edge obtained in Step 4 connects two different trees, then add it to the forest F • (for combining two trees into one tree). • ELSE • Discard the edge • Step 6: END
- 7. Prim's Algorithm Prim's algorithm is a greedy algorithm that starts from one vertex and continue to add the edges with the smallest weight until the goal is reached. • Start from any vertex. • Grow the MST by adding the smallest weight edge connected to the MST. • Continue until all vertices are included. Time Complexity: O(V2)O(V^2)O(V2) for simple version, or O(Elog V)O(E log V)O(ElogV) using priority queue.
- 8. • Step 1: Select a starting vertex • Step 2: Repeat Steps 3 and 4 until there are fringe vertices • Step 3: Select an edge 'e' connecting the tree vertex and fringe vertex that has minimum weight • Step 4: Add the selected edge and the vertex to the minimum spanning tree T • [END OF LOOP] • Step 5: EXIT Prim's Algorithm
- 9. Example of prim's algorithm • Suppose, a weighted graph is - • Step 1 - First, we have to choose a vertex from the above graph. Let's choose B. • Step 2 - Now, we have to choose and add the shortest edge from vertex B. There are two edges from vertex B that are B to C with weight 10 and edge B to D with weight 4. Among the edges, the edge BD has the minimum weight. So, add it to the MST. • Step 3- Now, again, choose the edge with the minimum weight among all the other edges. In this case, the edges DE and CD are such edges. Add them to MST and explore the adjacent of C, i.e., E and A. So, select the edge DE and add it to the MST. • Step 4 - Now, select the edge CD, and add it to the MST. • Step 5 - Now, choose the edge CA. Here, we cannot select the edge CE as it would create a cycle to the graph. So, choose the edge CA and add it to the MST. • So, the graph produced in step 5 is the minimum spanning tree of the given graph .Cost of MST = 4 + 2 + 1 + 3 = 10 units.
- 10. Time Complexity of the Algorithms Algorithm Time Complexity Kruskal's O(E log E) Prim's O(E log V) The time complexity depends on the data structures used, with Prim's having a slight edge for sparse graphs.
- 11. Feature Prim’s Algorithm Kruskal’s Algorithm Approach Vertex-based, grows the MST one vertex at a time Edge-based, adds edges in increasing order of weight Data Structure Priority queue (min-heap) Union-Find data structure Initialization Starts from an arbitrary vertex Starts with all vertices as separate trees (forest) Edge Selection Chooses the minimum weight edge from the connected vertices Chooses the minimum weight edge from all edges Complexity O(V^2) for adjacency matrix, O((E + V) log V) with a priority queue O(E log E) or O(E log V), due to edge sorting Suitable for Dense graphs Sparse graphs Memory Usage More memory for priority queue Less memory if edges can be sorted externally Example Use Cases Network design, clustering with dense connections Road networks, telecommunications with sparse connections Comparison of Kruskal’s and Prim’s Algorithms
- 12. Conclusion and Key Takeaways • MSTs are fundamental in optimizing connected networks. • Kruskal’s and Prim’s algorithms are key tools for finding MSTs. • Applications span diverse fields like networking, ML, and design. Final Thoughts: Emphasize MST’s real-world importance in cost reduction and resource optimization.