Minimum Spanning Tree in design and analysis
- 1. MINIMUM SPANNING TREE By Raksha kanwar
- 2. SPANNING TREE A spanning tree of an undirected graph is a subgraph that includes all the vertices of the original graph connected with the minimum number of edges, ensuring no cycles are present.
- 3. MINIMUM SPANNING TREE Given a connected, undirected graph G=(V,E) with weighted edges, a Minimum Spanning Tree is a spanning tree that: Connects all vertices: Ensures there is a path between any pair of nodes. Is acyclic: Contains no cycles, maintaining the tree structure. Has minimal total edge weight: The sum of the weights of its edges is the smallest among all possible spanning trees of the graph. For a graph with v vertices, any spanning tree, including the MST, will have exactly (V−1) edges. Definition
- 5. PROPERTIES:MST Uniqueness: If all edge weights are distinct, the MST is unique. Multiple MSTs: If edge weights are not unique, there may be multiple MSTs with the same total weight. Cycle Property: For any cycle in the graph, the edge with the highest weight in that cycle does not belong to the MST. Cut Property: For any cut in the graph, the edge with the smallest weight crossing the cut is included in the MST.
- 6. ALGORITHNMS Two primary greedy algorithms are used to compute the MST: 1. Kruskal’s Algorithm 2.Prim’s Algorithm
- 7. KRUSKAL’S ALGORITHM Below are the steps for finding MST using Kruskal’s algorithm: Sort all the edges in a non-decreasing order of their weight. Pick the smallest edge. Check if it forms a cycle with the spanning tree formed so far. If the cycle is not formed, include this edge. Else, discard it. Repeat step 2 until there are (V-1) edges in the spanning tree.
- 8. EXAMPLE Fig2:eg of kruskal’s algo
- 9. Greedy Approach: Selects the smallest-weight edge that doesn't form a cycle to construct the Minimum Spanning Tree (MST). Edge-Centric: Emphasizes edges rather than vertices, making it particularly effective for sparse graphs. Cycle Detection: Uses the Union-Find data structure to efficiently merge disjoint sets and prevent cycles. Time Complexity: Runs in O(E log V) time, thanks to edge sorting and near-constant time Union-Find operations. Key Characteristics
- 10. PRIM’S ALGORITHM The steps how prim’s algorithm works are below: Choose a starting point – Pick any vertex to begin, like 0. Find new connections – Look at all the vertices not yet included in the tree. Check available edges – Find edges connecting the tree to those new vertices. Pick the smallest edge – Choose the edge with the least weight. Add it to the tree – Bring the connected vertex into the growing MST. Since we only add edges that connect new vertices, there are no loops. Repeat until done – Keep adding edges until all vertices are included, then return the final MST.
- 12. Greedy approach – Always selects the smallest edge. Incremental growth – Expands the MST one edge at a time. No cycles – Ensures an acyclic spanning tree. Works with weighted graphs – Requires a connected, weighted, undirected graph. Efficient with priority queue – Optimized using a min-heap. Time complexity – Runs in O(V²) with an adjacency matrix, or O(E log V) with a priority queue. Key Characteristics
- 13. DIFFERENCE
- 14. APPLICATIONS OF MST Internet & Network Connections – Helps design cost-effective layouts for laying cables and setting up Wi-Fi networks. Road Planning – Used to find the shortest way to connect cities or towns with minimal road length. Electricity Distribution – Helps power companies create efficient grids to supply electricity with fewer cables. Water Supply Systems – Used to design pipeline networks that connect homes and cities with minimal material cost. Cluster Analysis in AI – Helps group similar data points, like organizing images or sorting customer preferences.
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