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(iii) Given below is the Kruskal Algorithm to find the minimum spanning tree. What is the time complexity of this algorithm using a sorting algorithm in the worst case scenario? How can you improve this algorithm for an average or best case scenario? G = (X.U) G' = (X',U') where X' = X,U' = Ø Sort all arcs eU in increasing order. For each arc u e U Do If U'U {u} does not contain a cycle Then U' +U'U{u} Endif EndFor

(iii) Given below is the Kruskal Algorithm to find the minimum spanning tree. What is the time complexity of this algorithm using a sorting algorithm in the worst case scenario? How can you improve this algorithm for an average or best case scenario? G = (X.U) G' = (X',U') where X' = X,U' = Ø Sort all arcs eU in increasing order. For each arc u e U Do If U'U {u} does not contain a cycle Then U' +U'U{u} Endif EndFor

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
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Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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### Kruskal Algorithm for Minimum Spanning Tree **Given Problem:** Given below is the Kruskal Algorithm to find the minimum spanning tree. What is the time complexity of this algorithm using a sorting algorithm in the worst case scenario? How can you improve this algorithm for an average or best case scenario? --- #### Notations and Initialization: 1. **Graph Representation:** - \( G = (X, U) \) - Here, \(G\) is a graph where: - \(X\) represents the set of vertices. - \(U\) represents the set of edges. 2. **Initialize New Graph:** - \( G' = (X', U') \) - Where: - \(X' = X\) (initially the vertices set \(X'\) is the same as \(X\)). - \(U' = \emptyset \) (initially, the edges set \(U'\) is empty). 3. **Sorting Edges:** - Sort all arcs \( u \in U \) (edges in \(U\)) in increasing order based on their weight. --- #### Algorithm Steps: 1. **For each edge \( u \in U \) Do:** - **Cycle Checking and Edge Addition:** - If \( U' \cup \{u\} \) does not contain a cycle, Then: - Update \( U' \leftarrow U' \cup \{u\} \) - EndIf - EndFor --- #### Explanation and Improvements: - **Sorting Step:** - The sorting of the edges takes \(O(E \log E)\) time, where \(E\) is the number of edges. - **Cycle Check:** - The cycle check, often implemented using Union-Find data structure, tends to take nearly constant time, \(O(\log V)\) per operation, where \(V\) is the number of vertices. - **Overall Time Complexity:** - The overall time complexity of Kruskal's Algorithm in the worst case, dominated by the sorting step, is \(O(E \log E)\). - **Improvement for Average/Best Case:** - In practice, using efficient data structures like Union-Find with path compression and union by rank, which improves the performance of cycle checks. - Using heuristics
Transcribed Image Text:### Kruskal Algorithm for Minimum Spanning Tree **Given Problem:** Given below is the Kruskal Algorithm to find the minimum spanning tree. What is the time complexity of this algorithm using a sorting algorithm in the worst case scenario? How can you improve this algorithm for an average or best case scenario? --- #### Notations and Initialization: 1. **Graph Representation:** - \( G = (X, U) \) - Here, \(G\) is a graph where: - \(X\) represents the set of vertices. - \(U\) represents the set of edges. 2. **Initialize New Graph:** - \( G' = (X', U') \) - Where: - \(X' = X\) (initially the vertices set \(X'\) is the same as \(X\)). - \(U' = \emptyset \) (initially, the edges set \(U'\) is empty). 3. **Sorting Edges:** - Sort all arcs \( u \in U \) (edges in \(U\)) in increasing order based on their weight. --- #### Algorithm Steps: 1. **For each edge \( u \in U \) Do:** - **Cycle Checking and Edge Addition:** - If \( U' \cup \{u\} \) does not contain a cycle, Then: - Update \( U' \leftarrow U' \cup \{u\} \) - EndIf - EndFor --- #### Explanation and Improvements: - **Sorting Step:** - The sorting of the edges takes \(O(E \log E)\) time, where \(E\) is the number of edges. - **Cycle Check:** - The cycle check, often implemented using Union-Find data structure, tends to take nearly constant time, \(O(\log V)\) per operation, where \(V\) is the number of vertices. - **Overall Time Complexity:** - The overall time complexity of Kruskal's Algorithm in the worst case, dominated by the sorting step, is \(O(E \log E)\). - **Improvement for Average/Best Case:** - In practice, using efficient data structures like Union-Find with path compression and union by rank, which improves the performance of cycle checks. - Using heuristics
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