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Published byTeresa Blackner Modified over 3 years ago

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O(N 1.5 ) divide-and-conquer technique for Minimum Spanning Tree problem Step 1: Divide the graph into N sub-graph by clustering. Step 2: Solve each sub-problem separately using Prim's algorithm (quadratic complexity) Step 3: Merge the sub-solutions: Construct a meta graph where each node corresponds to one cluster Solve MST for the meta graph Add links from the meta graph to the original graph to complete the solution

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Example 1: Algorithm finds sub-optimal solution 5 8 1 3 2 6 5 5 8 10 2 5 3 4 7 5 3 1 6

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Step 1: Divide the graph into N sub-graph by clustering 5 8 1 3 2 6 5 5 8 10 2 5 3 4 7 5 3 1 6

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5 8 1 3 2 6 5 5 8 2 5 3 4 7 5 3 1 6 Step 2: Solve each sub-problem by Prim’s algorithm

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5 8 1 3 2 6 5 8 10 2 5 3 4 7 3 1 6 Step 3.1 (a): Select center point for each cluster

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5 8 1 3 2 6 5 8 10 2 5 3 4 7 3 1 6 Step 3.1 (b): Connect the nodes of this meta graph

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5 8 1 3 2 6 5 8 10 2 5 3 4 7 3 1 6 Step 3.1 (c): Set the weights based on shortest distances 1 3 4

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5 8 1 3 2 6 5 8 10 2 5 3 4 7 3 1 6 Step 3.2: Solve MST for the meta graph 1 3 4

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5 8 3 2 6 5 8 10 2 5 4 7 3 1 6 Step 3.3: Select the corresponding links 1 3 3 1

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5 8 1 3 2 6 5 8 10 2 5 3 4 7 3 1 6 Step 3.3: Add links from the MST of the meta graph Total weight = 21

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5 8 1 3 2 6 5 8 10 2 5 3 4 7 3 1 6 Better solution: Total weight = 20 Optimality of the solution? Add Remove

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5 8 1 3 2 6 6 8 10 2 5 3 4 7 3 1 5 Total weight = 20 Example 2: Algorithm finds optimal solution

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