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Minimal Spanning Tree

Minimal Spanning Tree. In an undirected graph, find a minimal spanning tree Minimal means … Spanning means … Tree means … We can use Prim's Algorithm. Similar to Dijkstra's Algorithm start with initial vertex in MST find minimal edge from MST vertex to other vertex

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Minimal Spanning Tree

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  1. Minimal Spanning Tree • In an undirected graph, find a minimal spanning tree • Minimal means … • Spanning means … • Tree means … • We can use Prim's Algorithm. Similar to Dijkstra's Algorithm start with initial vertex in MST find minimal edge from MST vertex to other vertex add minimal edge/vertex to MST repeat until each vertex added

  2. Prim's Algorithm • How do we know this algorithm works? • Suppose not, some graph for which algorithm yields non minimal spanning tree, is it not a tree? Not spanning? … • Assume we have such a graph, call it G, now we reason … • Will lead to contradiction (exploit minimal property) • How efficient is Prim's Algorithm? • Depends on implementation, look at Dijkstra and modify • What did Dijkstra's algorithm do? Problem solved?

  3. Implementation, priority queue • Keep track of shortest distance from MST vertices to others, use chosen start vertex • Initially all are infinity except start is 0 from itself • Choose minimal edge between MST vertices and others • This is same as choosing minimal element in PQ • When choosing, modify distances by adding elements to PQ appropriately • When distance < current best modify current best (add new element to PQ) • How is this different from Dijkstra's algorithm? • How do we construct path?

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