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Minimum Spanning Trees

Minimum Spanning Trees. Easy. Terms. Node Edge Cut Cut respects a set of edges Light Edge Minimum Spanning Tree. Generic Algorithm. Start with an empty set of edges. Continuously add only edges that are part of a minimum spanning tree Stop when we have a minimum spanning tree.

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Minimum Spanning Trees

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  1. Minimum Spanning Trees Easy

  2. Terms • Node • Edge • Cut • Cut respects a set of edges • Light Edge • Minimum Spanning Tree

  3. Generic Algorithm • Start with an empty set of edges. • Continuously add only edges that are part of a minimum spanning tree • Stop when we have a minimum spanning tree.

  4. Kruskal’s Algorithm • Find the smallest edge in the graph. • If it connects two unconnected sets, add it. • Repeat for each edge. • What are the optimal data structures? • O( E lg V) • E = # edges • V = # nodes (vertices)

  5. Prim’s Algorithm • Start with a set of 1 nodes: V and empty set of edges A • Pick a light edge and add it to A. • Repeat until all nodes are in V. • Best Data Structures? • O(E lg V) or O(E+V lg V)

  6. Other Algorithms • Borůvka's algorithm, O( E lg V ) • Bernard Chazelle, O( E α(V)) • Randomized, expected O(E) • Karger, Klein, and Tarjan

  7. Distributed MST LAN 1 B3 B2 B1 LAN 3 B4 LAN 2

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