MST-class-new-handouts-2

MST-class-new-handouts-2 - Minimum Spanning Tree 4/3/2006...

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Minimum Spanning Tree 4/3/2006 2:48 PM 1 Minimum Spanning Trees 1 Minimum Spanning Trees JFK BOS MIA ORD LAX DFW SFO BWI PVD 867 2704 187 1258 849 144 740 1391 184 946 1090 1121 2342 1846 621 802 1464 1235 337 Minimum Spanning Trees 2 Outline and Reading Minimum Spanning Trees (§7.3) ± Definitions ± A crucial fact The Prim-Jarnik Algorithm (§7.3.2) Kruskal's Algorithm (§7.3.1)
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Minimum Spanning Tree 4/3/2006 2:48 PM 2 Minimum Spanning Trees 3 Minimum Spanning Tree Spanning subgraph ± Subgraph of a graph G containing all the vertices of G Spanning tree ± Spanning subgraph that is itself a (free) tree Minimum spanning tree (MST) ± Spanning tree of a weighted graph with minimum total edge weight Applications ± Communications networks ± Transportation networks ORD PIT ATL STL DEN DFW DCA 10 1 9 8 6 3 2 5 7 4 Minimum Spanning Trees 4 Cycle Property Cycle Property: ± Let T be a minimum spanning tree of a weighted graph G ± Let e be an edge of G that is not in T and C let be the cycle formed by e with T ± For every edge f of C, weight ( f ) weight ( e ) Proof: ± By contradiction ± If weight ( f ) > weight ( e ) we can get a spanning tree of smaller weight by replacing e with f 8 4 2 3 6 7 7 9 8 e C f 8 4 2 3 6 7 7 9 8 C e f Replacing f with e yields a better spanning tree
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Minimum Spanning Tree 4/3/2006 2:48 PM 3 Minimum Spanning Trees 5 UV Partition Property Partition Property: ± Consider a partition of the vertices of G into subsets U and V ± Let e be an edge of minimum weight across the partition ± There is a minimum spanning tree of G containing edge e Proof: ± Let T be an MST of G ± If T does not contain e, consider the cycle C formed by e with T and let f be an edge of C across the partition ± By the cycle property, weight ( f ) weight ( e ) ± Thus, weight ( f ) = weight ( e ) ± We obtain another MST by replacing f with e 7 4 2 8 5 7 3 9 8 e f 7 4 2 8 5 7 3 9 8 e f Replacing f with e yields another MST Minimum Spanning Trees 6 Prim-Jarnik’s Algorithm Similar to Dijkstra’s algorithm (for a connected graph) We pick an arbitrary vertex s and we grow the MST as a
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MST-class-new-handouts-2 - Minimum Spanning Tree 4/3/2006...

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