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CSC 213 – Large Scale Programming
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Minimum Spanning Tree (MST) Spanning subgraph Subgraph w/ all vertices ORD PIT ATL STL DEN DFW DCA 10 1 9 8 6 3 2 5 7 4
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Minimum Spanning Tree (MST) Spanning subgraph Subgraph w/ all vertices ORD PIT ATL STL DEN DFW DCA 10 1 9 8 6 3 2 5 7 4
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Minimum Spanning Tree (MST) Spanning subgraph Subgraph w/ all vertices Spanning tree Combines spanning subgraph + tree ORD PIT ATL STL DEN DFW DCA 10 1 9 8 6 3 2 5 7 4
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Minimum Spanning Tree (MST) Spanning subgraph Subgraph w/ all vertices Spanning tree Combines spanning subgraph + tree MST Spanning tree which minimizes sum of edge weights ORD PIT ATL STL DEN DFW DCA 10 1 9 8 6 3 2 5 7 4
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No Cycles in MST Edge in MST cheaper than one making cycle Assume there exists an edge e not in MST Cycle, C, occurs after adding e to min. spanning tree Assume that f in C heavier than e Shrink MST using e and dropping f from C 8 4 2 3 6 7 7 9 8 e f 8 4 2 3 6 7 7 9 8 e f
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Partition Property Given partitioning of vertices in a graph exactly1 Required that each vertex in exactly 1 partition Use smallest edge to connect each of the To complete following MST, can use either f or e 7 4 2 8 5 7 3 9 8
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Kruskal’s Algorithm Similar to Prim-Jarnik, including finding MST Also like Prim-Jarnik, adds edges greedily Edge s But Kruskal processes Edge s using PriorityQueue Check if Edge makes cycle before adding to MST No cycles in an MST, so add Edge if no cycle occurs Detecting cycles hard, however
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Kruskal’s Algorithm Similar to Prim-Jarnik, including finding MST Also like Prim-Jarnik, adds edges greedily Edge s But Kruskal processes Edge s using PriorityQueue Check if Edge makes cycle before adding to MST No cycles in an MST, so add Edge if no cycle occurs Detecting cycles hard, however
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Kruskal’s Algorithm Similar to Prim-Jarnik, including finding MST Also like Prim-Jarnik, adds edges greedily Edge s But Kruskal processes Edge s using PriorityQueue Check if Edge makes cycle before adding to MST No cycles in an MST, so add Edge if no cycle occurs Detecting cycles hard, however
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Structure for Kruskal’s Algorithm Kruskal’s needs to maintain collection of trees Accept Edge connecting 2 trees & reject it otherwise Data structure named Partition relied upon Store disjoint Set instances within a Partition For Kruskal’s, each Set holds tree from graph Methods supporting Set s defined by Partition find(u) : find and returns Set containing u union(p,r) : replace p & r with their union makeSet(u) : creates Set for u & adds to Partition
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Kruskal’s Algorithm Algorithm KruskalMST(Graph G) Q new PriorityQueue() T new Graph() P new Partition() for (Vertex v : G.vertices()) P.makeSet(v) T.insertVertex(v) for (Edge e : G.edges()) Q.insert(e.getWeight(), e); while (T.numEdges() < T.numVertices()-1) Edge e = Q.removeMin().value() Assign u, v to G.endpoints(e) if P.find(u) P.find(v) T.insertEdge(e) P.union(P.find(u),P.find(v)) return T
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Kruskal Example 867 2704 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO 187
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Kruskal Example 867 2704 187 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO
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Kruskal Example 867 2704 187 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO
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Kruskal Example 867 2704 187 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO
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Kruskal Example 867 2704 187 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO
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Kruskal Example 867 2704 187 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO
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Kruskal Example 867 2704 187 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO
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Kruskal Example 867 2704 187 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO
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Kruskal Example 867 2704 187 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO
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Kruskal Example 867 2704 187 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO
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Kruskal Example 867 2704 187 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO
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Kruskal Example 867 2704 187 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO
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Kruskal Example 867 2704 187 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO
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Kruskal Example 867 2704 187 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO
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Kruskal Example 867 2704 187 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO
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Kruskal Example 867 2704 187 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO
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Kruskal Example 867 2704 187 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO
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Kruskal Example 867 2704 187 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO
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Kruskal Example 867 2704 187 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO
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Kruskal Example 867 2704 187 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO
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Kruskal Example 867 2704 187 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO
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Kruskal Example 867 2704 187 1258 849 144 740 1391 946 1090 1121 2342 1846 621 802 1464 1235 337 BOS PVD JFK ORD BWI 184 MIA DFW LAX SFO
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