Transcription of Lecture 16 Network Models: Minimal Spanning Tree
1 Lecture 16 Network Models: Minimal Spanning TreeOctober 16, 2009 Lecture 16 Outline Network Models Minimal Spanning tree Problem - TODAY Shortest Route Problem Maximal-flow ProblemOperations research Methods1 Lecture 16 Bridges of K onigsbergThe city of K onigsberg in Prussia was established in can see the map of Prussia and the city onigsberg was set on the banks of Pregel River, and it included two largeislands which were connected to each other and the mainland by sevenbridges. The inhabitants of the city wanted to decide whether it is possibleto follow a path that crosses each bridge exactly once and returns to thestarting pointThe problem became know as the Seven Bridges of K research Methods2 Lecture 16 The Seven Bridges of K onigsbergOperations research Methods3 Lecture 16In 1736, Euler solved the problemshowing that such a path does notexist.
2 This solution is considered tobe the first theorem of graph theory,specifically of planar graph , such a tour is known asEulerian can learn more at: research Methods4 Lecture 16 Network ProblemsGiven a set of locations and possible roads to be built between pairs ofcities with the associated costs, we need to determine the minimum costroad Network connecting all the isMinimum Spanning tree ProblemNote that the graph is undirected. The weights on the links research Methods5 Lecture 16 Given a road Network and a starting nodes, we want to determine theshortest path to all the other nodes in the Network (or to a specifieddestination node).This isShortest Path ProblemNote that the graph is directed.
3 The weights on the links arecostsOperations research Methods6 Lecture 16 Given a capacitated Network connecting a supply node with destinationnodes, we want to determine the maximum amount of shipment to isMax-Flow ProblemNote that the graph is directed. The weights on the links arelink capacitiesOperations research Methods7 Lecture 16 Minimum Spanning tree ProblemWe are given a undirected graph(V, E)with the node setVand theedge setE. We are also given weight/costcijfor each edge{i, j} the minimum cost Spanning tree in the problem is solved by using the Minimal Spanning tree research Methods8 Lecture 16 Minimum Spanning tree ProblemStep any node in the Network , sayi.
4 LetS={i}, and let Sbe theset containing all the nodes other thani. Formally, S=V\ the edges{i, j}crossing fromSto S, find the edges with thesmallest cost (ties broken arbitrarily), , findi S, j Ssuch thatcij = mini S,j ScijBringj in the setS, ,newS=oldS {j }.Step testIf the setScontains all the nodes, stop - we are , go to Step research Methods9
