NETWORK MODELS

NETWORK MODELSNETWORKS MODELSA network is an arrangement of paths (branches) connected at various points (nodes) through which one or more items move from one point to another.

The network is drawn as a diagram providing a picture of the system, thus enabling visual representation and enhanced understanding.

A real-life systems can be modeled as networksThe Shortest Route ProblemThe Minimal Spanning Tree ProblemThe Maximal Flow Problem

NETWORKS MODELS

The Shortest Route Problem

Examples of shortest path problems: airline schedulingequipment replacementrouting in telecommunications networksreliability problemstraffic routing

The Maximal Flow ProblemIn the maximum flow problem our goal is to send the largest amount of flow possible from a specified origin node to a specified destination node subject to arc capacities.

The Minimal Spanning TreeA tree is a connected undirected graph that has no cycles/ no loops.A spanning tree for a graph is a sub graph that includes every nodes of the original.A minimum spanning tree, (MST) is a sub graph whose sum of edge weights is minimized.

The Minimal Spanning Tree ProblemProblem: Each node in a network represents a computer in a computer network, arc(i,j) might represent an underground cable that connects computer i to computer j. Connect all nodes in a network so that the total of the arc lengths are minimized.

STEPS:Select any starting node. Select the node closest to the starting node to join the spanning tree.Select the closest node not currently in the spanning tree.Repeat step 3 until all nodes have joined the spanning tree.EXAMPLE:A company has seven computers. The distances between computers are given in the figure below. What is the minimum length of cable required to interconnect the computers?

Copyright 2013 Pearson Education, Inc. Publishing as Prentice HallSolution: step 1

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Copyright 2013 Pearson Education, Inc. Publishing as Prentice HallSolution: step 211

Copyright 2013 Pearson Education, Inc. Publishing as Prentice HallContinue to select the closest node not presently in the spanning area.Solution: step 312

Copyright 2013 Pearson Education, Inc. Publishing as Prentice HallSolution: step 413Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall

Solution: step 514Copyright 2013 Pearson Education, Inc. Publishing as Prentice HallOptimal SolutionSolution: step 6

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Minimum length = 72A pipeline is to be built that will link six cities. The cost (in millions) of constructing each potential link depends on distance and terrain as shown in the weighted graph in figure below. Find a system of pipelines to connect all the cities and yet minimize the total cost.160260LipisBentongKuantanRompinPekanRaubTemerlohMaran9010023085130605013633