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16_Spanning_Trees

# 16_Spanning_Trees - 15.082 and 6.855 The Minimum Cost...

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1 15.082 and 6.855 The Minimum Cost Spanning Tree Problem

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2 Communications Systems Consider a communications company, such as AT&T or GTE that needs to build a communication network that connects n different users. The cost of making a link joining i and j is c ij . What is the minimum cost of connecting all of the users? 1 6 3 7 5 8 9 4 2 10 Common assumption: the only links possible are the ones directly joining two nodes.
3 Electronic Circuitry Consider a system with a number of electronic components. In order to make two pins i and j of different components electrically equivalent, one can connect i and j by a wire. How can we connect n different pins in this way to make them electrically equivalent to each other so as to minimize the total wire length. 1 2 3 4 5

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4 Minimum Cost Spanning Tree Problem Undirected network G = (N, A). (i, j) is the same arc as (j, i). We associate with each arc (i, j) A a cost c ij . A spanning tree T of G is a connected acyclic subgraph that spans all the nodes. A connected graph with n nodes and n – 1 arcs is a spanning tree. The minimum cost spanning tree problem is to find a spanning tree of minimum cost.
5 A Minimum Cost Spanning Tree Problem 35 10 30 15 25 40 20 17 8 15 11 21 1 2 3 4 5 6 7

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6 A Minimum Cost Spanning Tree 35 10 30 15 25 40 20 17 8 15 11 21 1 2 3 4 5 6 7
7 The Traveling Salesman Problem Consider the traveling salesman problem of finding a minimum cost tour linking n cities. One way of formulating this problem is using minimum spanning trees. 1

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8 Representing the TSP problem A collection of arcs is a tour if 1. There are two arcs incident to each node 2. The red arcs (those not incident to node 1) form a spanning tree in G\1. 1 We will see this again in the lectures on lagrangian relaxation.
9 Definitions Let T* be a spanning tree. We refer to the arcs in T* as tree arcs and to those arcs not in T* as nontree arcs .

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