{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
1 15.082 and 6.855 The Minimum Cost Spanning Tree Problem
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 2
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
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 4
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
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 6
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
Image of page 7

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 8
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 .
Image of page 9

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern