lec12 - 15.053 March 22, 2007 Introduction to Networks...

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1 15.053 March 22, 2007 Introduction to Networks Announcement: no recitations this week Comment on Excel
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2 Quotes for today "A journey of a thousand miles begins with a single step." -- Confucius “You cannot travel the path until you have become the path itself” -- Buddha
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3 Network Models z Optimization models that exhibit a very special structure. z For special cases, this structure to dramatically reduce computational complexity (running time). z First widespread application of LP to problems of industrial logistics z Addresses huge number of diverse applications z Today’s lecture (first of 3): introductory material, Eulerian tours, the Shortest Path Problem
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1 Notation and Terminology Note: Network terminology is not (and never will be) standardized. The same concept may be denoted in many different ways. Called: NETWORK directed graph digraph graph Class Handouts (Ahuja, Magnanti, Orlin) Node set N = {1, 2, 3, 4} Arc Set Network G = (N, A) {(1,2), (1,3), (3,2), (3,4), (2,4)} E={1-2,1-3,3-2,3-4,2-4} Graph G = (V,E) Edge set: Also Seen Vertex set V = {1,2,3,4} 2 1 4 3 Personally, I find introducing all of the network notation and terminology to be boring. At the same time, it is critical to any further understanding in this area. I use the notation in the text Network Flows by Ahuja, Magnanti, and Orlin. Tom Magnanti is currently Dean of Engineering at MIT. Ravi Ahuja and I have worked together more than 20 years, and have co - authored around 50 papers together as well as this book.
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2 Directed and Undirected Networks 2 3 4 1 a b c d e A Directed Graph The field of Network Optimization concerns optimization problems on networks Networks are used to transport commodities physical goods (products, liquids) communication electricity, etc. 2 3 4 1 a b c d e An Undirected Graph Networks are universal, and are applied to a wide range of situations. Almost any system can be described in large part by a network that describes relationships between parts of a system. This includes communication systems, electrical systems, manufacturing systems, social networks, and much more.
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6 Networks are Everywhere z Physical Networks Road Networks Railway Networks Airline traffic Networks Electrical networks, e.g., the power grid z Abstract networks organizational charts precedence relationships in projects z Others?
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7 Overview: z Networks and graphs are powerful modeling tools. z Most OR models have networks or graphs as a major aspect z Next five lectures: we will develop models that are efficiently solvable. Help form a toolkit for those problems that are harder to solve z Next: representations of networks A great insight from computer scientists: how data is represented is important to how it is used
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8 The Adjacency Matrix (for directed graphs) 2 3 4 1 a b c d e A Directed Graph 0101 0010 0000 0110 Have a row for each node 1 2 3 4 1 2 3 4 Have a column for each node Put a 1 in row i- column j if (i, j) is an arc What would happen if (4, 2) became (2, 4)?
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lec12 - 15.053 March 22, 2007 Introduction to Networks...

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