01introduction - 15.082 and 6.855J Spring 2003 Network...

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1 15.082 and 6.855J Spring 2003 Network Optimization J.B. Orlin
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2 WELCOME! ± Welcome to 15.082/6.855J ± Introduction to Network Optimization ± Instructor: James B. Orlin ± Textbook: Network Flows: Theory, Algorithms, and Applications by Ahuja, Magnanti, and Orlin referred to as AMO
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3 Quick Overview ± Next: The Koenigsberg Bridge Problem z Introduces Networks and Network Algorithms ± Some subject management issues ± Network Flows and Applications ± Computational Complexity ± Overall goal of today’s lecture: set the tone for the rest of the subject z provide background z provide motivation z handle some class logistics
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4 On the background of students ± Requirement for this class z Either Linear Programming (15.081J) z or Data Structures ± How many have had linear programming? z There will be a “review” lecture later in the term ± How many have had a subject in data structures? z The “review” lecture for data structures is Thursday
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5 Some aspects of the class ± Fondness for Powerpoint animations ± Cold-calling as a way to speed up learning of the algorithms ± Talking with partners (the person next to you in in the classroom.) ± Class time: used for presenting theory, algorithms, applications, theory z mostly outlines of proofs illustrated by examples (not detailed proofs) z detailed proofs are in the text
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6 The Bridges of Koenigsberg: Euler 1736 ± “Graph Theory” began in 1736 ± Leonard Eüler z Visited Koenigsberg z People wondered whether it is possible to take a walk, end up where you started from, and cross each bridge in Koenigsberg exactly once z Generally it was believed to be impossible
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7 The Bridges of Koenigsberg: Euler 1736 A D C B 12 4 3 7 6 5 Is it possible to start in A, cross over each bridge exactly once, and end up back in A?
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8 The Bridges of Koenigsberg: Euler 1736 A D C B 12 4 3 7 6 5 Conceptualization: Land masses are “nodes”.
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9 The Bridges of Koenigsberg: Euler 1736 12 4 3 7 6 5 A C D B Conceptualization: Bridges are “arcs.”
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10 The Bridges of Koenigsberg: Euler 1736 12 4 3 7 6 5 A C D B Is there a “walk” starting at A and ending at A and passing through each arc exactly once?
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11 Notation and Terminology Network terminology as used in AMO. 2 3 4 1 a b c d e A Directed Graph or Directed Network 2 3 4 1 a b c d e An Undirected Graph or Undirected Network Network G = (N, A) Node set N = {1, 2, 3, 4} Arc Set A = {(1,2), (1,3), (3,2), (3,4), (2,4)} In an undirected graph, (i,j) = (j,i)
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Path: Example: 5, 2, 3, 4. (or 5, c, 2, b, 3, e, 4) No node is repeated. Directions are ignored. 2 3 4 a b c 1 5 d e Directed Path . Example: 1, 2, 5, 3, 4 (or 1, a, 2, c, 5, d, 3, e, 4) No node is repeated. Directions are important. 2 3 4 a b c 1 5 d e Cycle (or circuit or loop) 1, 2, 3, 1. (or 1, a, 2, b, 3, e) A path with 2 or more nodes, except that the first node is the last node.
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01introduction - 15.082 and 6.855J Spring 2003 Network...

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